## Chapter 6: PERCENTAGE

### Introduction

Percentage Calculations

• Suppose you have a product of two variables say 10 × 10.

• If the first variable changes to 11 and the second variable changes to 12, what will be the percentage change in the product? [Note there is a 10% increase in one part of the product and a 20% increase in the other part.]

• The formula given for this situation goes as: (a + b + ab/100)

• Hence, Required % change = 10 + 20 + $$\frac{10 * 20}{100}$$

• If two quantities A and B are given then saying A is increase by X% to get B is same as saying B is reduced by Y% to get A. This relation is given in below table.

A is increased by B is reduced by
10 9.09
20 16.66
25 20
33.33 25
50 33.33
60 37.5
75 42.85
100 50

If the price of a commodity increases by R%, then the reduction in consumption so  as not to increase the expenditure is

 New consumption = [R / ( 100 + R )) * 100] %

If the price of the commodity decreases by R%,then the increase in consumption so as to decrease the expenditure is

 New consumption = [ ( R / ( 100 - R ) * 100] %

Results on Population : Let the population of the town be P now and suppose it increases at the rate of
R% per annum, then :

 Population after n years = P * [1+(R/100)]^n Population n years ago = P / [1+(R/100)]^n

Results on Depreciation :  Let the present value of a machine be P. Suppose  it depreciates  at the rate
R% per annum.

 Value of the machine after n years = P * [1-(R/100)]^n Value of the machine n years ago = P / [1-(R/100)]^n

Ans .

700% of 9

1. Explanation :

It can be clearly seen that 700% of 9 = 63 is the highest number.

Ans .

135

1. Explanation :

0.25 N = 75 so N = 300. Thus, 0.45 × 300 = 135

Ans .

5.25

1. Explanation :

20% of 50% of 75% of 70 = 20/100 × 50/100 × 75/100 × 70 = 0.2 × 0.5 × 0.75 × 70 = 5.25.

Ans .

7/17

1. Explanation :

41 (3/17)% = 700/17%. As a fraction, the value = 700/(17 × 100) = 7/17

Ans .

28.56

1. Explanation :

New consumption = [R / ( 100 + R )) * 100] % were R = 40.

Ans .

25

1. Explanation :

100 -> 140 (after a 40% increase) -> 105. The reduction from 140 to 105 is 25% and hence, it means that he needs to reduce his consumption by 25%

Ans .

40

1. Explanation :

Assume Ram as 100. Shyam will be 133.33 and Bram will be 80
Thus, Bram’s goods are 40% cheaper than Shyam’s (53.33/133.33)

Ans .

1575

1. Explanation :

Total votes = 6000. Valid votes = 75% of 6000 = 4500. Bhiku gets 65% of 4500 votes and Mhatre gets 35% of 4500. Hence, Mhatre gets: 0.35 × 4500 = 1575 votes

Ans .

20

1. Explanation :

If the candidate has inadvertently increased his height by 25% the correction he would need to
make to go back to his original height would be to reduce the stated height by 20%

Ans .

104.34

1. Explanation :

Let Arjit’s height be H. Then, H × 1.15 = 120 so H = 120/1.15 = 104.34

Ans .

96

1. Explanation :

Let the number be N. Then, 5N should be the correct outcome. But instead the value got is 0.2N.
Change in value = 5N – 0.2N = 4.8N. The percentage change in the value = 4.8N × 100/5N =
96%.


Ans .

cant say

1. Explanation :

The percentage difference would be given by thinking of the percentage change between two
numbers: (x – 5) to (x + 5) [‘What he wanted to get’ to ‘what he got by mistake’].
The value of the percentage difference in this case depends on the value of x. Hence, this cannot
be answered. Option (d) is correct.

Ans .

20

1. Explanation :

From the first statement we get that out of 80 litres of the mixture, 20 litres must be milk. Since,
we are adding water to this and keeping the milk constant, it is quite evident that 20 litres of milk
should correspond to 20% of the total mixture. Thus, the amount in the total mixture must be 100,
which means we need to add 20 litres of water to make 100 litres of the mixture

Ans .

none

1. Explanation :

earlier if it was 10 * 10 = 100 now it was 11 * 12 = 132. so answer is 32. which is (d)

Ans .

11.42

1. Explanation :

area of triangle is 0.5* b * h. Assuming both b and h are 10. we get area of 50. Now h becomes 14 and area can be 80. so base can be around 11.42

Ans .

98

1. Explanation :

cuboid = l*b*h. assume all are 10 initially we get area as 1000. now increase them to 11*12*15 we get 1980. so an increase 98%

Ans .

