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1
RATIO, PROPORTION AND
PERCENTAGE
Unit Structure:
1.0 Objectives
1.1 Ratio
1.2 Proportion
1.3 Percentage
1.4 Check your progress
1.5 Unsolved problems
1.0 OBJECTIVES:
After going through this unit, learner will be able to:
Define and distinguish between concepts of Ratio, Proportion and
Percentage.
Apply ratio, proportion and percentage to solve real life problems.
1.1 RATIO
1.1.1 Introduction:
Ratio is defined as t he relation between two quantities of the same kind,
same type, which expresses on e quantity ( ) as a multiple or part of the
other quantity ( ).
i.e., one cannot compare quantity with unit as kilometre with the quantity
with unit as litre. Both quantities must have same unit. Even, one quantity
in kilo meter and another in centimetre are not comparable. We need to
convert both quantities in same unit, either in kilometre or in centimetre .
Ratio has no unit. i.e., if we compare two quantities of 4 kg and 20 kg,
then ratio will be 4:20 without unit. It means that, whenever t here is ratio
: , both and have same unit.
The ratio of quantities and is written as : or
(read as is to ),
where, (first quantity) is calledas antecedent and (second quantity) is
called as consequent . munotes.in
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If terms in ratio i.e., antecedent and consequent multiplied or divided by
same non -zero number then resulting ratio remains same as that of original
ratio.
Ex. If original ratio is 3:7 then
×
× =
=
1. If = , then ratio : is called ratio of equality.
2. If > , then ratio : is called ratio of greater inequality.
3. If < , then ratio : is called ratio of lesser inequality.
1.1.2 Compound Ratio:
Ratios are compounded by multiplying together their antecedents to form
a new antecedent and their consequent to form a new consequent.
If : and : are two ratios then : i.e.,
is compound ratio of :
and : .
Ex: 2:3 and 11:7 are two ratios then compound ratio is 2×11∶3×
7 i.e.,22:21
One can compound more than two ratios.
Ex. For ratios : , : , : ,..., : the compound ratio will be
× × × .......×
× × × ........×
For any ratio : ,
1. ratio
, is called duplicate ratio of :
2. ratio
is called triplicate ratio of :
3. ratio √
√ is called sub -duplicate ratio of :
4. ratio √
√ is called sub -triplicate ratio of :
Ex. For ratio 4:5
1) duplicate ratio is 4 :5 i.e.,4×4∶5×5i.e.,16:25
2) triplicate ratio is 4 :5 i.e.,4×4×4∶5×5×5 i.e.,64:125
3) sub -duplicate ratio is √4:√5
4) sub -triplicate ratio is √4 ∶ √5 munotes.in
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1.1.3 Inverse Ratio:
If : is a ratio then : is inverse ratio or reciprocal ratio of : .
Ex. original ratio is 17: 25 then inverse ratio is 25 : 17
1.1.4 Continued Ratio:
If more than two quantities of the same kind are compared , it is known as
continued ratio.
Ex. quantities a, b, c, d is said to be in continued ratio then it is written as
a:b:c:d.
1.1.5 Solved examples:
Ex: Divide a profit of Rs. 25,828 between two partners in the ratio 4:7.
Solution:
Since the ratio of profit is 4:7.
Let the shares of two partners in profit be 4x and 7x respectively.
∴4 +7 =25,828
∴11 =25,828
∴ =25,8258
11
∴ =2,348
Hence, share of first partner in profit is 4 =4×2348 =Rs.9392
Share of second partner in profit is 7 =7×2348 =Rs.16436
Ex: Find three positive numbers in the ratio 3:5:2 such that the sum of
their squares is equal to 2432.
Solution :
Given that three numbers are in ratio 3:5:2.
Let these numbers be 3 ,5 and 2 .
∴ (3 ) +(5 ) +(2 ) =2432
∴ 9 +25 +4 =2432
∴38 =2432
∴ =
∴ =64 munotes.in
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⇒ =8
So, required positive numbers are
3 =3×8=24,
5 =5×8=40 and
2 =2×8=16.
Ex: If the ratio of A:B is 4:5 and the ratio of B:C is 3:2, find the ratio of
A:B:C .
Solution:
Since A : B is 4:5,B:C is 3:2
Since we have to find ratio A:B:C i.e., to make contribution of B same in
both ratios.
Therefore, we multiply first ratio 4:5 by 3
i.e.,
= ×
× =
also, we multiply second ratio 3:2 by 5
i.e.,
= ×
× =
Therefore, ratio A:B:C is 12: 15 :10.
Ex: Ratio of the present ages of Suresh and h is mother is 2:5. Suresh was
born when h is mother was 27 years old. Find their present ages.
Solution:
Since, ratio of the present ages of Suresh and h is mother is 2:5
Let, their present ages be 2 and 5 respectively.
Before 2 years, Mothers age was 27
So, 5 −2 =27
3 =27
=9.
So present age of Suresh = 2 =2×9=18yrs.
Present age of h is mother = 5 =5×9 =45yrs.
Ex: There are 30fruits in a basket, and the ratio of the number of apples to
the numb er of oranges is 1:1. How many more orange to be added to the
basket to make the ratio 1:2? munotes.in
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Solution:
Total fruits in a basket = 30
Current ratio of apples to orange is 1:1
∴ Number of apples =
×30=15 and
∴ Number of oranges =
×30=15
Let more oranges to be added to basket ,
Then as per given condition new ratio becomes 1:2
∴ 15
15+ =1
2
∴ 15×2 =15+
∴30=15+
∴ =15.
Hence, 15 more oranges to be added to basket to make ratio 1:2
Ex: Monthly incomes of A and B are in the ratio 7:4 and their
expenditures are in the ratio 9:5. Each of them saves Rs. 10,000. Find their
incomes.
Solution:
Since, Monthly incomes of A and B are in the ratio 7:4
Let their incomes be Rs. 7 and 4 .
We know that, saving = income – expenditure
∴ expenditure = income – saving
Also, each of them saves Rs. 10,000
Therefore, their expenditures be 7 −10,000 and 4 −10,000
It is gives that;ratio of their expenditure s is 9:5.
i.e. ,
, =
∴ 5(7 −10,000)=9(4 −10,000)
∴35 −50,000 =36 −90,000
∴90,000 −50,000 =36 −35
∴ =40,000 munotes.in
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Hence,
Income of A= 7 −10,000 =7×40,000 = .2,80,000
Income of B =4 −10,000 =4×40,000 = .1,60,000
Ex: The total marksof Simran and Romain examination are in the ratio
11:13. The difference between their marks is 56. What are their marks?
Solution:
Given that, marks of Simran and Roma are in ratio 11: 13
Let, their marks be 11 and 13 .
Since , the difference between their marks is 56.
∴13 −11 =56
∴2 =56
∴ =28
Hence,
Marks of Simran =11×28=308 and
Marks of Roma =13×28=364 .
1.2 PROPORTION:
1.2.1 Introduction:
Proportion indicates equality between two ratios. It can also extend to
equality of more than two ratios.
Let a, b, c, d be four quantities such that a and b of the same kind and c
and d are of the same kind. We say that a, b, c, d is in proportion , if two
ratios a:b and c: dis equal .
It is denoted by : :: : or
=
this implies = i.e., product of
last two terms is equal to product of middle two trems.
In this relation first term and last term are called ‘ Extremes ’ and middle
terms are called ‘Means’.
1.2.2 Continued proportion:
If three quantities a, b, c of same kind such that : = : i.e.
=
⇒ =
⇒ =√ munotes.in
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1.2.3 Direct proportion:
When two ratios are in direct proportion with each other means they are in
same direction. i.e., the terms of ratio are increasing or decreasing
simultaneously.
One can express this relatio n in the form
= .
i.e.,
is k times of
1.2.4 Inverse proportion:
When two ratios are in inverse proportion with each other means they are
in opposite direction. i.e., the terms of ratio are increasing or decreasing
inversely.
One can express this relation in the form
= .
i.e.,
is times of
1.2.5 Solved Examples:
Ex: find fourth proportional to 5, 10 and 7.
Solution:
Let, fourth proportional be .
Therefore, w e have 5, 10, 7 and in proportion.
∴ 5
10=7
∴ 5 =7×10
∴ =70
5
∴ =14
Hence, required forth term in given proportion is 14.
Ex: if cost of 9 toys is Rs. 180 , how much would 15toys cost?
Solution:
There is r atio of chocolates to cost in both statements, so they are in
proportion.
Let, cost of 15 toys be Rs. .
Then we have 9, 180, 15, are in proportion and we have to find .
∴9
180=15
∴9 =15×180 munotes.in
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∴ =15×180
9=300
Hence the cost of 15 toys is Rs. 300.
Ex: ( +12), ( +4), ( +5) ( −1) are in proportion. Find .
Solution:
∴ +12
+4= +5
−1
∴( +12)( −1)=( +4)( +5)
∴ +11 −12= +9 +20
∴11 −9 =20+12
∴2 =32
∴ =16
Ex: What number must be subtracted from each of the numbers 8, 16 and
40. So that resulting numbers are in continued proportion?
Solution:
Let be that number to subtracted from each of 8, 16 and 40.
So let, resulting number be 8− , 16− , 40− which are in continued
proportion.
∴(8− )×(40− )=(16− )
∴320 −48 + =256 −32 +
∴320 −256 =48 −32
∴64=16
∴ =4
Hence 4 must be added from 8, 16 and 40 to get resulting numbers in
continued proportion.
Ex: 15 workers can finish a project in 60days. If the project must be
finished in 36 days, how many more worker will be required?
Solution:
Let, the number of workers required to finish project in 36 days be .
number of workers and days are in inverse proportio n. munotes.in
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9
i.e.
=
and we have two ratios, 15 : 60 and : 36
Therefore,
∴15
=36
60
∴15×60=36
∴15×60
36=
∴ =900
36
∴ =25
i.e. 25 workers will finish work in 36 days.
So, needed 10 more workers(already 15 workers are there) to finish work
in 36 days.
Ex: How to bearloss of Rs. 25,828 between two partners in the ratio 6:5.
Solution:
Let, their shares in loss be 6 and 5 .
∴6 +5 =25828
∴11 =25828
∴ =25828
11
∴ =2348
Therefore, partners have to bear loss of Rs.14,088 and Rs. 11,740
respectively.
Ex: Manoj’s saving box contains a total of Rs. 248 in the form of coins of
Rs.10, Rs.5, Rs.2, Rs.1 in the ratio 9:3:7:5. Find the number of coins of
each denomination.
Solution:
Let, number of coins be 9 ,3 ,7 ,5 respectively .
i.e., Rs. 10 coins be 9 , Rs. 5 coins be 3 , Rs. 2 coins be 7 and Rs. 1
coin be 5 .
Therefore, Amount of Rs.10 = 10×9 =90 , munotes.in
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Amount of Rs.5 = 5×3 =15 ,
Amount of Rs.2 = 2×7 =14 and
Amount of Rs.1 = 1×5 =5
∴Total amount = 90 +15 +14 +5
∴248 =90 +15 +14 +5
∴124 =248
∴ =248
124=2
Therefore,
coins of Rs.10 = 9 =9(2)=18
Coins of Rs.5= 3 =3(2)=6
Coins of Rs.2= 7 =7(2)=14
Coins of Rs.1 = 5 =5(2)=10
Ex: A, B and C invested Rs.70,000, Rs.50,000 and Rs.80,000 respectively
in a business. At the end of the year, C received Rs.16,000 as his share in
the profit. Find A's and B's share in the profit.
Solution:
Since, ratio of their investment is 70,000 :50,000:80,000
We can divide all terms by 10,000 as they in proportion.
∴ ration becomes 7:5:8
Profit is distributed in same ratio that of investment .
Let, share of partner's A, B, C in profit be 7 ,5 ,8 respectively.
Given that, share of C is 8 =16,000
∴ =16,000
8
∴ =2,000
Therefore, share of A = 7 =7×2000 =14,000
Share of B = 5 =5×2000 =10,000
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1.3 PERCENTAGE:
1.3.1 Introduction:
Percentage means per cent (cent means hundreds) i.e., per hundred.
One can convert any ratio to percentage and vice versa.
1) Multiply fraction (ratio) by 100 it will get converted to percentage.
2) Divide percentage by 100 it will get converted to fraction.
Ex.:
1)
=
×100 % =75 %
2)
=10 %
3) 25%=
=
4) 33% 240 =
×240 =79.2
1.3.2 Solved examples:
Ex: A person's monthly income increased from Rs. 8,000 to Rs. 8,560.
Find the percentage increase.
Solution:
Income increased from Rs. 8000 to Rs. 8560, i.e. inceased by Rs. 560.
So we take ratio of incement to base. i.e. 560 ∶8000
To convert in percentage we multiply it by 100.
∴560
8000×100 =7%
Hence, a person’s monthly income increased by 7%.
Ex: 8% of certain amount is Rs.20. Find the amount.
Solution:
Let, that certain amount be .
∴8% =20
∴8
100× =20
∴ =20×100
8
∴ =250 munotes.in
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Hence, required amount is Rs. 250.
Ex: Aniruddha spends 60% of his income. If in a month, saved Rs.20,928
then find his earning for that month.
Solution:
Let, Aniruddha’s earning for that month be .
He spends 60% of his income, it means he saves 40% of his income .
∴ As per given information,
Rs.20,928 is 40% of his income i.e.
∴40 % =20928
∴40
100× =20928
∴ =20928 ×100
40
∴ =52320
Hence, aniruddha’s income for that month is Rs.52,320.
Ex: If akshy's profit decreased from 28,000 to 26,880. find the percentage
of decrease.
Solution:
We first find amount decreased and then convert it to percentage.
Since, akshy's profit decreased from 28,000 to 26,880 , i.e. decreased by
Rs. 1120.
We take ratio of amount decrease to base. i.e. 1120 : 28000
To convert to percentage, multiply by 100.
∴1120
28000×100 =4%
Hence, akshay’s profit decreased by 4%.
1.4 CHECK YOUR PROGRESS:
1) If 4, 20, z is in continued proportion, then z = _________.
A) 100 B) 110 C) 120 D) 130
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2) The percentage of marks secured by a student scoring 438 out of 600 is
_____.
A)70 B) 71 C) 72 D) 73
3) The duplicate ratio of 4:3 is _________.
A)16: 9 B) 9: 16 C) 3: 4 D) 4: 9
4) The percentage of marks secured by a student scoring 360 out of 600 is
_____.
A) 70 B) 60 C) 80 D) 90
5) The compound ratio of 3/2 , 1/5 and 2/9 is
A) 3/8 B) 3/8 : 1/5: 2/9 C) 1/15 D) 8/3
6) The duplicate ratio of 3:2 is
A) 6:4 B) 9:4 C) 5:2 D) 6:1
7) The sub -triplicate ratio of 1:64 is
A) 64:1 B) 3:192 C) 1:16 D) 1:4
8) The inverse ratio of 9:4 is
A) 4:9 B) 1/4 : 1/9 C) 3:2 D) 1:36
9) If a:b = 2:3 and b:c = 2:3 then a:b:c is
A) 2:4:3 B) 4:6:9 C) 2:6:9 D) 9:6:4
10) If the angels of a triangle are in the ratio 3:8:9 then their respective
degree measures are
A) 27 , 72 , 81 B) 32 , 70 , 78 C) 24, 64 , 92
11) 20% of 360 is
A) 7.2 B) 720 C) 72
12) The fourth proportion to 21, 30 and 35 is
A) 50 B) 18 C) 24.5
13) Two numbers are in the ratio 3 : 4. If 6 is added to each term of the
ratio, the new ratio is 4 : 5. So, the given numbers are
A) 9 , 12 B) 18 , 24 C) 15 , 20
14) Present ages of Soham and Soha are in the ratio 7 : 12. If two years
ago, the rat io of their ages was 3 : 8 , then their present ages are
A) 7yrs , 12yrs B) 3yrs , 6yrs C) 3yrs 6 month, 6yrs munotes.in
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Answers:
1) 100 2) 73 3) 16:9 4) 60 5) 1/15 6) 9:4
7) 1:4 8) 4:9 9) 4:6:9 10) 27, 72, 81 11) 72
12) 50 13) 18, 24 14) 3yrs 6months, 6yrs
1.5 UNSOLVED PROBLEMS:
1) Find the value of ′ ′ if :3∷60:15
Ans: 12
2) A table costs a carpenter Rs. 720 to make. He sells it for Rs. 920. What
percentage of profit does he earn?
Ans: 27.78 %
3) The price of 15 suits is Rs.6750. How many such suits can be
purchased by an amount of Rs. 4050?
Ans: 9
4) 6 men can paint a house in 4 days. How long it would take to paint the
house if 3 men are employed?
Ans: 8 days
5) Sagar spends 25% of his income on house rent, 60% of the rest amount
on household expenditure. If he saves Rs. 2100, what is his total
income?
Ans: Rs. 7000
6) Ex: A college collected Rs. 6,02,136 for charity which is to be divided
between an orphanage and a school in the ratio 7:11. what amount did
each institution receive ?
Ans: Rs. 2,34,164 &Rs.3,67,972
7) Three numbers are in the ratio 2: 3:5 and the sum of their squares is
950. Find the numbers.
Ans: 10, 15, 25
8) A ratio is equal to 5: 7. If its antecedent is 35, what is the consequent?
Ans: 49
9) Two numbers are in the ratio 3: 5. If 8 is added to each number, the
ratio becomes 2:3. Find the numbers.
Ans: 24, 40
10) What number must be added to each term of the ratio 3 : 5 to make it
11 : 12 ?
Ans: 19
11) Divide 3,600 among A, B, C in the ratio 1/3:1/4:1/6 .
Ans. 1600, 1200, 800 munotes.in
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12) Divide 3,740 into three parts in such a way that half of the first part ,
one third of the second part, and one -sixth of the third part are equal.
Ans: 680, 1020, 2040
13) In a mixture of 63 litres, the ratio of milk and water is 5:2. How much
water must be added tothis mixture to make the ratio 3:2 ?
Ans: 12 litres
14) An empl oyer reduces the number of employees in the ratio 10:7 and
increases their wages in the ratio of 14:15. In what ratio, the wage bill
is increased or decreased?
Ans: Decreased in ratio 4:3
15) A purse contains one rupee, 50 paise and 25 paise coins in the ratio 2:
3:4. If the total amount in the purse is Rs. 180, find the number of
coins of each kind.
Ans: 80, 120, 160
16) Ritika reduces her weight in the ratio 3:2. What is her weight now, if
originally it was 81 kg?
Ans: 54 kg
17) The length and width of a rectangle are in the ratio 5: 8. If the
perimeter of the rectangle is 156 feet. What are the length and width
of the rectangle?
Ans. length =30 feet ,width =48 feet
18) A, B, C start a busin ess by investing 2 20,000, 35,000 and 45,000
respectively and share the profit of 10,000 at the end of the year. Find
the share in profit
Ans: A =2,000, B = 3,500, C =4,500
19) Mr. Manish, Mr. Gopal and Mr. Kumar started a transport business of
investing 1 lakh each. Mr. Manish left after 5 month from the
commencement of business and Mr. Gopal left 3 months later. At the
end of the year the business realized a profit of 75,000. Find the share
ofprofit of each partner.
Ans: Manish =15,000, Gopal = 24,000, Kumar =36,000
20) The difference of squares of two numbers which are in the ratio 3:5 is
144. Find the numbers.
Ans: 9 and 15
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21) Find the fourth proportional to 6, 10 and 12.
Ans: 120
22) Find the third proportional to 8 and 16.
Ans: 32
23) Find the mean proportion of 3.6 and 4.9.
Ans: 4.2
24) What least number must be added to each of the numbers 16, 7, 79 and
43, so that the resulting numbers are in proportion?
Ans: 5
25) If (x-2), (X+2), (2X+1) and (2X+19) are in proportion, find the value
of X.
Ans:x =4
26) The ratio of income to the expenditure of a company is 5: 3. If the
company spends 2,700, what is its income? How much is its saving?
Ans:Rs. 4,500
27) If 4 men can do a work in 2 days, in how many days 2 men can do
this work?
Ans: 4 days
28) If 48 men can dig a pit in 14 days, how long will it take 28 men to dig
the similar pit.
Ans: 24 days
29) A garrison is provided with food for 80 soldiers to last for 60 days.
Find how long would the food last if 20 additional soldiers join them
after 15 days.
Ans: 36 days
30) The rent for 3 rooms for 9 months is 135. What will be the rent for 9
rooms for 3 months?
Ans:135
31) If 10 horses consume 18 bushels in 36 days, how long will 24 bushels
last for 30 horses?
Ans: 16 days
32) There are 20 eggs in a fridge and 6 of them are brown. What percent
of eggs are not brown? munotes.in
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Ans: 70%
33) 44% of the students of a class are girls. If the number of girls is 6 less
than the number of boys, how many students are there in the class?
Ans: 50
34) A man donated 5% of his monthly income to a charity and deposited
12% of the rest in bank. If he has Rs. 11,704 left with him, what is
his monthly income?
Ans:Rs. 14,000
35) The passing marks in a examination is 35%. If a student gets 190
marks, falls short by 20 marks, then find the total marks of the
examination conducted.
Ans: 600
36) Samir bought a bicycle for 1.724 and after three month s he sold it at a
profit of 25% for what price did he sell the bicycle?
Ans:Rs. 2,155
37) By selling an article for 4,200, the shopkeeper lost 20%. At what
price should he have sold it to gain 16%?
Ans: Rs.6,090
38) By selling a soft set for 3,825, a shopkeeper loses 15% on it. Find the
price at which it was bought.
Ans:Rs. 4,500
39) Rajnish sells a pair of shoes at a profit of 30%. Find its cost price if
the selling price is 650 .
Ans: 500
40) In an examination a stud ent scored 176 marks and failed by 34 marks.
Find the maximum marks of the examination if a student must score
at least 40% to pass the examination.
Ans: 525
41) A shopkeeper purchases two calculators A and B at a total cost of
550. He sells calculator A a t 15% profit and calculator B at a loss of
20% and gets the same selling price for both the calculators. Find the
cost price of each one of the two calculators.
Ans. A = 225.64 , B=324.36 munotes.in
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42) Soham bought grapes worth Rs. 500 and sold half of them at a gain of
8%. At whatgain percent must he sell the remaining grapes so as to
get a gain of 20% on the whole?
Ans: 32%
43) A dealer buys a table listed at 1,800 and gets a discount of 25%. He
spends 150 on transportation and se lls it at a profit of 10%. Find the
selling price of table.
Ans: 1,650
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PROFIT, LOSS, DISCOUNT, COMMISSION
AND BROKERAGE
Unit Structure :
2.1 Profit and Loss
2.2 Exercises
2.3 Discount
2.4 Trade discount and cash discount
2.5 Exercises
2.6 Commission and brokerage
2.7 Exercises
2.1 PROFIT AND LOSS
Traders, shopkee pers earn by selling goods generally. Traders earn by
buying and selling goods. The price at which an item is purchased is called
its cost price (CP). The price at which an item is sold is called its Net
Selling Price (NSP).
If the Net Selling Price is g reater than cost price, then a profit is earned. If
the selling price is less than the cost price, then a loss is incurred. When
the selling price is equal to cost price then neither profit nor loss is made.
This is called a Break even point.
Profit = NSP - CP it NSP > CP
Loss = CP - NSP if NSP < CP
Profit percentage (loss percentage) is calculated when profits (or loss) is
compared with the cost price.
Profit % Profit= ×100CP
Loss % Loss= ×100CP
This gives profit 100CP= Profit% ×
Now, since, NSP = CP + Profit
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20 Hence, NSP = CP + 100CPProfit% ×
NSP = 100Profit%1+ ×CP
NSP = 100100+ Profit%×CP
Similarly we can show that,
NSP = 100100 - Loss%×CP
NSP = 100100+ Profit%×CP
when profit is earned
NSP = 100100 - Loss%×CP
when loss is incurred
Solved Examples : (Profit and Loss)
1) Ms. Divya bought a property per Rs. 14,00,000 and sold it for
Rs. 14,72,000. Find her perc entage profit or loss.
Solution :
Cost Price = 14,00,000
Selling price = 14,72,000
Since selling price > cost price, profit is earned
Profit = Selling price - Cost Price
= 14,72,000 - 14,00,000
= 72,000
% Profit Profit= ×100CP
14,00,00072,000= ×100 = 5.14
2) Mr. Harish bought a wardrobe for Rs. 4,50,000 and sold it for
Rs. 4,36,000. Find his percentage gain or loss.
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Cost Price = 4,50,000
Selling Price = 4,36,000
Since cost price > s elling price, Loss is incurred
Loss = Cost Price - Selling Price
= 4,50,000 - 4,36,000
= 14,000
Loss % Loss= ×100CP
4,50,00014,000= ×100= 3.11
3) An article is bought for Rs. 4,800 and sold at 15% profit. What was the
selling price?
Solution :
Selling price = ?
Cost price = 4,800, Profit % = 15
Selling price = Cost Price 100 %100+ Profit
100 154800100
1154800100
5520+
4) An article was bought for Rs. 7,000 and sold at 2% loss. Find its
selling pric e.
Solution :
Cost Price = 7,000, Loss % = 2
Selling Price = Cost Price 100 %100- Loss munotes.in
Page 22
22 100 27000100987000100
6860-
5) Printer was sold at Rs. 8,980 at a 5% loss. Find its cost Price.
Solution :
Selling Price = 8,980, Loss % = 5
Selling Price = Cost Price 100 %100- Loss
8,980 = Cost Price 100 5100-
8,980 = Cost Price (0.95)
Cost Price = 8,9809,452.630.95
6) Mr. Tushar earned a profit at 25% on cost by selling an article for
Rs. 6,140. What would have been the percentage profit or loss if he
had to sold the article for Rs. 7,040?
Solution :
Selling Price = 6,140, Profit % = 25
Selling Price = Cost Price 100 %100+ Profit
6,140 = Cost Price 100 25100
6,140 = Cost Price (1.25)
Cost Price = 6,1404,9121.25
Now Cost Price = 4,912, and selling price = 7,040 then profit % or
Loss % = ?
Since selling price > cost price, profit is earned
Profit = Selling Price - Cost Price
= 7,040 - 4,912
= 2,128
% Profit = 2,128100 100 43.324,912Profit
CP
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Page 23
23 7) Mr. Vinay made a 7% loss by selling the article for Rs. 13,625. What
would have been his percen tage loss or gain if he had sold it for
Rs. 15,250?
Solution :
Selling Price = 13,625, Loss % = 7, CP = ?
Selling Price = Cost Price 100 %100- Loss
13,625 = Cost Price 100 7100-
13,625 = Cos t Price 93100
Cost Price = 13,62514,650.540.93
Selling Price = 15,250 and cost price = 14,650.54.
Since selling price > cost price, Profit is earned
Profit = Selling Price - Cost Price
= 15,250 - 14,650.54
= 599.46
% Profit = 599.46100 100 4.0914,650.54Profit
CP
8) A shopkeeper brought two tables at Rs. 8,800 each. He sold one table
at 10% profit and the other at 10% loss. Find his total percentage gain
or loss.
Solution :
For the first table, selling price =?
Selling price = Cost Price 100 %100+ Profit
100 108,800100+
= 9,680
For Second table, Selling Price =?
Selling price = Cost Price 100 %100- Loss
100 108,800100-
= 7,920
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24 Total Cost Price = 8,800 + 8,800 = 17,600
Total Selling Price = 9,680 + 7,920 = 17,600
Total Cost Price = Total Selling Price
No Profit No Loss
9) A person sold two tables for Rs. 990 each and thus earned 10% profit
on one and incurred a 10% loss on the other. If both tables are
considered together, find his percentage profit or loss.
Solution :
Total selling price = 990 + 990 = 1980
For first table, CP = ?
Selling Price = Co st Price 100 %100+ Profit
990 Cost Price 100 10100+
990 = Cost price (1.1)
Cost Price 990900101
For Second table, CP = ?
Selling Pri ce = Cost Price 100 %100- Loss
990 Cost Price 100 10100-
Cost Price 99011000.9
Total Cost Price = 900 + 1100 = 2000
Total Selling Price = 990 + 990 = 1980
Since Total Selling Price < Total Cost Price, Loss is incurred.
Loss = Total Cost Price - Total Selling Price
= 2000 - 1980
= 20
Loss % 100LossTotal Cost Price
20100 12000
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Page 25
25 10) When sold at a p rofit of 6% an article gives Rs. 715 more than when it
is sold at a loss of 5%. Find its cost price?
Solution :
Let Cost Price = Rs. x
If the article is sold at 6% profit, then
Selling Price = Cost Price 100 %100+ Profit
100 + 61001.06xx
If the article is sold at 5% loss, then
Selling Price = Cost Price 100 %100- Profit
100 -51000.95xx
The difference 1.06 0.95 0.11xxx
But it given as 715.
0.11 71571565000.11xx
Cost Price = 6500
2.2 EXERCISES
1) Ms. Sheena bought a property for Rs. 12,00,000 and sold it for
Rs. 12,72,000. Find her percentage profit or loss.
2) Mr. Madhukar bought a sofa set for Rs. 3,50,000 and sold it for
Rs. 3,36,000. Find his percentage profit or loss.
3) A Lamp shade was bought for Rs. 2,800 and sold at 15% profit, what
was the selling price?
4) A cupboard was sold for Rs. 7,906 at 18% profit, what was the cost
price?
5) An article was boug ht for Rs. 5,000 and sold at 5% loss. Find its
selling price.
6) A printer was sold at Rs. 7,980 at 5% loss. Find its cost price.
7) By selling an article at Rs. 3,000, a person earned 20% profits. What
would have b een the percentage profit or loss is he has sold it at
Rs. 2,750? munotes.in
Page 26
26 8) A person earned 12% profit by selling an article at Rs. 4,144. What
would have been the selling price if he had sold it at 16% profit?
9) By selling an article at Rs. 24,288, a loss of 8% on cost was incurred.
Had the article been sold at 8% profit, what would have been the
selling price?
10) A shopkeeper bought 2 chairs at Rs. 4,400 each. He sold one chair at
10% profit and the other at 10% loss. Find his total percentage gain or
loss.
11) When sold at a profi t of 6%, an article fetches Rs. 715 more than when
it is sold at a loss of 5%. Find its cost price.
Answers :
1) 6% Profit 2) 4% loss 3) Rs. 3,220 4) Rs. 6,700 5) Rs. 4,900
6) Rs. 8,400 7) 10% Profit 8) Rs. 4,292 9) Rs. 28,512
10) No Profit No Loss 11) Rs. 6,500
2.3 DISCOUNT
In the market each good / item for sale has a marked price or printed price.
The prices of all goods forms a list or catalogue, hence this rice is also
called as the List Price or listed price or catalogue price.
LP List Price
In order to attract customers, the seller (manufacturer or trader) offers a
reduction in the list price. This reduction is called as Discount. After
deducting the discount, an item is sold at the Net Selling Price (NSP).
Thus,
NSP = LP - Discount
Usually discount is expressed as a percentage on the list price.
2.4 TRADE DISCOUNT AND CASH DISCOUNT
When a trader is selling goods to another trader (e.g. wholesaler to
retailer) usually a two discounts structure is followe d.
First at all, trade discount (TD) is given to all traders. T.D. is a percentage
on the List price. List price minus the Trade Discount is called Invoice
Price (IP) or reduced List Price.
If an item is purchased for immediate cash payment then an ext ra discount
which is called as cash discount is offered. It is calculated against the
invoice price.
munotes.in
Page 27
27 Thus,
Trade discount = x% at List Price.
Invoice Price = List Price - Trade Discount
Cash discount = 9% at Invoice Price.
Net Selling Price = I nvoice Price - Cash discount
Profit = NSP - CP
Profit % 100ProfitCP
Loss = CP - NSP
Loss %100LossCP
Solved Examples : (Discount)
1) After giving a 15% discount on the list price, a pant is sold for
Rs. 1,500, find the list price.
Solution :
Trade discount = 15% (on list price)
Net selling price = 1500
Let List Price = Rs. x
Trade discount = 15% of x = 0.15 x
No Cash discount is given,
Net Selling Price = Invoice Price
Invoice Price = Net Selling Price = List Price - Trade discount
1500 = x - 0.15 x
1500 = 0.85 x
15001764.710.85x
2) A trader bought an article for Rs. 4000 and listed it for Rs. 9000. He
gave 10% discount on the list price. What was the his pro fit
percentage?
Solution :
Cost price = 4000
List Price = 9000
Trade discount = 10%
Invoice Price = List Price - Trade discount
= 9000 - 10% at 9000
= 9000 - 900
= 8100
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Page 28
28 No cash discount is given
Invoi ce Price = Net Selling Price
Net Selling Price = 8100
Profit = Net Selling Price - Cost Price
= 8100 - 4000
= 4100
Profit % 100ProfitCP
41001004000102.5
3) A trader purchased a cupboard and listed it two times his purchase
price and then allowed 50% discount on the list price. Find the profit
percentage.
Solution :
Let Cost Price = Rs. 100
List Price = 200
Trade discount = 50% on LP
Invoice Price = List Price - Trade discount
= 200 - 50% of 200
= 200 - 100
= 100
Cash discount is not given Invoice Price = Net Selling Price
Net Selling Price = 100
Profit = Net Selling Price - Cost Price
= 100 - 100
= 0
4) A trad er purchased an item for Rs. 4000. After giving 20% discount on
the list price, he made 44% Profit on his cost. Find the list price.
Solution :
Cost Price = 4000
Trade discount = 20% on List Price
Let List Price = Rs. x
Invoice Price = List Price - Trade discount
= x - 20% of x
= 0.8 x
Profit % = 44 munotes.in
Page 29
29 Net Selling Price = Invoice Price
Net Selling Price = Cost Price 100 %100+ Profit
0.8 x = 4000 100 44100+
0.8 x = 4000 (1.44)
0.8 x = 5760
x 576072000.8
List Price = `7200
5) A trader gave 30% discount on List Price and made 50% profit on his
cost. If his list price was Rs. 7200, find hi s cost price.
Solution :
Trade discount = 30%
Profit % = 50
List Price = 7200
Invoice Price = List Price - Trade discount
= 7200 - 30% of 7200
Invoice Price = 7200 - 2160 = 5040
Net Selling Price = Invoice Price
Net Selling Price = Cost P rice 100 %100+ Profit
5040 = Cost Price 100 50100+
5040 = Cost Price (1.5)
Cost Price 504033601.5
6) After giving a 20% discount, a pen was sold for Rs. 608 and 45%
profit on cost was made. Fi nd the list price and the cost price.
Solution :
Trade discount = 20%
Net Selling Price - 608
Profit % = 45
Let List Price = Rs. x
Invoice Price = List Price - Trade discount
= x - 20% of x
= 0.8 x
Net S elling Price = Invoice Price munotes.in
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30 Net Selling Price = 0.8 x
0.8x = 608
x 6087600.8
List Price = Rs. 760
Net Selling Price = Cost Price 100 %100+ Profit
608 = Cost Price 100 45100+
Cost Price 608419.311.45
7) A trader gave 20% discount on list price and a further 2% cash
discount on the invoice price. If the list price was Rs. 3000 then fi nd
net selling price.
Solution :
List Price = 3000
Trade discount = 20% on List Price
Invoice Price = List Price - Trade discount
= 3000 - 20% of 3000
= 3000 - 600
= 2900
Cash discount is given on Invoice P rice =
2% of 2400 = 12 2400 48100
Net Selling Price = Invoice Price - Cash discount
= 2400 - 48
= 2352
8) A trader gave 15% trade discount and further 1% cash discount for
cash payment and sold article for Rs. 50000. Find the List Price.
Solution :
Let List Price = x
Invoice Price = List Price - Trade discount
= x - 15% of x
= 0.85 x
Cash discount = 1% of Invoice Price
Net Selling Price = Invoice Price - Cash disc ount
= 0.85 x - 1% of 0.85 x
= 0.85 x - 0.0085 x munotes.in
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31 Net Selling Price = 0.8415 x
But Net Selling Price = 50000
50000 = 0.8415 x
5000059417.710.8415x
List Price = Rs. 59417.71
9) Shopkeeper purchased article for Rs. 4000 and listed it at 4 times the
purchase price. He then allowed 30% trade discount and further 3% for
cash payment. Find the profit percentage.
Solution:
Cost Price = Rs. 4000
List Price = Rs. 16000
Trade discount = 30% on List Price
Invoice Price = List Price - Trade discount
= 16000 - 30% of 16000
= 16000 - 4800
= 11200
Cash discount = 3% on invoice price
Net Selling Price = I nvoice Price - Cash discount
= 11200 - 3% of 11200
= 11200 - 336
Profit Profit = Net Selling Price - Cost Price
= 10864 - 4000
= 6864
Profit % 100ProfitCost Price
68641004000171.6
10) A trader gave 30% trade discount and 5% cash payment and made
20% Profit on his cost price of Rs. 8000. Find the list price.
Solution :
Cost Price = 8000
Let List Price = Rs. x
Invoice Price = List Price - Trade di scount
= x - 30% of x
= 0.7 x munotes.in
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32 Cash discount is given on Invoice Price = 5%
Net Selling Price = Invoice Price - Cash discount
= 0.7 x - 5% of 0.7 x
= 0. 7x -0.035 x
= 0.665 x
Net Selling Price = Cost Price 100 %100+ Profit
0.665 x = 8000 100 20100+
0.665 x = 8000 (1.2)
0.665 x = 9600
960014436.090.665x List Price = 14436.09
11) A shopkeeper gave 30% trade discount and a further 2% cash discount
on an item and sold it for a net price of Rs. 40000 and still made 25%
profit on cost. Find the list price and cost price.
Solution :
Let List Price = Rs. x
Trade discount = 30% on List Price
Invoice Price = List Price - Trade discount
= x - 30% of x
= 0.7 x
Cash discount is 2% on Invoice Price
Net Selling Price = Invoice Price - Cash discount
= 0.7 x - 2% of 0.7 x
= 0.7 x - 0.014 x
= 0.686 x
But Net Selling Price = 40000
40000 0.6864000058309.040.686xx
Net Selling Price = Cost Price 100 %100+ Profit
40000 = Cost P rice 100 25100+
Cost Price 400001.25
32000
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Page 33
33 2.5 EXERCISES
1) After giving a 12% discount on the list price, a pant is sold for
Rs. 1056, Find the List price.
2) A Trader bought gift item for Rs. 3,600 and listed it for Rs. 8,500. He
gave 9% discount on the list price, what was his profit percentage?
3) A trader purchased a cupboard and listed it four times his purchase
price and then allowed a 50% discount on the list price. Find the profit
percentage.
4) A trader purchased an item for Rs. 4,000 and after giving 20%
discount on the list price, he made 44% profit on his cost. Find the list
price.
5) A trader gave 20% disc ount on list price made 44% profit on his cost.
If the list price was Rs. 720, find his cost price.
6) After giving 20% discount, a pen was sold for Rs. 304 and 52% profit
on cost was made. Find the list price and cost price.
7) A trader gave 10 trade di scount on the list price and a further 1% cash
discount on the invoice price, if the list price was Rs. 2430, then find
net selling price.
8) A trader gave 10% trade discount and a further 1% discount for cash
payment and sold a bathroom fitting for Rs. 216513 . Find the list
price.
9) A trader purchased a gift item for Rs. 4,000 and listed it 2.5 times the
purchase price she then allowed 40% discount and further 4% for cash
payment. Find the percentage profit.
10) After giving 25% trade discount and a fu rther 4% discount for cash
payment a merchant made 19.52% profit. If the list price was
Rs. 16600 find the merchant’s cost price.
11) A merchant allowed 30% trade discount and a further 2% cash
payment discount on an item and sold it for a net price of Rs. 30870
and still made 20% profit on cost. Find the merchant’s list price and
cost price.
Answers :
1) Rs. 1200 2) 114.896 % 3) 100% 4) Rs. 7200
5) Rs. 400 6) Rs. 380, Rs. 200 7) Rs. 2165.13
8) Rs. 243000 9) 44% 10) Rs. 10000
11) Rs. 45000 12) Rs. 25725
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Page 34
34 2.6 COMMISSION AND BROKERAGE
Most of the manufacturers do not sell their goods directly to the
customers. They employ other to sell goods for them. Traders who by and
sell goods also sometimes emplo y others to buy and or / sell goods for
them.
An agent or a commission agent is a person who buys and or / sells goods
for another person. The person who employs the services of the agent is
called the principal and the remuneration given by the principa l to the
agent is called commission.
A commission agent may sells goods for cash or on credit. If the sale takes
place on credit, then there could be a risk of payment default.
A Del credere agent sells goods and guarantees for the collection of dues
from the customers to the principal. for this he or she charges extra
commission which is also known as del credere.
A broker is an agent who brings together buyer and seller and negotiated
the sale. For this he or she charges a commission called brokerag e from
buyer and the seller.
An auctioneer sells the goods by public audition to the highest bidder.
Solved Examples : (Commission and Brokerage)
1) An agent charged 10% commission on sales and thus earned Rs. 5000.
Find the value of sales.
Solution :
Sales = ?, Commission = 10% of sales
But commission = 5000 (Given)
Commission = 10% of sales
5000 = 10% of sales
5000 = 1100sales
5000 10010 Sales
Sales = Rs. 50000
2) An agent sold goods worth Rs. 30000 and after deducting his
commission, remitted Rs. 27125 to the principal. Find the rate of
commission charged by the agent.
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Page 35
35 Solution :
Total Sales = Rs. 30000
Amount remitted = 27125
Commission = 30000 - 27125
= 2875
Late rate of commission = x%
Commission = x% of 30000
2875 =130000100x
2875 =300 x
28759.58 10%300x ;
4) A manufactured gave his agent 7% commission plus 4% del credere.
The agent sold goods worth Rs. 25000. What amount should the agent
give principal after deducting his commission?
Solution :
Total Sales = Rs. 25000
Ordinary Commission = 7% of 25000
17 250001001750
Del credere = 4% of 25000
14 250001001000
Total Commission = Ordinary Commission + Del credere
1750 10002750
Amount remitted after deducting his commission
= 25000 - 2750
= 22250
4) A salesman receives 4% commission on sales upto Rs. 6000 and 5%
commission on sales above Rs. 6000. If he sold goods worth Rs. 8000
in a week, find the commission earned by him.
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Page 36
36 Solution :
Commission at 4% on 46000100
= 240
Additional Commission at 5% on (8000 - 6000)
= 5% on 2000
15 2000100
= 100
Total Commission = 240 + 100 = 34 0
5) A company fixed the rate of commission to its salesman as follows :
5% on the first 12000,
9% on next 8000,
10% on next 9000 and 6% on balance.
Find the commission of a Salesman who received Sales worth
Rs. 33000.
Solution :
Total Sales = 33000 = (12000 + 8000 + 9000 + 4000)
Commission = 5% on 12000 + 9% on 8000 + 10% on 9000 + 6% on
4000 (balance)
111 15 12000 9 8000 10 9000 6 4000100 100 100 100600 720 900 240
= `2460
6) A Company pays its salesman a monthly salary of Rs. 10000 and
commission as follows : 2% on sales above Rs. 10000 and upto
Rs. 16000 2.5% on sales over Rs. 16000 and upto Rs. 20000 3% on
sales over Rs. 20000.
Find the total monthly remuneration of a salesman who sold goods
worth Rs. 35000 in a month.
Solution :
Monthly Salary = Rs. 10000
Commission @2% between 10000 & 16000 = (16000 - 10000) 2.5% on 4000 = 100
3% on sales over 20000 i.e. (35000 - 20000)
= 3% on 15000 = 450
Total Commission = 120 + 100 + 450 = 670
Total monthly remune ration = 10000 + Total Commission
= 10000 + 670
= 10670 munotes.in
Page 37
37 7) A Salesman is allowed 8% commission on total sales plus a bonus of
2% on the sales exceeding Rs. 15000. If he earned Rs. 1600 on
commission done, find his total earnings.
Solution :
Commissi on = Rs. 1600
Let Total Sales = Rs. x
Commission = 8% of x
11600 81001600 0.08xx
16000.08x Rs. 20,000
Total Sales = 20000
If sales exceeds Rs. 15000 then bonus will be @2%
= 2% of (2000 0 - 15000)
= 2% of 5000
= 12 5000 100100
Total earnings = 1600 + 100 = 1700
8) A salesgirl is allowed 8% commission on total sales plus 3% bonus on
sales above Rs. 24000. If her total earnings are Rs. 3440 then find the
value of her sales.
Solution : Commission + bonus = 3440
Let total Sales be Rs. x
Commission @ 8% on x = 8% of x
181000.08xx
Commission + bonus = 3440
Bonus = 3% of ( x - 24000)
Commission + 3% of ( x - 24000) = 3440
3% 24000 3440of x Commission munotes.in
Page 38
38 13 24000 3440 0.08100
113 3 24000 3440 0.08100 100
0.03 720 3440 0.08
0.03 0.08 3440 720
0.11 4160
416037818.190.11xxxxxx
xx
x
x
9) A house was sold through a broker for Rs. 90,00,000 who charged
2.5% from the buyer and 1.5% from the seller. Find the amount paid
by the buyer. Also find the amounts received by the seller and t he
broker.
Solution : - Total Sales = Rs. 90,00,000
Amount paid by buyer = Total purchase + brokerage
= 90,00,000 + 2.5% of 90,00,000
= 90,00,000 + 225000
= 92,25,000
Amount received by seller
= Total Sales - brokerage
= 90,00,000 - 1.5% of 90,0 0,000
= 90,00,000 - 1,35,000
= 88,65,000
Amount received by broker
= 2,25,000 + 1,35,000
= 3,60,000
10) A Trader instructed his agent to buy 600 caps and sell them 50%
above the purchase price. The agent charged 2% commission on the
purchase and 3% commission on sales and thus earned Rs. 2000 as
total commission what was the purchase price of a single cap?
Solution :
Let purchase price of a single cap = Rs. x
Selling price of 600 caps = 600x + 900x
= 900x
Commission of 2% on purchase = 2% of 600x
12 60010012xx munotes.in
Page 39
39
Commission of 3% on Sales = 3% of 900x
13 90010027xx
Total Commission = 12x + 27x
39x
This is given to be 2000
39 20002000
3951.28 52xxx ;
Purchase price of a single cap = 52
11) A del credre agent charges 3% commission on cash sales and 5%
commission on credit sales. In a particular year, he earned on an
average 4.5% commission on total sales. Find the ratio of his cash
sales to credit sales.
Solution :
Let Cash Sales = Rs. x
Credit Sales = Rs. y
Commission at 3% on Cash Sales 3100x
0.03x
Commission at 5% on Credit Sales = 0.05y
Total Commission = 0.03x + 0.05y
But given that
Total Commission = 4.5% on Sales
4.5100(Cash Sales + Credit Sales)
= 0.045 (x + y)
= 0.045x + 0.045y
Thus we have,
Total Commission
0.03 0.05 0.045 0.0450.005 0.015
0.005
0.015
1
3xy x yyx
x
y
x
y
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Page 40
40 12) After deducting his commission at 6% on first Rs. 90,000 and 9% on
balance of sales made by him, an agent remitted Rs. 96,000 to his
principal. Find the value of goods sold by him.
Solu tion :
Let Total Sales be Rs. x
Commission at 6% on first Rs. 90000
690000100 `5400
Commission at 9% on remaining Sales i.e. (x - 90000)
9900001000.09 8100xx
Total Commiss ion = 5400 + 0.09x - 8100
=0.09x - 2700
Agent remitted to the principal
= Sales Value - Total Commission
= x - (0.09x - 2700)
= x - 0.09x + 2700
=0.91x + 2700
This is given to be 96000
0.91 2700 960000.91 96000 270093300
0.91102527.47x
x
xx
13) A Salesman is paid a fixed monthly salary plus a commission at a
certain rate on sales. The salesman received Rs. 1130 and Rs. 1360 as
remuneration for two successive months and his sales were Rs. 17100
and Rs. 21700 respected. Find the fixed m onthly salary and the rate of
commission.
Solution :
Remuneration = Salary + Commission
Let Salary Rs. x
Commission rate = y%
1130 %xy of 17100
1130 171xy ………….. (1)
1360 = x + y% of 21700
1360 217xy ………….. (2)
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Page 41
41 (1) - (2)
1130 1711360 217230 462305%46xyxyyy
Rate of commission = 5%
Putting y = 5 in (1),
1130 = x + 5% of 17100
i.e. 1130 = x + 5(171)
1130 = x + 855
1130 855275x
x
Salary = `275
2.7 EXERCISE S
1) An agent earned 4794 after charging 8.5% commission on sales. Find
the value of sales.
2) An agent sold goods worth Rs. 25000 and after deducting his
commission, remitted Rs. 24125 to the principal. Find the rate of
commission charged by the agent.
3) A manufacturer gave his agent 6% commission plus 3% del credere.
One agent sold goods worth Rs. 22000. What amount should the agent
give the principal after deducting his commission?
4) A salesgirl receives 3.5% commission on sales upto Rs. 5000. If s he
sold goods worth Rs. 7500 in a week, find the commission earned by
her.
5) A company fixed the rate of commission to its salesman as follows :
3% on first 1000
4% on next Rs. 9000
5% on next Rs. 8000
6% on balance
Find the remuneration of a salesm an who secured sales worth Rs.
32500.
6) A Salesman is allowed 7% commission on total sales plus a bonus at
2.5% on the sales above Rs. 15000. If he earned Rs. 1400 on
commission, alone, Find his total earnings. munotes.in
Page 42
42 7) A sales girl is allowed 8% commission on total sales plus 2% bonus on
sales above Rs. 24000. If her total earnings are 2420 than find the
value of her sales.
8) A house was sold through a broker for 80,00,000 who charges 2.5%
from the buyer and 1.5% from the seller. Find the amount paid by the
buyer. Also find the amount received by the seller and the broker.
9) A trader instructed his agent to buy 400 caps and sell them 50% above
the purchase price. The agent charged 1% commission on the purchase
and 2% commission on the sales and thus earned Rs. 1000 as total
commission what was the purchase price of a single cap?
10) A del credere agent charges 3% commission on cash sales and 5%
commission or credit sales. In a particular year, he earned on an
average 4.5% commission on total sales find the r atio of his cash sales
to credit sales.
11) An agent was paid Rs. 22275 as commission on total sales. If the rate
of commission was 12% and the price of each TV was Rs. 6875, find
the number of TV’s he sold.
12) A Company fixed the rate of commission to its salesman as follows
3% on first Rs. 5000, 5% on the next Rs. 8000, 8% on next Rs. 10000
and 11% on the balance. The company had agreed to pay 114% of
total sales as bonus it the sales crossed Rs. 30000. A salesman of the
company secured sales worth Rs. 32000. Calculate total earnings of
the salesman.
Answers :
1) Rs. 56,400 2) 3.5% 3) Rs. 20000 4) Rs. 287.5
5) Rs. 1,390 6) Rs. 1,525 7) Rs. 29,000
8) Rs. 82,00,000, Rs. 78,80,000, Rs. 3,20,000 9) 100 10) 1:3
11) 27 12) Rs. 2,420
Glossary :
Agent or Commission Agent : A person or a firm that buys and / or sells
goods for another person or firm, for a remuneration (which is called
commission and which is usually a percentage on the sales value.)
Broker : An agent who brings together prospective buyer and seller and
negotiates a deal, charging a commission from both the buyer and the
seller.
Cash Discount (C.D) : A reduction given on the invoice price for cash
payment, usually by a manufacturer or a trader to another trader, thereby
lowering the invoice price to the net selling price. Cash discount is
specified as a percentage on the invoice price.
Cost Price (C.P.) : The price at which an article is purchased. munotes.in
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43 Del Credere agent : A commission agent who guarantees the principal
the collection of dues from the customers for an extra remuneration (above
the usual commission) which is called as del credere commission.
Discount : A reduction in the price given to attract customers. Discount is
specified as a percentag e on the price.
Invoice Price (1. P) or Reduced List Price : The list price minus the
trade discount.
List Price (L.P.) or Printed Price or Marked Price or Catalogue Price:
It is the price printed on goods, which is found in the price list or
catalogue , before giving any discount.
Loss : A negative profit.
Net Selling Price (N.S.P.) : The price at which an article is actually sold.
Principal : The person or firm which employs the commission agent.
Profit : The difference between the net selling price and the cost price. If
the profit is negative, it is called a loss.
Trade Discount (T.D.) : A reduction on the list price given by a
manufacturer or a trader to a trader, thereby lowering the list Price to the
invoice price. Trade Discount is speci fied as a percentage on the list price.
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44 3
SIMPLE INTEREST AND
COMPOUND INTEREST
Unit Structure :
3.0 Objectives
3.1 Introduction
3.2 Definitions of Terms Used In This Chapter
3.3 Simple Interest
3.4 Compound Interest
3.0 OBJECTIVES
After reading this chapter you will be able to:
Define interest, principal, rate of interest, period.
Find simple interest (SI), rate of S.I., period of investment.
Find Compound Interest (CI), rate of C.I., Amount accumulated at the
end of a period.
Compound interest compounded yearly, half -yearly, quarterly or
monthly.
3.1 INTRODUCTION
In every day life individuals and business firms borrow money from
various sources for different reasons. This amount of money borrowed has
to be returned from the lender in a stipulated time by paying some interest
on the amount borrowed. In this chapter we are going to study the two
types of interests viz. simple and compound interest. We start with some
definitions and then proceed with the formula related to both the types of
interests.
3.2 DEFINITIONS OF TERMS USED IN THIS CHAPTER
Principal : The sum borrowed by a person is called its principal . It is
denoted by P.
Period : The time span for which money is lent is called period . It is
denoted by n.
Interest : The amount paid by a borrower to the lender for the use of
money borrowed for a certain period of time is called Interest . It is
denoted by I. munotes.in
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45 Rate of Interest : This is the interest to be paid on the amount of Rs. 100
per annum (i.e. per year). This is denoted by r.
Total Amount : The sum of the principal and inter est is called as the total
amount (maturity value ) and is denoted by A. Thus, A = P + I .
i.e. Interest paid I = A – P.
3.3 SIMPLE INTEREST
The interest which is payable on the principal only is called as
simple interest (S.I.). For example the intere st on Rs. 100 at 11% after one
year is Rs.11 and the amount is 100 + 11 = Rs. 111.
It is calculated by the formula: I =100Pnr= P x n x 100r
Amount at the end of nth year = A = P + I = P + 100Pnr= P1100nr
Remark : The period n is always taken in ‘years’. If the period is given in
months/days, it has to be converted into years and used in the above
formula. For example, if period is 4 months then we take n = 4/12 = 1/3 or
if period is 60 days then n = 60/365.
Example 1: If Mr. Sagar borrows Rs. 500 for 2 years at 10% rate of
interest, find (i) simple interest and (ii) total amount.
Ans: Given P = Rs. 500, n = 2 and r = 10 %
(i) I =100Pnr=500 x 2 x 10100= Rs. 100
(ii) A = P + I = 500 + 100 = Rs. 600
3.3.1 Problems involving unknown factors in the formula I =100Pnr
The formula I =100Pnrremaining the same, the unknown factor in
the formula is taken to the LHS and i ts value is computed. For example, if
rate of interest is unknown then the formula is rewritten as x 100P x Irn .
Example 2: If Mr. Prashant borrows Rs. 10 00 for 5 years and pays an
interest of Rs. 300, find rate of interest.
Ans: Given P = 1000, n = 5 and I = Rs. 300
Now, I = 100Pnr x 100P x Irn = 300 x 1001000 x 5 = 6
Thus, the rate of interest is 6%. Simple Interest = Prinicpal x period x rate of interest
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46 Example 3: Find the period for Rs. 2500 to yield Rs. 900 in simple
interest at 12% .
Ans: Given P = Rs. 2500, I = 900, r = 12 %
Now, I =100Pnr x 100 x InPr = 900 x 1002500 x 12 = 3
Thus, the period is 3 years.
Example 4: Find the period for Rs. 1000 to yield Rs. 50 in simple interest
at 10%.
Ans: Given P = Rs. 1000, I = 50, r = 10%
Now, I =100Pnr x 100 x InPr = 50 x 1001000 x 10 = 0.5
Thus, the period is 0.5 years i.e. 6 months.
Example 5: Mr. Ak ash lent Rs. 5000 to Mr. Prashant and Rs. 4000 to Mr.
Sagar for 5 years and received total simple interest of Rs. 4950. Find (i)
the rate of interest and (ii) simple interest of each.
Ans: Let the rate of interest be r.
S.I. for Prashant = 5000 x 5 x 100r= 250r … (1)
and S.I. for S agar = 4000 x 5 x 100r= 200 r … (2)
from (1) and (2), we have,
total interest from both = 250 r + 200 r
= 450 r
But total interest received be Mr. Ak ash = Rs. 4950 450r = 4950 r =4950450= 11 the rate of interest = 11%
Example 6: The S. I. on a sum of money is one -fourth the principal. If the
period is same as that of the rate of interest then find the rate of interest.
Ans: Given I = 4Pand n = r
Now, we know that I =100Pnr 4P= x x 100Prr 1004= r2 r2 = 25 r = 5. the rate of interest = 5%
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47 Example 7: If Rs. 8400 amount to Rs. 110 88 in 4 years, what will Rs.
10500 amount to in 5 years at the same rate of interest?
Ans:
(i) Given n = 4, P = Rs. 8400, A = Rs. 11088 I = A – P = 11088 – 8400 = Rs. 2688
Let r be the rate of interest.
Now, I = 100Pnr 2688 = 8400 x 4 x 100r r = 8%
(ii) To find A when n = 5, P = Rs. 10500, r = 8
A = P 1100nr= 10500 x 5 x 81100= 10500 x 140100= 14700 the required amount = Rs. 14,700
Example 8: Mr. Shirish borrowed Rs. 12,000 at 9% interes t from Mr.
Girish on January 25, 2007. The interest and principal is due on August
10, 2007. Find the interest and total amount paid by Mr. Shirish.
Ans: Since the period is to be taken in years, we first count number of
days from 25th January to 10th Aug ust, which is 197 days.
Now, I = 100Pnr= 12000 x 197365x 9100 I = Rs. 582.9
Total amount = P + I = 12000 + 582.9
A = Rs. 12,582.9
Check your progress 10.1
1. Find the SI and amount for the following data giving principal, rate
of interest and number of years:
(i) 1800, 6%, 4 years. (ii) 4500, 8%, 5 years
(iii) 7650, 5.5%, 3 years. (iv) 6000, 7.5%, 6 years
(v) 25000, 8%, 5 year s (vi) 20000, 9.5%, 10 years.
Ans: (i) 432, 2232 (ii) 1800, 6300, (iii) 1262.25, 8912.25
(iv) 2700, 8700 (v) 10000, 35000 (vi) 19000, 39000
2. Find the S.I. and the total amount for a principal of Rs. 6000 for 3
years at 6% rate of interest.
Ans: 1080, 7080
January 6
February 28
March 31
April 30
May 31
June 30
July 31
August 10
Total 197 munotes.in
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48 3. Find the S.I. and the total amount for a principal of Rs. 3300 for 6
years at 3½ % rate of interest.
Ans: 693, 3993
4. Find the S.I. and the total amount for a principal of Rs. 10550 for 2
years at 10¼ % rate of interest.
Ans: 2162.75, 12712.75
5. Find the rate of interest if a person invests Rs. 1000 for 3 years and
receives a S.I. of Rs. 150.
Ans: 5%
6. Find the rate of interest if a person invests Rs. 1200 for 2 years and
receives a S.I. of Rs. 168.
Ans: 7%
7. A person invests Rs. 4050 in a bank which pa ys 7% S.I. What is the
balance of amount of his savings after ( i) six months, ( ii) one year?
Ans: 141.75, 283.5
8. A person invests Rs. 3000 in a bank which offers 9% S.I. After how
many years will his balance of amount will be Rs. 3135?
Ans: 6 months
9. Find the principal for which the SI for 4 years at 8% is 585 less than
the SI for 3½ years at 11%.
Ans: 9000
10. Find the principal for which the SI for 5 years at 7% is 250 less than
the SI for 4 years at 10%.
Ans: 5000
11. Find the principal for which the SI for 8 years at 7.5% is 825 less
than the SI for 6½ years at 10.5%.
Ans: 10000
12. Find the principal for which the SI for 3 years at 6% is 230 more
than the SI for 3½ years at 5%.
Ans: 46000
13. After what period of investment would a principal of Rs. 12,350
amount to Rs. 17,043 at 9.5% rate of interest?
Ans: 4 years
14. A person lent Rs. 4000 to Mr. X and Rs. 6000 to Mr. Y for a period
of 10 years and received total of Rs. 3500 as S.I. Find ( i) rate of
interest, ( ii) S.I. from Mr. X, Mr. Y.
Ans: 3.5%, 1400, 2100
15. Miss Pankaj Kansra lent Rs. 2560 to Mr. Abhishek and Rs. 3650 to
Mr. Ashwin at 6% rate of interest. After how many years should he
receive a total interest of Rs. 3726?
Ans: 10 years munotes.in
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49 16. If the rate of S.I. on a certain principal is same as that of the period
of investment yields same interest as that of the principal, find the
rate of interest.
Ans: 10%
17. If the rate of S.I. on a certain principal is same as that of the period
of investment yields interest equal to one -ninth of the principal, find
the rate of intere st.
Ans: 313 years
18. Find the principal and rate of interest if a certain principal amounts
to Rs. 2250 in 1 year and to Rs. 3750 in 3 years.
Ans: 1500, 50%
19. Find the principal and rate of interest if a certain principal amounts
to Rs. 3340 in 2 years and to Rs. 4175 in 3 years. Ans: 1670, 50%
20. If Rs. 2700 amount Rs. 3078 in 2 years at a certain rate of interest,
what will Rs. 7200 amount to in 4 years at the same rate on interest?
Ans: 7%, 9216
21. At what rate on interest will certa in sum of money amount to three
times the principal in 20 years?
Ans: 15%
22. Mr. Chintan earns as interest Rs. 1020 after 3 years by lending Rs.
3000 to Mr. Bhavesh at a certain rate on interest and Rs. 2000 to Mr.
Pratik at a rate on interest 2% more than t hat of Mr. Bhavesh. Find
the rates on interest.
Ans: 6%, 8%
23. Mr. Chaitanya invested a certain principal for 3 years at 8% and
received an interest of Rs. 2640. Mr. Chihar also invested the same
amount for 6 years at 6%. Find the principal of Mr. Chaitanya and
the interest received by Mr. Chihar after 6 years.
Ans: 11000, 3960
24. Mr. Ashfaque Khan invested some amount in a bank giving 8.5%
rate of interest for 5 years and some amount in another bank at 9%
for 4 years. Find the amounts invested in both the bank s if his total
investment was Rs. 75,000 and his total interest was Rs. 29,925.
Ans: 45000, 30000
25. Mrs. Prabhu lent a total of Rs. 48,000 to Mr. Diwakar at 9.5% for 5
years and to Mr. Ratnakar at 9% for 7 years. If she receives a total
interest of Rs. 25, 590 find the amount she lent to both.
Ans: 18000, 30000
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50 3.4 COMPOUND INTEREST
The interest which is calculated on the amount in the previous year is
called compound interest .
For example, the compound interest on Rs. 100 at 8% after one year is Rs.
8 and after two years is 108 + 8% of 108 = Rs. 116.64
If P is the principal, r is the rate of interest p.a. then the amount at the end
of nth year called as compound amount is given by the formula:
The compound interest is given by the formula:
Note :
1. The interest may be compounded annually (yearly), semi -annually
(half yearly), quarterly or monthly. Thus, the general formula to calculate
the amount at the end of n years is as follows:
Here p: number of times the interest is compounded in a year.
p = 1 if interest is compounded annually
p = 2 if interest is compounded semi -annually (half-yearly )
p = 4 if interest is compounded quarterly
p = 12 if interest is compounded monthly
2. It is easy to calculate amount first and then the compound intere st as
compared with finding interest first and then the total amount in case of
simple interest.
Example 9: Find the compound amount and compound interest of Rs.
1000 invested for 10 years at 8% if the interest is compounded annually.
Ans: Given P = 1000, r = 8, n = 10.
Since the interest is compounded annually, we have
A = 1100nrP= 1000 x 1081100= 1000 x 2.1589 = Rs. 2158.9
Example 10: Find the principal which will amount to Rs. 11,236 in 2
years at 6% compound int erest compounded annually.
A = P 1100nr
CI = A – P
A = P 1 x 100npr
p
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51
Ans: Given A = Rs. 11236, n = 2, r = 6 and P = ?
Now, A = 1100nrP 11236 = P 261100= P x 1.1236 P = 112361.1236= 10,000 the required principal is Rs. 10,000.
Example 11: Find the compound amount and compound interest of
Rs. 1 200 invested for 5 years at 5% if the interest is compounded
(i) annually, ( ii) semi annually, ( iii) quarterly and ( iv) monthly.
Ans: Given P = Rs. 1 200, r = 5, n = 5
Let us recollect the formula A = P1 x 100npr
p
(i) If the interest is compounded annually , p = 1:
A = 1100nrP= 1200 x 551100= 1200 x 1. 2763 = Rs. 1 531.56
CI = A – P = 1531.56 – 1200 = Rs. 33 1.56
(ii) If the interest is compounded semi -annually , p = 2:
A = 212 x 100nrP= 1200 x 1051200= 1200 x 1. 28 = Rs. 1 536
CI = A – P = 1536 – 1200 = Rs. 3 36.
(iii) If the interest is compounded quarterly , p = 4:
A = 414 x 100nrP= 1200 x 2051400= 1200 x 1. 2820 =
Rs. 1 538.4
CI = A – P = 1538.4 – 1200 = Rs. 3 38.4
(iv) If the interest is compounded monthly , p = 12 :
A = 12112 x 100nrP= 1200 x 60511200= 1200 x 1. 2834 = Rs. 1 540
CI = A – P = 1540 – 1200 = Rs. 340
Example 12: Mr. Santosh wants to invest some amount for 4 years in a
bank. Bank X offers 8% interest if compounded half yearly while bank Y
offers 6% interest if compounded monthly. Which bank should Mr.
Santosh select for better benefits?
Ans: Given n = 4.
Let the principal Mr. Santosh wants to invest be P = Rs. 100 munotes.in
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52 From Bank X: r = 8 and interest is compounded half -yearly, so p = 2. A = 212 x 100nrP= 100 x 481200= 116.9858 … (1)
From Bank Y: r = 6, p = 12
A = 12112 x 100nrP= 100 x 48611200= 127.0489 … (2)
Comparing (1) and (2), Dr. Ashwinikumar should invest his amount in
bank Y as it gives more interest at the end of the period.
Example 13: In how many years would Rs. 75,000 amount to
Rs. 1,05,794.907 at 7% compound interest compounded semi -annually?
Ans: Given A = Rs. 105794. 907, P = Rs. 75000, r = 7, p = 2
A =212 x 100nrP 105794.907 = 75000 x 271200n
105794.90775000 = (1.035) 2n
1.41059876 = (1.035) 2n
(1.035) 10 = (1.035) 2n 2n = 10
Thus, n = 5
Example 14: A certain principal amounts to Rs. 4410 after 2 years and to
Rs.4630.50 after 3 years at a certain rate of interest compounded annually.
Find the principal and the rate of interest.
Ans: Let the principal be P and rate of interest be r.
Now, we know that A = P1100nr
From the given data we have,
4410 = P21100r and 4630.5 = P31100r
4410 = P(1 + 0.01 r) 2 … (1)
Do not write ‘1 + 0.01 r’ as 1.01r munotes.in
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53 4630.5 = P(1 + 0.01 r)3 … (2)
Dividing (2) by (1), we have
4630.54410= 32(1 0.01 )(1 0.01 )PrPr
1.05 = 1 + 0.01 r 0.05 = 0.01 r
Thus, r = 5%
Example 15: Find the rate of interest at which a sum of Rs. 2000 amounts
to Rs. 2690 in 3 years given that the interest is compounded half yearly.
(61.345= 1.05)
Ans: Given P = Rs. 2000, A = Rs. 2680, n = 3, p = 2
Now, A = 212 x 100nrP 2690 = 2000 x 61200r 26902000 = 61200r 1.345 = 61200r 61.345 = 1 + 200r 1.05 = 1 + 200r
r = 0.05 x 200 = 10%
Thus, the rate of compound interest is 10 % .
Example 16: If the interest compounded half yearly on a certain principal
at the end of one year at 8% is Rs. 3264, find the principal.
Ans: Given CI = Rs. 3264, n = 1, p = 2 and r = 8
Now, CI = A – P = P281200 – P
i.e. 3264 = P[ (1.04) 2 – 1] = 0.0816 P P =32640.0816= 40000
Thus, the principal is Rs. 40,000.
Check your progress 10.2
1. Compute the compound amount and interest on a principal of Rs.
21,000 at 9% p.a. after 5 years.
Ans: 32,311.10, 11,311.10
2. Compute the compound amount and interest on a principal of Rs.
6000 at 11% p.a. after 8 years.
Ans: 13827.23, 7827.23 munotes.in
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54 3. Compute the compound amount and compound interest of Rs. 5000
if invested at 11% for 3 years and the interest compounded
i) annually, ( ii) semi annually, ( iii) quarterly and ( iv) monthly.
Ans: (i) 6838.16, 1838.16 (ii) 6894.21, 189421
(iii) 6923.92, 1923.92 (iv) 6944.39, 1944.39
4. Compute the compound amount and compound interest of Rs. 1200
if invested at 9% for 2 years and the interest compounded
i) annually, ( ii) semi annually, ( iii) quarterly and ( iv) monthly.
Ans: (i) 1425.72 , 225.72 (ii) 1431.02, 231.02
(iii) 1433.8, 233.8 (iv) 1435.7, 235.7
5. Miss Daizy invested Rs. 25,000 for 5 years at 7.5% with the interest
compounded semi -annually. Find the compound interest at the end
of 5 years. Ans: 11,126.10
6. Mr. Dayanand b orrowed a sum of Rs. 6500 from his friend at 9%
interest compounded quarterly. Find the interest he has to pay at the
end of 4 years? Ans: 2779.54
7. Mr. Deepak borrowed a sum of Rs. 8000 from his friend at 8%
interest compounded annually. Find the int erest he has to pay at the
end of 3 years? Ans:
2077.70
8. Mr. Deshraj borrowed Rs. 1,25,000 for his business for 3 years at
25% interest compounded half yearly. Find the compound amount
and interest after 3 years. Ans: 2,53,410.82;
12,8410.82
9. Mrs. Hemlata bought a Sony Digital Camera for Rs. 15,800 from
Vijay Electronics by paying a part payment of Rs. 2,800. The
remaining amount was to be paid in 3 months with an interest of 9%
compounded monthly on the due amount. How much amount did
Mrs. Hemlat a paid and also find the interest.
Ans: 13294.70, 294.70
10. Mr. Irshad bought a Kisan Vikas Patra for Rs. 10000, whose
maturing value is Rs. 21,000 in 4½ years. Calculate the rate of
interest if the compound interest is compounded quarterly.
Ans: 16.8%
11. Wha t sum of money will amount to Rs. 11236 in 2 years at 6% p.a.
compound interest? Ans: 10,000
12. Find the principal which will amount to Rs. 13468.55 in 5 years at
6% interest compounded quarterly. [ (1.015)20 = 1.346855]
Ans: 10000
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55 13. Find the principal which will amount to Rs. 30626.075 in 3 years at
7% interest compounded yearly. Ans: 25000
14. Find the principal if the compound interest payable annually at 8%
p.a. for 2 years is Rs. 1 664. Ans: 10000
15. If Mr. Sagar wants to earn Rs. 50 000 afte r 4 years by investing a
certain amount in a company at 10% rate of interest compounded
annually, how much should he invest? Ans:
34150.67
16. Find after how many years will Rs. 4000 amount to Rs. 4494.40 at
6% rate of interest compounded yearly. Ans: n = 2
17. Find after how many years Rs. 10,000 amount to Rs. 12,155 at 10%
rate of interest compounded half -yearly . Ans: n = 1
18. Find the rate of interest at which a principal of Rs.10000 amounts to
Rs. 11236 after 2 years. Ans: 6%
19. Find the rate of interest at which a principal of Rs.50000 amounts to
Rs. 62985.6 after 3 years. (31.259712 = 1.08) Ans: 8%
20. Mrs. Manisha Lokhande deposited Rs. 20,000 in a bank for 5 years.
If she received Rs.3112.50 as interest at the end of 2 years, find the
rate of interest p.a. compounded annually. Ans: 7.5%
21. A bank X announces a super fixed deposit scheme for its customers
offering 10% interest compounded half yearly for 6 years. Another
bank Y offers 12% simple interest for the same period. Which bank’s
scheme is more beneficial for the customers? Ans: Bank X
22. ABC bank offers 9% interest compounded yearly while XYZ bank
offers 7% interest compounded quarterly. If Mr. Arunachalam wants
to invest Rs. 18000 for 5 years, w hich bank should he choose?
Ans: Bank ABC
23. Mangesh borrowed a certain amount from Manish at a rate of 9% for
4 years. He paid Rs. 360 as simple interest to Manish. This amount
he invested in a bank for 3 years at 11% rate of interest compounded
quarterly. Find the compo und interest Mangesh received from the
bank after 3 years. Ans:
1384.78
24. On a certain principal for 3 years the compound interest
compounded annually is Rs. 11 25.215 while the simple interest is
Rs. 1050, find the principal and the rate of interest.
Ans: 5000, 7%
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56 25. On a certain principal for 4 years the compound interest
compounded annually is Rs. 13923 while the simple interest is Rs.
12000, find the principal and the rate of interest.
Ans: 30000, 10%.
26. Which investment is better for Mr. Hariom Sharma (i) 6%
compounded half yearly or (ii) 6.2% compounded quarterly?
Ans:
27. Which investment is better for Mr. Suyog Apte (i) 9% compounded
yearly or (ii) 8.8% compounded quarterly?
Ans:
28. A bank X offers 7% interest compounded semi -annual ly while
another bank offers 7.2% interest compounded monthly. Which bank
gives more interest at the end of the year?
Ans:
29. Mr. Nitin Tare has Rs. 10000 to be deposited in a bank. One bank
offers 8% interest p.a. compounded half yearly, while the other
offers 9% p.a. compounded annually. Calculate the returns in both
banks after 3 years. Which bank offers maximum return after 3
years?
Ans:
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57 4
ANNUITIES AND EMI
Unit Structure :
4.0 Objectives
4.1 Introduction
4.2 Annuity
4.3 Types of Annuities
4.4 Sinking Fund
4.5 Equated Monthly Installment (Emi)
4.0 OBJECTIVES
After reading this chapter you will be able to:
Define annuity, future va lue, present value, EMI, Sinking Fund.
Compute Future Value of annuity due, Present Value of an ordinary
annuity.
Compute EMI of a loan using reducing balance method and flat
interest method.
Compute Sinking Fund (periodic payments).
4.1 INTRODUCTION
In the previous chapter we have seen how to compute compound interest
when a lump sum amount is invested at the beginning of the investment.
But many a time we pay (or are paid) a certain amount not in lump sum
but in periodic installments. This series of e qual payments done at periodic
intervals is called as annuity .
Let us start the chapter with the definition of an annuity .
4.2 ANNUITY
A series of equal periodic payments is called annuity . The payments are
of equal size and at equal time interval .
The common examples of annuity are: monthly recurring deposit schemes,
premiums of insurance policies, loan installments, pension installments
etc. Let us understand the terms related to annuities and then begin with
the chapter.
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58 Periodic Payment:
The amount of payment made is called as periodic payment .
Period of Payment:
The time interval between two payments of an annuity is called as the
period of payment .
Term of an annuity:
The time of the beginning of the first payment period to the end of t he last
payment period is called as term of annuity . An annuity gets matured at
the end of its term.
4.3 TYPES OF ANNUITIES
Though we will be discussing two types of annuities in detail, let us
understand different types of annuities based on the dura tion of the term or
on the time when the periodic payments are made. On the basis of the
closing of an annuity, there are three types of annuities:
1. Certain Annuity :
Here the duration of the annuity is fixed (or certain), hence called
certain annuity . We w ill be learning such annuities in detail.
2. Perpetual Annuity :
Here the annuity has no closing duration, i.e. it has indefinite duration.
Practically there are rarely any perpetuities.
3. Contingent Annuity :
Here the duration of the annuity depends on an ev ent taking place. An
example of contingent annuity is life annuity . Here the payments are to
be done till a person is alive, like, pension, life insurance policies for
children (maturing on the child turning 18 years) etc.
On the basis of when the perio dic payments are made we have two types
of annuities: ordinary annuity and annuity due.
4.3.1 Immediate (Ordinary) Annuity:
The annuity which is paid at the end of each period is called as immediate
(ordinary ) annuity . The period can be monthly, quarte rly or yearly etc. For
example, stock dividends, salaries etc.
Let us consider an example of an investment of Rs. 5000 each year is to be
made for four years. If the investment is done at the end of each year then
we have the following diagrammatic exp lanation for it: munotes.in
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59
4.3.2 Present Value :
The sum of all periodic payments of an annuity is called its present value .
In simple words, it is that sum which if paid now will give the same
amount which the periodic payments would have given at the end of the
decided period. It is the one time payment of an annuity.
The formula to find the present value ( PV) is as follows:
Let i = x 100r
p, the rate per period, then the above formula can be
rewritten as follows:
4.3.3 Future Value (Accumulated value) :
The sum of all periodic payments along with the interest is called the
future value (accumulated amount ) of the annuity.
The formula to find the future value ( A) of an immediate annuity is as
follows:
PV=11
1 x 100 x 100npP
r r
p p Where
P: periodic equal payment
r: rate of interest p.a.
p: period of annuity
PV =
11
1npP
i i munotes.in
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60
Example 1: Find the future value after 2 years of an immediate annuity of
Rs. 5000, the rate of interest being 6% p.a compounded annually.
Ans: Given n = 2, P = Rs. 5000, r = 6 and p = 1 i = 6100= 0.06
A = Pi11npi= 50000.0621 0.06 1= 50001.1236 10.06 A = 5000 x 2.06 = Rs. 10300
Example 2: Find the amount for an ordinary annuity with periodic
payment of Rs. 3000, at 9% p.a. compoun ded semi -annually for 4 years.
Ans: Given n = 4, P = Rs. 3000, r = 9 and p = 2 i = 92 x 100= 0.045
Now, A = Pi11npi= 30000.0452 x 4[(1 0.045) 1]= 30000.045x 0.4221
Thus, A = Rs. 28,140
Example 3: Mr. Ravi invested Rs. 5000 in an annuity with quarterly
payments for a period of 2 years at the rate of interest of 10%. Find the
accumulated value of the annuity at the end of 2nd year.
Ans: Given n = 2, P = Rs.5000, r = 10 and p = 4 i = 104 x 100= 0.025
Now, A = Pi11npi= 50000.025 2 x 4(1.025) 1= 50000.025x 0.2184
Thus, A = Rs. 43,680
Example 4: Mr. Ashok Rane borrowed Rs. 20,000 at 4% p.a. compounded
annually for 10 years. Find the periodic payment he has to make.
Ans: Given PV = Rs. 20,000; n = 10; p = 1 and r = 4 i = 0.04
Now to find the periodic payments P we use the following formula:
PV =
11
1npP
i i 20000 =
10110.04 1 0.04P=0.04Px 0.3244 P =20000 x 0.040.3244= 2466.09
Thus, Mr. Rane has to make the periodic payments of Rs. 2466.09 A = 11 x 100
x 100npPr
r p
p
A = Pi11npi Here,
P: periodic equal payment
r: rate of interest p.a.
p: period of annuity i.e. yearly,
half yearly, quarterly or
monthly
and i = x 100r
p munotes.in
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61
Example 5: Find the future value of an immediate annuity after 3 years
with the periodic payment of Rs. 12000 at 5% p.a. if the period of
payments is (i) yearly, (ii) half -yearly, (iii) quarterly and (iv) monthly.
Ans: Given P = Rs. 1200, n = 3, r = 5
(i) period p = 1 then i =5100= 0.05
A = Pi11ni= 120000.0531 0.05 1= 120000.051.1576 1 A = 12000 x 3.1525 = Rs. 37,830
(ii) period p = 2 then i =52 x 100= 0.025
A = Pi211ni= 120000.0256(1 0.025) 1= 120000.025x 0.1597 A = 12000 x 6.388 = Rs. 76,656
(iii) period p = 4 then i =54 x 100= 0.0125
A = Pi411ni=120000.0125[(1 + 0.0125)12 – 1] = 120000.0125 x 0.16075
A = 12000 x 12.86 = Rs. 1,54,320
(iv) period p = 12 then i =512 x 100= 0.00417
A = Pi1211ni=120000.00417[(1 + 0.00417)36 – 1] = 120000.00417 x 0.1615
A = 1200 x 38.729 = Rs. 4,64,748
Example 6: Mr. Nagori invested certain principal for 3 years at 8%
interest compounded half yearly. If he received Rs.72957.5 at the end of
3rd year, find the periodic payment he made. [(1.04)6 = 1.2653]
Ans: Given n = 3, r = 8, p = 2 i = 82 x 100= 0.04
Now, A = Pi11npi 72957.5 = 0.04P[(1 + 0.04)6 – 1]= 0.04P x 0.2653 72957.5 = P[6.6325] P = 72957.56.6325 = 11000
Thus, the periodic payment is Rs. 11,000
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62 4.4 SINKING FUND
The fund (money) which is kept aside to a ccumulate a certain sum in a
fixed period through periodic equal payments is called as sinking fund .
We can consider an example of a machine in a factory which needs to be
replaced after say 10 years. The amount for buying a new machine 10
years from no w may be very large, so a proportionate amount is
accumulated every year so that it amounts to the required sum in 10 years.
This annual amount is called as sinking fund . Another common example is
of the maintenance tax collected by any Society from its me mbers.
A sinking fund being same as an annuity, the formula to compute the
terms is same as that we have learnt in section 2.3.3
Example 7: A company sets aside a sum of Rs. 15,000 annually to enable
it to pay off a debenture issue of Rs. 1,80,000 at the end of 10 years.
Assuming that the sum accumulates at 6% p.a., find the surplus after
paying off the debenture stock.
Ans: Given P = Rs. 15000, n = 10, r = 6 i = 0.06 A = Pi11ni= 150000.06x [(1 + 0.06) 10 – 1] = 150000.06 x 0.7908 A = Rs. 1,97,700
Thus, the surplus amount after paying off the debenture stock is
= 197712 – 180000 = Rs. 17712.
Example 8: Shriniketan Co -op Hsg. Society has 8 members and collects
Rs. 250 0 as maintenance charges from every member per year. The rate of
compound interest is 8% p.a. If after 4 years the society needs to do a
work worth Rs. 100000, are the annual charges enough to bear the cost?
Ans: Since we want to verify whether Rs. 2500 y early charges are enough
or not we assume it to be P and find its value using the formula:
A = Pi11ni
Here A = Rs. 100000, n = 4, r = 8 i = 0.08
P =
x 11nAii=
4100000 x 0.081 0.08 1= 22192
Thus, the annual payment of all the members i.e. 8 members should be Rs.
22192.
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63 the annual payment per member =221928= Rs. 2774
This payment is less than Rs. 2500 which the society has decided to take
presently. Thus, the soc iety should increase the annual sinking fund.
4.5 EQUATED MONTHLY INSTALLMENT (EMI)
Suppose a person takes a loan from a bank at a certain rate of interest for a
fixed period. The equal payments which the person has to make to the
bank per month are ca lled as equated monthly installments in short EMI.
Thus, EMI is a kind of annuity with period of payment being monthly
and the present value being the sum borrowed .
We will now study the method of finding the EMI using reducing
balance method and flat interest method.
(a) Reducing balance method :
Let us recall the formula of finding the present value of an annuity.
PV =
11
1npP
i i
The equal periodic payment ( P) is our EMI which is denoted it by E.
The present value ( PV) is same as the sum ( S) borrowed.
Also the period being monthly p = 12 , i = 1200ras we are interested in
finding monthly installments and n is period in years .
Substituting this in the above formula we have:
S =
1211
1nE
i i
Thus, if S is the sum borrowed for n years with rate of interest r % p.a.
then the EMI is calculated by the formula:
E = 12 x
11(1 )nSii
(b) Flat Interest Method:
Here the amount is calculated using Simple Interest for the period and the
EMI is computed b y dividing the amount by total number of monthly
installments.
Let S denote the sum borrowed, r denote the rate of interest and n denote
the duration in years, then as we know the amount using simple interest munotes.in
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64 formula is A = S1100nr. The total number of monthly installments for
duration of n years is 12n. Hence the EMI is calculated as
E = 12An
Example 9: Mr. Sudhir Joshi has taken a loan of Rs. 10,00,000 from a
bank for 10 years at 11% p.a. Find his EMI usi ng (a) reducing balance
method and (b) Flat interest method.
Ans: Given S = Rs. 1000000, n = 10, r = 11 i = 111200= 0.0092
(a) Using flat interest method :
A = S1100nr= 10000001101100= 2100000
Thus, E = 12An= 2100000120= 17,500 … (1)
(b) Using reducing balance method :
Now, E =12 x
11(1 )nSii=1201000000 x 0.009211(1 0.0092) = 13797.65 E = Rs. 13,798 approximately … (2)
Comparing (1) and (2), we can see that the EMI using flat interest method
is higher than by reducing balance method .
Example 10: Mr. Prabhakar Naik has borrowed a sum of Rs. 60,000 from
a person at 6% p.a. and is due to return it back in 4 monthly installments.
Find the EMI he has to pay and also prepare the amortization table of
repayment.
Ans: Given S = Rs. 60,000; n = 4 months;
r = 6% i = 61200= 0.005
Now, E = x 11(1 )nSii=460000 x 0.00511(1 0.005) =3000.01975
Now, we will prepare the amortization table i.e. the table of repayment of
the sum borrowed using reducing balance method.
In the beginning of the 1st month the outstanding principal is the sum
borrowed i.e. Rs. 60000 and the EMI paid is Rs. 15187.97
The interest on the outstanding principal is 0.005 x 60000 = Rs. 300 … (1)
Thus, the principal repayment is 15187.97 – 300 = Rs. 14887.97 … (2) E = Rs. 15,187.97 munotes.in
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65 The outstanding principal ( O/P) in the beginning of the 2nd month is now
60000 – 14887.97 = 45112.03.
Note :
(1) is called the interest part of the EMI and (2) is called as the
principal part of the EMI.
As the tenure increases the interest part reduces and the principal part
increases.
This calculation can be tabulated as follows:
O/P EMI Interest Part Principal
Part Month
(a) (b) (c) = (a) x i (b) - (c)
1 60000 15187.97 300 14887.97
2 45112.03 15187.97 225.56 14962.45
3 30141.02 15187.97 150.75 15037.22
4 15111.80 15187.97 75.56 15112.41
In the beginning of the 4th month the outstandin g principal is Rs. 15111.80
but the actual principal repayment in that month is Rs. 15112.41. This
difference is due to rounding off the values to two decimals, which leads
the borrower to pay 61 paise more!!
Example 11: Mr. Shyam Rane has borrowed a sum of Rs. 100000 from a
bank at 12% p.a. and is due to return it back in 5 monthly installments.
Find the EMI he has to pay and also prepare the amortization table of
repayment.
Ans: Given S = Rs. 100000; n = 5 months;
r = 12% p.a. = 1212= 1% p.m i = 0.0 1
Now, E = x 11(1 )nSii=5100000 x 0.0111(1 0.01) =10000.0485343= 20603.98
The amortization table is as follows:
O/P EMI Interest Part Principal
Part Month
(a) (b) (c) = (a) x i (b) - (c)
1 100000 20603.98 1000 19603.98
2 80396.02 20603.98 803.96 19800.02
3 60596 20603.98 605.96 19998.02
4 40597.98 20603.98 405.98 20198
5 20399.98 20603.98 204 20399.98
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66 Check your progress
1. An overdraft of Rs. 50,000 is to be paid back in equal annual
installments in 20 years. Find the installments, if the interest is 12%
p.a. compounded annually. [(1.12) 20 = 9.64629]
2. A man borrows Rs. 30,000 at 6% p.a. compounded semi -annually
for 5 years. Find the periodic payments he has to make.
3. What periodic payments Mr. Narayanan has to make if he has
borrowed Rs. 1,00,000 at 12% p.a. compounded annually for 12
years? [(1.12)12 = 3.896]
4. Find the future value of an immediate annuity of Rs. 1200 at 6% p.a.
compounded annually for 3 years.
5. Find the future value of an immediate annuity of Rs. 500 at 8% p.a.
compounded p.m. for 5 years.
6. Find the accumulated value after 2 years if a sum of Rs. 1500 is
invested at the end of every year at 10% p.a. compounded quarterly.
7. Find the accumulated amount of an immediate annuity of Rs. 1000 at
9% p.a. compound ed semi -annually for 4 years.
8. Find the future value of an immediate annuity of Rs. 2800 paid at
10% p.a. compounded quarterly for 2 years. Also find the interest
earned on the annuity.
9. Find the sum invested and the accumulated amount for an ordinary
annui ty with periodic payment of Rs. 2500, at the rate of interest of
9% p.a. for 2 years if the period of payment is (a) yearly, (b) half -
yearly, (c) quarterly or (d) monthly.
10. Find the sum invested and the accumulated amount for an ordinary
annuity with period ic payment of Rs. 1500, at the rate of interest of
10% p.a. for 3 years if the period of payment is (a) yearly, (b) half -
yearly, (c) quarterly or (d) monthly.
11. Mr. Banerjee wants to accumulate Rs. 5,00,000 at the end of 10
years from now. How much amount sh ould he invest every year at
the rate of interest of 9% p.a. compounded annually?
12. Find the periodic payment to be made so that Rs. 25000 gets
accumulated at the end of 4 years at 6% p.a. compounded annually.
13. Find the periodic payment to be made so that Rs. 30,000 gets
accumulated at the end of 5 years at 8% p.a. compounded half
yearly.
14. Find the periodic payment to be made so that Rs. 2000 gets
accumulated at the end of 2 years at 12% p.a. compounded quarterly.
15. Find the rate of interest if a person depositin g Rs. 1000 annually for
2 years receives Rs. 2070.
16. Find the rate of interest compounded p.a. if an immediate annuity of
Rs. 50,000 amounts to Rs. 1,03,000 in 2 years. munotes.in
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67 17. Find the rate of interest compounded p.a. if an immediate annuity of
Rs. 5000 amounts to Rs. 10400 in 2 years.
18. What is the value of the annuity at the end of 5 years, if Rs. 1000
p.m. is deposited into an account earning interest 9% p.a.
compounded monthly? What is the interest paid in this amount?
19. What is the value of the annuity at the end o f 3 years, if Rs. 500 p.m.
is deposited into an account earning interest 6% p.a. compounded
monthly? What is the interest paid in this amount?
20. Mr. Ashish Gokhale borrows Rs. 5000 from a bank at 8% compound
interest. If he makes an annual payment of Rs. 150 0 for 4 years,
what is his remaining loan amount after 4 years?
(Hint : find the amount using compound interest formula for 4 years and
then find the accumulated amount of annuity, the difference is the
remaining amount.)
21. Find the present value of an immed iate annuity of Rs. 10,000 for 3
years at 6% p.a. compounded annually.
22. Find the present value of an immediate annuity of Rs. 100000 for 4
years at 8% p.a. compounded half yearly.
23. Find the present value of an immediate annuity of Rs. 1600 for 2
years at 7% p.a. compounded half yearly.
24. A loan is repaid fully with interest in 5 annual installments of Rs.
15,000 at 8% p.a. Find the present value of the loan.
25. Mr. Suman borrows Rs. 50,000 from Mr. Juman and agreed to pay
Rs. 14000 annually for 4 years at 10% p.a. Is this business profitable
to Mr. Juman?
(Hint: Find the PV of the annuity and compare with Rs. 50000)
26. Mr. Paradkar is interested in saving a certain sum which will amount
to Rs. 3,50,000 in 5 years. If the rate of interest is 12% p.a., how
much should he save yearly to achieve his target?
27. Mr. Kedar Pethkar invests Rs. 10000 per year for his daughter from
her first birthday onwards. If he receives an interest of 8.5% p.a.,
what is the amount accumulated when his daughter turns 18?
28. Dr. Wakankar, a dentist has started his own dispensary. He wants to
install a machine chair which costs Rs. 3,25,000. The machine chair
is also available on monthly rent of Rs. 9000 at 9% p.a. for 3 years.
Should Dr. Wakankar buy it in cash or rent it?
29. A sum of Rs. 50,000 is req uired to buy a new machine in a factory.
What sinking fund should the factory accumulate at 8% p.a.
compounded annually if the machine is to be replaced after 5 years?
30. The present cost of a machine is Rs. 80,000. Find the sinking fund
the company has to ge nerate so that it could buy a new machine after
10 years, whose value then would be 25% more than of today’s
price. The rate of compound interest being 12% p.a. compounded
annually. munotes.in
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68 31. Mr. Mistry has to options while buying a German wheel alignment
machine fo r his garage: (a) either buy it at Rs. 1,26,000 or (b) take it
on lease for 5 years at an annual rent of Rs. 30,000 at the rate of
interest of 12% p.a.. Assuming no scrap value for the machine which
option should Mr. Mistry exercise?
32. Regency Co -op. Hsg. So ciety which has 50 members require
Rs. 12,60,000 at the end of 3 years from now for the society repairs.
If the rate of compound interest is 10% p.a., how much fund the
society should collect from every member to meet the necessary
sum?
33. Mr. Lalwaney is of 40 years now and wants to create a fund of Rs.
15,00,000 when he is 60. What sum of money should he save
annually so that at 13% p.a. he would achieve his target?
34. If a society accumulates Rs. 1000 p.a. from its 200 members for 5
years and receives 12% inte rest then find the sum accumulated at the
end of the fifth year. If the society wants Rs. 13,00,000 for society
maintenance after 5 years, is the annual fund of Rs. 1000 per
member sufficient?
35. How much amount should a factory owner invest every year at 6%
p.a. for 6 years, so that he can replace a mixture -drum (machine)
costing Rs. 60,000, if the scrap value of the mixture -drum is Rs.
8,000 at the end of 6 years.
36. If a society accumulates Rs. 800 p.a. from its 100 members for 3
years and receives 9% interest then find the sum accumulated at the
end of the third year. If the society wants Rs. 2,50,000 for society
maintenance after 3 years, is the annual fund of Rs. 800 per member
sufficient?
37. Mr. Kanishk wants clear his loan of Rs. 10,00,000 taken at 12% p.a.
in 240 monthly installments. Find his EMI using reducing balance
method.
38. Using the reducing balance method find the EMI for the following:
Loan amount
(in Rs.) Rate of Interest
(in % p.a.) Period of Loan
(in years)
i) 1000 6 5
ii) 50000 6 10
iii) 8000 7 6
iv) 12000 9 10
v) 1000 9.5 10
vi) 1100000 12.5 20 munotes.in
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69 39. Mr. Vilas Khopkar has taken a loan of Rs. 90,000 at 11% p.a. Find
the EMI using (a) reducing balance method and (b) Flat interest
method, if he has to return the loan in 4 years.
40. Find the EMI using reducing balance method on a sum of Rs. 36,000
at 9%, to be returned in 6 monthly installments.
41. Find the EMI using reducing balance method on a sum of Rs. 72,000
at 12%, to be returned in 12 installments.
42. Mr. Sachin Andhale has borrowed Rs. 10,000 from his friend at 9%
p.a. and has agreed to return the amount with interest in 4 months.
Find his EMI and also prepare the amortization table.
43. Mr. Arvind Kamble has borrowed Rs. 30,000 from his friend at 14%
p.a. If he is to return this amount in 5 monthly instal lments, find the
installment amount, the interest paid and prepare the amortization
table for repayment.
44. Mrs. Chaphekar has taken a loan of Rs. 1,25,000 from a bank at 12%
p.a. If the loan has to be returned in 3 years, find the EMI, Mrs.
Chaphekar has to pay. Prepare the amortization table of repayment
of loan and find the interest she has to pay.
45. A loan of Rs. 75,000 is to be returned with interest in 4 installments
at 15% p.a. Find the value of the installments.
46. A loan of Rs. 60,000 is to be returned in 6 equal installments at 12%
p.a. Find the amount of the installments.
47. Find the sum accumulated by paying an EMI of Rs. 11,800 for 2
years at 10% p.a.
48. Find the sum accumulated by paying an EMI of Rs. 1,800 for 2 years
at 12% p.a.
49. Find the sum accumulated by paying an EMI of Rs. 12,000 for 3
years at 9% p.a.
50. Find the sum accumulated by paying an EMI of Rs. 11,000 for 8
years at 9.5% p.a.
Hints & Solutions to Check your progress
(1) 6694 (2) 3517 (3) 16,144 (4) 3820.32
(5) 36555.65 (6) 13104 (7) 9380 (8) 24461
(9)
Period Sum Invested Accumulated
Amount
Yearly 5000 5225
Half-yearly 10000 10695.5
Quarterly 20000 21648
Monthly 60000 65471
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70 (10)
Period Sum Invested Accumulated
Amount
Yearly 4500 4965
Half-yearly 9000 10203
Quarterly 18000 20693
Monthly 54000 62635
(11) 32910 (12) 5715 (13) 2498.72 (14) 225
(15) 7% (16) 6% (17) 8% (18) 75424, 15424
(19) 19688 , 1688 (20) 4719 (21) 26730
(22) 673274.5 (23) 5877 (24) 59890.65 (25) 44378, Yes
(26) 97093.4 (27) 393229.95 (28) 283021.25, take it on rent
(29) 12523 (30) 17698.42 (31) 108143.28 < 126000, Mr. Mistry
should use the second option. (32) 16245 (33) 18530
(34) 1270569.47, not sufficient (35) 7454.86 (36) 2,62,248; yes
(37) 11,011
(38)
Loan amount
(in Rs.) Rate of interest
(in % p .a.) Period of Loan
(in yrs.) EMI
(in Rs.)
i) 1000 6 5 19
ii) 50000 6 10 555
iii) 8000 7 6 136
iv) 12000 9 10 152
v) 1000 9.5 10 13
vi) 1100000 12.5 20 12498
(39) 2326, 2700 (40) 6158.48 (41) 6397.11
(42)
O/P EMI Interest Part Princ ipal
Part Month
(a) (b) (c) = (a) x i (b) - (c)
1 10000 2547.05 75 2472.05
2 7527.95 2547.05 56.45 2490.6
3 5037.35 2547.05 37.78 2509.27
4 2528.08 2547.05 18.96 2528.09
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71 (43)
O/P EMI Interest Part Principal
Part Month
(a) (b) (c) = (a) x i (b) - (c)
1 30000 6212.23 351 5861.23
2 24138.77 6212.23 282.42 5929.81
3 18208.96 6212.23 213.04 5999.19
4 12209.77 6212.23 142.85 6069.38
5 6140.39 6212.23 71.84 6140.39
(45) 19339.57 (46) 16353 (47) 3,12,673.60
(48) 48552.24 (49) 4,93,832.6 (50) 15,72,727
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72 5
SHARES AND MUTUAL FUNDS
Unit Structure :
5.0 Objectives
5.1 Introduction
5.1.1 Bonus Shares
5.1.2 Splitting of shares
5.2 Mutual Funds
5.3 Systematic Investment Plan (SIP)
5.0 OBJECTIVES:
After going through this chapter you will be able to:
Define a share, face value, market value, dividend, equity shares
preferential shares, bonus shares.
Understand the concept of Mutual fund.
Calculate Net Income after considering entry load, dividend,
change in Net Asset Value (N.A.V) and exit load.
Understan d the Systematic Investment Plan (S.I.P).
5.1 INTRODUCTION
In day -to-day life we hear about shares, share market etc. Here we will
see, exactly what these terms deal with .
When a group of persons plan to establish a company, they form a
company under the companies Act 1956. Now this company is an
established company . The people who establish this company are
called promoters of the company . These promoters can now raise a
certain amount of cap ital to start (run) the company . They divide this
required a mount into small parts called shares .
A share is the smallest unit of capital of a company. Stock is the
American term for share. Usually a share is of value Rs. 100 / - or
Rs.50/ - or Rs. 10 / -or Rs. 5/ - or Rs. 2/ - or Rs./ - 1 . This value is called
the fa ce value of the share. These shares are sold to the public. (usually
face value is Rs. 10/ - , unless otherwise specified ) . This sale is called
the Initial Public Offer (IPO) of the company.
The company issues share certificates to the persons from whom i t
accepts the money to raise the capital. Persons who have paid money
to form the capital are called share holders. Now -a- days the shares are
not in the form of paper, but in the electronic dematerialised (De mat)
form, hence the allotment of shares is do ne directly in the demat
account, without a certificate.
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73 Face value or n ominal value or Par value is the value printed on the
share certificate. Since shares exist in electronic demat form, we can
say that the face value is the value stated in the I.P.O. subscription
form.
The shareholders enjoy the profits (if any) of the company, after
providing for the taxes and other reserve funds . This is called as
dividend .
Types of shares ; Mainly the shares are of two types i) Preference
shares and ii) Equity s hares or common shares or ordinary shares .
i) Preference shares : These shares have a priority over the equity
shares. From the profits made by a company , a dividend at a fixed rate
is paid to them first, before distributing any profit amount t o the equ ity
shareholders. Also , if and when the company is closed down then
while returning of the capital, these shareholders get a preference.
Again, preference shares are mainly of two types:
a) Cumulative Preference shares: In case of loss or inadequate
profit , The preference shareholders are not paid their fixed rate of
dividend , then the dividend is accumulated in the subsequent years
to these shareholders & is paid preferentially whenever possible .
b) Non-cumulative Preference shares: As in the case o f cumulative
preference shares, here the unpaid dividends do not accumulate.
ii) Equity shares : These are the shares for whom the dividend and
the return of capital is paid after paying the preference shareholders.
In case of equity shares , the rate of dividend is not fixed and it is
decided by the Board of Directors .
Share Market
Shareholders are allowed to buy or sell shares like commodities.
Selling or buying a share for a price higher than its face value is legal.
The share prices are allowed to be subject to the market forces of
demand and supply and thus the prices at which shares are traded can
be above or below the face value.
The place at which the shares are bought and sold is called a share
market or stock Exchange and the price at whi ch a share is traded is
called its Market Price (MP) or the Market value. If the market price of
a share is same as its face value, then the share is said to be traded at
Par.
If M.P. is greater tha n face value of a share , then the share is said to
be available at a premium or above par and is called premium share or
above par share.
If M.P. is lower th an face value of a share , then the share is said to be
available at a discount or below par & the share is called a discount
share or below par share .
The purchase and sale of shares can take place through brokerage firms
and depositary P articipant (DP). e.g. Sharekhan .com, Kotak Securities munotes.in
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74 ltd , ICICI direct .com etc. They charge a commission on the purchase
and sale of shares, which is called as a b rokerage. The brokerage is
charged as a percentage of the M.P. of the share. Normally it is below
1%.
5.1.1 Bonus Shares
Sometimes, when a company's free reserves are high, it may choose to
capitalize a part of it by converting it into shares. This is do ne b y
issuing bonus shares to existing shareholders. These bonus shares are
issued free of cost. The ratio of bonus shares to the existing shares is
fixed.
Getting bonus shares increases the number of shares of shareholders.
But since this applies to all the shareholders in a fixed ratio, hence the
percentage of a shareholder's ownership of the company remains same
as before.
Now, we will study some examples based on the above concepts :
Example 1
Mr. Prashant invested Rs. 75,375/ - to purchase equit y shar es of a
company at market price of Rs. 250 / - through a brokerage firm,
charging 0.5% brokerage. The face value of a share is Rs. 10/ -. How
many shares did Mr. Prashant purchase?
Solution : Brokerage per share = 250 x 0.5100 = 1.25
cost of purchasing one share = 250+1.25 =251.25
Number of shares purchased = 75375251 .25 = 300
Example 2
Mr. Sandeep received Rs. 4,30,272 / - after selling shares of a company
at market price of Rs. 720 / - through Sharekhan Ltd., with brokerage
0.4%. Find the number of shares he sold.
Solution : Brokerage per share = 720 x 0.4 = 2.88
100
selling price of a share = 720 - 2.88 = 717.12
Number of shares sold = 430272717.12= 600
Example 3
Ashus Beauty World ' has issued 60,000 shares of par value of
Rs. 10/ - each. The company declared a total dividend of Rs. 72,000 / - .
Find the rate of dividend paid by the company.
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75 Solution : Face value of 60,000 shares = 60,000 x 10 = 6,00,000
Rate of Dividend = Total DividendFace value of 60,000 sharex100
= 72000600000x 100 = 12
The rate of dividend paid by the company is 12%
Example 4
The capital of ABC Company consists of Rs. 15 lakh s in 6 %
cumulative preference shares of Rs. 100 each and Rs. 30 lakh s in
equity shares of Rs.10/ - each. The dividends on cumulative preference
shares for earlier year was not paid . This year , the company has t o
distribute profit of Rs . 3 lakh after keeping 20 % as reserve fund. Find
the perc entage rate of dividend paid to the equity shareholders.
Solution: Reserve fund = 20100x 300000 = Rs. 60,000/ -
Profit to be distrib uted = 3,00,000 - 60,000 = 2,40,000
Annual dividend for 6 % cumulative preference shar eholders
= 6100x 1500000 = 90,000
This needs to be paid for 2 years (last year & current year ) as the
preference shares are cumulative & last year's dividend was not paid .
Total Dividend paid to Preference shareholders
= 2 x 90000= 1,80,000/ -
Now , dividend to be distributed to the equity shareholders
= 2,40,000 - 1,80,000 = Rs. 60,000/ -
Rate of dividend = 60,00030,00,000x 100 = 2
The rate of dividend to the equity shareholders is 2 %
Example 5
Mr. Dinesh bought some shares of a company which had a face value
of Rs.100 / -. The company declared a dividend of 15 % but Mr.
Dinesh's rate of return on investment was only 12% . At what market
price did he purchase the shares ? There was no brokerage involved.
Solution:
Dividend on one share = Rate of Dividend100x face value of one share
= 15 x 100 = Rs. 15/ -
Rate of Return on investment = Dividend on one sharepurchase price of 1 sharex 100
12 = 15purchase of 1 share x 100
purchase price of 1 share = 1512x 100 = 125 munotes.in
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76 Example 6: Comparison of two stocks
Mr. Subu invested Rs. 20,000 / - in Rs. 100/ - shares of company A at
the rate of Rs. 125 /- per share . He received 10 % dividend on these
shares. Mr. Subu also invested Rs. 24,000/ - in Rs. 10/ - shares of
company B at Rs.12/ - per share and he received 15 % dividend. Which
investment is more ben eficial?
Solution : Company A
Rate of return =Dividend on one sharepurchase price of 1 share x 100 = 10125x 100 = 8%
125
Company B
Rate of return = 1.512x 100 = 12.5 %
Investment in company B is more profitable .
Example 7
Ms. Ashma Mehta bought 300 shares of a company of face value Rs.
100 / - each at a market price of Rs. 240 / - each . After receiving a
dividend at 8 % , she sold the shares at Rs . 256 / - each. Find her rate
of return on investment. There was no brokerage involved.
Solution : Difference in the market price = 256 -240= 16
Dividend on 1 share = Rate of dividend100x face value of 1 share
= 8100x 100= 8
Rate of Return on Investment
= (Price change) (Dividend on 1 share)purchase price of 1 share= 16 8240x 100
= 2400240= 10
The rate of return on investment was 10 % .
5.1.2 Splitting of shares:
Sometimes companies split the face value of a share & break it
up into smaller units . For e.g. a Rs. 100 / - share can be split into 10
shares each of face value Rs. 10 / - or a Rs. 10/ - share can be split into
two shares of face value Rs. 5/ - each . Usually this does not affect a
shareholder's wealth . However , it can make selling of a part of the
holdings easier.
Example 8
Mr. Joshi purchased 30 shares of Rs. 10/ - each of Medi computers Ltd.
on 20th Jan . 2007, at Rs. 36/ - per share. On 3rd April 2007, the
company decided t spli t their shares from the face value of Rs. 10/ - per
share to Rs. 2/ - per share. On 4th April 2007, the market value of each
share was Rs. 8/ - per share. Find Mr. Joshi's gain or loss, if he was to munotes.in
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77 sell the shares on 4th April 2007? (No brokerage was invol ved in the
transaction) .
Solution : On 20th Jan 2007
Purchase cost of 30 shares = 30 x 36 = 1080 /-
On 3rd April 2007, each Rs. 10/ - share became 5 shares of Rs. 2/ -
each.
No. of shares = 30 x 5 =150
On 4th April 2007, market value of 150 shares wa s @ Rs. 8 each
= 150 x 8 = 1200
His gain = 1200 -1080 = 120/ -
Example 9
Rahul purchased 500 shares of Rs. 100 of company A at Rs. 700 / -.
After 2 months, he received a di vidend of 25 % . After 6 months , he
also got one bonus share for every 4 shares held. After 5 months, he
sold all his shares at Rs. 610/ - each . The brokerage was 2% on both ,
purchases & sales. Find his percentage return on the investment.
Solution : For purchase:
Face value = Rs. 100 / - , No. of shares = 500, market price = Rs.700/ -
Dividend = 25 %, brokerage = 2%
Purchase price of one share = 700 + 2100x700= 714
Total purchase = 500 x 714 = Rs.3,57,000/ -
Dividend = 25100of 100 i.e. Rs. 25 / - per share
Total dividend = 500 x25 = Rs. 12,500 / -
Now, bonus shares are 1 for every 4 shares .
No. of bonus shares = 14x 500 = 125
Total No. of shares = 50 0 +125 =625
For sales,
No. of shares = 625, market price = 610 , Brokerage 2%
Sale price of one share = 610 - 2% of 610 = 597.8
Total sale value = sale price of one share x No. of shares
= 597.8 x 625 = Rs. 3, 73,625/ -
Net profit = sale value + Dividend - purchase value
= 3,73,625 + 12500 - 3,57,000
= Rs . 29,125/ - .
% gain = 29,1253,57,000x 100 = 8.16
= 8.16
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78 EXERCISE :
1) Mr. Am ar invested Rs 1,20,480/ - to buy equity shares of a company
at market price of Rs . 480 / - at 0.4 % brokerage. Find the N o. of
shares he purchased.
Ans: 250
2) Aditi invested Rs. 19,890 / - to purchase shares of a company with
face value of Rs. 10/- each , at market price of Rs. 130/ - . She received
dividend of 20 % as well Afterwards , she sold these shares at market
price of Rs. 180/ - . She had to pay brokerage of 2 % for both purchase
and sales of shares. Find her net profit.
Ans: No. of shares =150, sales = 26460 , Divi dend = 300 ,
purchase = 19,890, profit= 6870
3) Amol wants to invest some amount in company A or company B ,
by purchasing equity shares of face value of Rs. 10 / - each , with
market price of R. 360/ - and Rs. 470/ - respectively . The companies
are expected to declare dividends at 20 % and 45% respectively .
Advise him on the choice of sh ares of company.
Ans: company B is a better choice .
4) Find the percentage gain or loss if 200 shares of face value Rs. 10/ -
were purchased at Rs . 350/ - each and sold later at Rs. 352 / - , the
brokerage being 0.5 % on each of the transaction .
Ans: -0.43 % i.e. a loss of 43 %
5) Find the number of shares if the total dividend at 8% on the shares
with face value Rs.10/ - was Rs. 240. Ans :- 300
5.2 MUTUAL FUNDS
In the previous unit shares, we have studied how one can transact in
share s. Now, we will study what are the mutual funds and how they
function.
An investor can invest money directly in shares or he can invest his
money through mutual funds . Mutual funds are managed by large
financial se rvices with a professional team of fund M anagers &
research experts.
Mutual fund is a pool of money , drawn from investors .The amount
collected is invested in dif ferent portfolios of securities , by the fund
managers and the profits (returns), proportional to the investment , are
passed back to th e investors.
At a given time, the total value is divided by the total number of units
to get the value of a single unit a given time . This is called Net Asset
Value (NAV).
NAV = Net Assets of the scheme
Total No. of units outstanding munotes.in
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79 or NAV = Total Assets - liabilitie s____
Total No. of units outstanding
There are mixed or hybrid funds which invest in both de bt and equity.
The offer document s give the guidelines / constraints under which the
fund managers would operate. e.g. investment in equity 80 % to
100 %, investment in money markets 0 % to 20 % etc.
In India , the mutual funds are gover ned by SEBI ( Securities and
Exchange Board of India ) .There are different companies , called the '
Fund Houses ' (like SBI or Reliance or HDFC) which float different
mutua l funds. Each such fund is called a 'scheme', e .g. HDFC has a
scheme ' HDFC Tax s aver ' etc.
Like IPO of a company's share , a mutual fund scheme starts by
having a N.F.O. (New Fund Offer) . Investors can invest by
purchasing Units of the mutual funds .Usually a unit is of Rs. 10/ - . A
share is the smallest unit of a company's capital , whereas in mutual
funds , even a fraction of a unit can be purchased after the N.F. O .
Let us study the following example to understand this concept :
Example 10
A mu tual fund 's scheme shows the following on 01/01/2007
Total v alue of securities
(Equity , Bonds etc.) Rs. 1500 crores
Cash Rs . 100 crores
Liabilities Rs . 200 crores
Total No. of units outstanding Rs. 100 crores
NAV = Rs. 1500 crores + Rs. 100 cro res - Rs. 200 crores
100 crores
= Rs. 1400 crores = Rs . 14 per unit .
100 crores
The NAV of a mutu al Fund scheme is calculated and disclosed to the
publc for evey work ing day . The NAV changes daily. Investors try to
invest when NAV is low and sell the units and get profits when the
NAV is high .
Most mutual fund schemes are not traded at stock marke t. Thus,
investor purchases as well as sells t he units to AMC i.e. Asset
management company, This sale is called redemption of units.
Basically funds are of two types : -
1) close ended funds 2) open ended funds .
1) Close ended mutual funds : - These are offered with a fixed date of
maturity and can be purchased from mutual fund companies during a
specific period . The investor can get the amount after expiry date of
the fund . If an investor wants to exit before the maturity date , he can munotes.in
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80 sell the units on the stock exchange at a discount or through a buy -
back option by the fund .
2) Open ended funds : These have no fixed date of maturity and the
units can be sold or repurchased at any time .The no. of units & its
capital changes daily .
Entry load & Exit load : Some mutua l fund schemes collect a charge
when investors purchase or redeem units . These are usually
percentage of NAV . The charge levied while purchasing a unit is
called the entry load & the charge collected on redemption is called
exit load .
Usually , the de bt funds have not loads . When there are no charges
while purchasing or selling of units , these funds are called No Load
Funds .
Mutual Funds can be broadly categorised into two types : 'Dividend '
funds which offer a dividend and 'Growth ' funds whic h do not offer a
dividend .
In mutual unds , the dividend given has no direct relation to the profit
earned . The mutual fund invests the money in different shares that
may or may not give a dividend at different times & different rates .
The fund ma nager may at any arbitrary point , decide to give a part of
the unit s' value back to the investors . This is called dividend .
For a growth fund , the NAV does not com e down due to dividends . It
moves up or down purely on the basis of the gains or loss es of the
securities that the fund has invested in.
For a growth fund , the gains per unit are purely from the difference
between the redemption price and the purchase price i.e. the total gain
is purely the capital gain . For a dividend fund , the to tal gain is the
addition of the capital gain & the dividend .
Capital gain = Amount received aft er redemption - Amount invested .
Rate of Return = C hange in NAV + Dividend x 100
NAV at the beginning of the period
(This is for a given period) .
Annualised rate of Return = Rate of Return x 365
n
where n is the number of days .
Some important Te rms :
i) Assets : - It refers to market value of investment of M.F. in
government securities , bonds etc. , its receivable s , accrued income &
other assets .
munotes.in
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81 ii) liabilities : - It includes all expenses like accrued expenses , payables
and other liabili ties for the M.F. scheme .
iii) Net Assets : - Total Assets - liabilities
iv) The valuation Date is the date on which NAV is calculated .
Example 11
Mr. Deore invested Rs. 25,000/ - to purchase 2,500 uits of ICICI MF -
B plan on 4th April 2007 . He decided to sell the units on 14th Nov.
2007 at NAV of Rs. 16.4 / -. The exit load was 2.5 % . Find his profit
(Calculations are upto 2 decimal points )
Solution :
No. of units =2500 , purchase cost = Rs. 25,000/ -
NAV on the date of sale = RS. 16. 4/- , exit load =2.5%= of 16.4 =
0.41
selling price of 1 unit = 16.4 - 0.41 = 15.99
sale value = 2500 x 15.99
= Rs. 39,975/ -
Profit = 39,975 - 25 ,000
= Rs. 14,975 .
Example 12
Ragini invested Rs. 94,070/ - in mutual Fund when NAV was Rs. 460 / -
with entry load of 2.25 % . She received a dividend of Rs. 5/ - per unit .
She, later sold all units of fund with an exit lo ad of 0.5 % . If her gain
was Rs. 1654/ -, find NAV at which she sold the units .
(Calculations are upto 2 decimal points)
Solution : purchase price of one unit = 460 + 2.25% of 460
= 460 +10.35 = 470.35
No. of units purchased = 94,070 = 200
470.35
Total dividend = 200 x5 = 1000
Ga in = Profit + Dividend
1654 = Profit + 1000
Profit = 1654 - 1000= 654
While selling l et NAV of one unit be y
sale price of one unit = NAV - exit load
= y - 0.5% of y
= 0.995 y
sale price of 200 units = 200 x 0.995 y= 199 y
Also , profit = Total sale - Total purchase
654 = 199y - 94,070 munotes.in
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82 199y = 654 + 94,070
199y = 94724
y= 476
NAV at which she sold units = Rs. 476/ - .
Example 13
If a mutual fund had NAV of Rs. 28 / - at the beginning of the year and
Rs. 38/ - at the end of the year , find the absolute change and the
percentage change in NAV during the year .
Solution : NAV at the beginning = Rs. 28/ -
NAV at the end = Rs. 38/ -
Absolute change in NAV = in 38 -28 = Rs. 10/ -
% change = Absolute change x 100 = 10 x100 = 35.71 %
NAV at the beginning 28
Example 14
If NAV was Rs. 72/ - at the end of the year , with 12.5 % increase
during the year , find NAV at the beginning of the year .
Solution : Let 'x' be the NAV at the beginning of the year .
Absolute change in NAV = 12.5 % of x = 12.5 x x = 0.125 x
100
NAV at the end of the year = x + 0.125 x = 1.125 x
1.125 x = 72
x = 72
1.125
= 64
NAV's initial value was Rs. 64 / - .
Example 15
Rohit purchased some units in open en d equity fund at Rs. 16/ - . The
fund distributed interim dividend of Rs. 5/ - per unit , and the NAV of
the fund at the end of the year was Rs. 25/ - . Find the total percentage
return . (Calculations are upto 2 decimal points)
Solution : Total gain = chang e in NAV + Dividend
= (25 -16) + 5
= 9+5
= 14
Total percentage gain = Total gain x 100
NAV at the beginning
= 14 x 100 = 87.5 %
16
Example 16
Mr. Hosur purchased some units in open - end fund at Rs. 30/ - and its
NAV a fter 18 months was Rs. 45/ - . Find the annualised change in
NAV as a percentage .
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83 Solution : change in NAV for 18 months = 45 -30 = Rs. 15/ -
annualised change = change in NAV x 12 x 100
NAV at beginning No. of months
= 15 x 12 x 100
30 18
= 33.33 %
Check your progress :
1) Mr. Kamble purchased 5 86.909 units of Kotak cash plus retail
Growth on 1st June 2007 when the NAV was RS. 20.4461. Its NAV as
on 3rd Dec ember, 2007 was Rs. 21.1960/ - . The fund has neither entry
load nor an exit load. Find the amount invested on 1st June 2007 and
the value of Mr. Kamble's investment on 3rd December 2007 .
Ans . 12,000 , 12440.12 .
2) Ms . Kannan purchased 113.151 units of 'FT India Prima Plus' on
9th April 2007 and redeemed all the units on 7th Aug 2007 when the
NAV was Rs. 35.5573 . The entry load was 2.2 5 % and the exit load
was 1 % . If she gained Rs. 483.11 , find the NAV on 9th April 2007
. (Calculations are upto 2 decimal points)
Ans . 30.2514
3) Mr. Pandit invested Rs. 10,000/ - in Birla Sunlife Equity Fund -
Divjdend plan ' on 10/07/2007 , wh en the NAV was Rs. 78.04 ,and
redeemed all the units on 12/11/2007 when the NAV was Rs. 84.54 . In
the meanwhile , on 31/08/2007 , she had received a dividend @ Rs. 10
per unit . Find her total gain and the rat e of retu rn considering loads as
follows:
The entry load was 2.25 % and the exit load was 0.5 % The number of
units were calculated correct upto 3 decimal places.
Ans . Total gain = Rs . 1794.46 , Rate of return = 17.94%
4) Given the following information , calculate NAV o f the mutual
fund : -
No. of units =15000
Market value of investments in Govt . securities = Rs. 20 lakhs
Market value of investments in corporate Bonds = Rs. 25 lakhs
Other Assets of the fund = Rs. 15 lakhs
Liabilities of the fund = 6 lakhs
Ans . Rs. 360/ - .
5) Mumtaz purchased 1200 units of TATA BIG Bond - G Rs. 12,000
/- on 14th April 2007 . She sold her units on 9th Dec 2007 at NAV of
Rs. 15.36/ - . The short term gain tax (STGT) was 10% of the profit .
Find her net profit . (Calculations upt o 2 decimal points )
Ans . profit = 6432 , STGT = 643.2 , Net profit = 5788.8
(profit - STGT) .
5.3 SYSTEMATIC INVESTMENT PLAN (SIP)
In SIP an investor invests a fixed amount (e.g. say 1000/ -) every
month on a fixed date (e.g. 1st of every month ) . In general the munotes.in
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84 minimum amount is Rs. 1000/ - per month , in diversified equity
schemes . It can be even Rs. 500/ - as well in ELSS schemes . If this is
done for many months , then each time units are p urchased at a
different NAV . Over a period of few months, an investor gets the
benefit of a phenomenon called 'Rupee cost Averaging' .
Rupee -cost- averaging : - If NAV increases , the no. of units
decreases & if NAV decreases , the no. of units purchas ed increases .
Thus on the whole , it lowers the average cost of units because
indirectly ,the investor buys more units when NAV prices are low &
he buys less units , when NAV prices are high . It is called Rupee -cost-
Averaging .
Consider the following example : -
Mr. Shaikh keeps Rs. 5000/ - on 3rd of every month for 4 months as
follows : -(Calculations are correct to 2 points of decimal)
Month Amount (in Rs.) NAV No. of units he gets
1 5000 109.48 5000/109.48=45.67
2 5000 112.36 5000/112.36=44.50
3 5000 108.14 5000/108.14 =46.24
4 5000 105.62 5000/105.62=47.34
Total 20,000 183.75
Avg price of units = 20,000 / 183.75 = 108.84
If Mr. Shaikh would have invested the entire amou nt of Rs. 20,000/ -
0n 3rd of first month only , with NAV Rs. 109.48/ - , the no. of units
purchased would have been 20,000/ 109 .48 = 182.68
Thus he gained more units and average price of units also was
Rs.108.84 instead of Rs.109.48 which was NAV on 3rd of the first
month
If SIP is followed for a long period of time , it can create wealth to
meet a person's future needs like housing , higher education etc .
Now , we will study the following examples to understand SIP .
Example 17
Mr. Patil investe d in a SIP of a M.F. , a fixed sum of Rs. 10,000/ - on
5th of every month , for 4 months . The NAV on these dates were Rs.
34.26 , 46.12 , 39.34 and 41.85 . The entry load was 2.25 % through
out the period . Find the average price , including the entry loa d , using
the Rupee -cost-Averaging method .How does it compare with the
Arithmetic mean of the prices ? (Calculations are correct to 4 digits
decimal)
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85 Solution :
Month NAV Entry load =
2.25% Total price No.ofunits=100 0/
Total price
1 34.26 00 0.7708 35.0308 285.4627
2 46.12 00 1.0377 47.1577 212.0544
3 39.34 00 0.8851 40.2252 248.6006
4 41.85 00 0.9141 42.7916 233.6906
TOTAL 165.2053 979.8083
By using Rupee -cost-Averaging method : -
Avg Price = Total amount
Total No. of units
= 40,000 = 40 .8243
979.8083
A.M. of price = Total price = 165.2053 = 41.3013
4 4
Avg. price using Ru pee-cost- Averaging method is less than A.M.
of prices .
Example 18
Mr. Desai invested Rs. 5000/ - on 1st of every month for 5 months in a
SIP of a M.F. with NAV's as 48.15 , 52.83, 41.28, 35.44 & 32.65
respectively . There was no entry load charged. Find the average
price , Mr. Desai paid using the Rupee -cost-Averaging method . After 6
months ,he sold all his units , when NAV was Rs. 51.64 with contingent
deferre d sales charge (CDSC) as 2.25 % . Find his net gain.
(Calculations are correct to 2 di gits decimal)
Solution : consider the following table : -
Month Amount (in Rs.) NAV No. of units
1 5000 48.15 5000/48.15=103.84
2 5000 52.83 5000/52.83=94.64
3 5000 41.28 5000/41.28=121.12
4 5000 35.44 5000/35.44=141.08
5 5000 32.65 5000/32.65=153.14
TOTAL 25000 613.82
Avg . price of units = 25000 = 40.73
613.82 0
For selling :
selling price of one unit = 51.64 - 2.25% of 51.64 = 50.48
Total sales = 50.48 x 613.82= 30,991.77
Net gain = 30,991.77 -25,000 = Rs. 5991.77 .
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86 Check your progress :
1) Mr. Thomas started a SIP in 'HDFC long term advantage Fund ' . On
the 10th July , Aug and Sept 2007 he invested Rs. 1000/ - each at the
NAVs Rs. 44.100 , Rs. 43.761/ -, s. 45.455 respectively . The entry load
was 2.25% . Find h is average acquisition cost per unit upto 3 decimal
places . (Calculations are up to 3 decimal points) .
Ans. Rs. 45.427/ - .
2) Maneesha d Rs. 20,000/ - on 2nd of every month for 5 onths in a
SIP of a mutual fund , with NAVs as Rs. 53.12 , Rs. 56.26 , Rs. 48.86
,Rs.50.44 and Rs. 54.62 respectively . The entry load was 2.25 %
throughout this period .Find average price , including the entry load ,
using the Rupee -cost -Averaging method and compare it with
Arithmetic mean of prices .
(Calculate up to 2 decimal points )
Ans . 53.70 , 53.84 .
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