23

1. Explanation :

assume amit salary is 130 and varuns is 100. so 100 becomes 23% of 130.

Ans .

60

1. Explanation :

assume earlier sugar cost 10 per kg and now it cost 7.5 but aayush buys sugar of rs 12. which means he spends 4.5 more so 4.5/12 is 60%,

Ans .

25

1. Explanation :

If the price of rice has fallen by 20% the quantity would be increased by 25% (if we keep the expenditure constant.)
This means that 20 kgs would increase by 25% to 25 kgs.

Ans .

1500

1. Explanation :

The winning candidate gets 56% of the votes cast and the losing candidate gets 44% of the votes
cast. Thus, the gap between the two is 12% of the votes cast = 144 votes. Thus, the votes cast =
1200. Since, this is 80% of the number of voters on the voting list, the number of people on the
voting list = 1200/0.8 = 1500.


Ans .

1,21,000

1. Explanation :

100000 increased by 10% is 110000 increased by 10% is 121000

Ans .

12,540

1. Explanation :

10000 becomes 11000 (after a 10% increase) becomes 10450 (after a 5% decrease) becomes 12540 (after a 20% increase)

Ans .

1950

1. Explanation :

His investments are 2000, 3000 and 5000 respectively. His dividends are: 200, 750 and 1000, which means total dividend = 1950.

Ans .



1. Explanation :

30% students got a final score of 13. 10% students got a final score of 33 (inclusive of grace marks.) 35% students got a final score of 60. Hence, average score of the class is 37.6 as (0.35*60 + 0.1 *33 + 0.3*13)/0.75 = 37.6

Ans .

6000

1. Explanation :

The only logic for this question is that Hans’ salary would be more than Bhaskar’s salary. Thus,
only option (a) is possible for Hans’ salary.


Ans .

2500

1. Explanation :

By using options, you can easily see that option (a) satisfies.
2500 females means 3000 males.
Increase = 2500 × 0.2 + 3000 × 0.11 = 830

Ans .

96

1. Explanation :

If Ashu’s salary =100, then Vicky’s salary = 175. Ashu’s new salary = 125, Vicky’s new salary =
175 × 1.4 = 245. Percentage difference between Vicky’s salary and Ashu’ salary now = 120 ×
100/125 = 96%.

Ans .

250

1. Explanation :

Let the second row contain 100 books. Then, the first row would contain 125 books and the third
row would contain 75 books. The total number of books would be 100 + 125 + 75 = 300. But this
number is given as 600 which means that the total number of books would be double the assumed
value for each row. Thus, the first row would contain 125 × 2 = 250 books

Ans .

266.66

1. Explanation :

Since the only iron contained in the ore is 90% of 25%, the net iron percentage would be 22.5%.
Thus, 60 kg should be 22.5% of the ore so 60/0.225 = 266.66

Ans .

cant say

1. Explanation :

The data is insufficient since the number of matches to be played by India this year is not given.
(You cannot assume that they will play 40 matches.)

Ans .

79

1. Explanation :

Total people present = 700 + 500 + 800 = 2000.
Indians = 0.2 × 700 + 0.4 × 500 + 0.1 × 800 = 420 = 21% of the population. Thus, 79% of the
people were not Indians

Ans .

800/x

1. Explanation :

1 Bottle gives 0.5x metres
x Bottles gives 400 meters
Using unitary method, we get no. of bottles = 400/0.5x = 800/x Bottles

Ans .

3212.5

1. Explanation :

The total cost for a year = 2,50,000 + 2% of 2,50,000 + 2000
= 2,55,000 + 2000 = 2,57,000
To get a return of 15% he must earn: 2,57,000 × 0.15 = 38,550 in twelve months.
Hence, the monthly rent should be 38550/12 = 3212.5.


Ans .

second shirt, 14.28%

1. Explanation :

The sales price of the first shirt is 8/7 × 42 =  48.
Hence, I am being offered a discount of  6 on a price of  48 – a 12.5% discount.
The sales price of the second shirt is 7/6 × 36 =  42.
Hence, I am being offered a discount of  6 on  42 – a 14.28% discount. Hence, the second shirt is
a better bargain.

Ans .

112.5

1. Explanation :

The sum says assume initial sales as 100. 100 with sales increase of 50% gives 150. What is the price drop percentage to get 82.5 from 150. Hence, the required price drop is 67.5/150 = 45% drop. Thus there is a drop of 250 × .45 = 112.5

Ans .

35

1. Explanation :

In 2001, BMW = 15%, Maruti = 50% and hence Honda = 35%

Q. If the sales tax  reduced from  3 1/2 % to  3 1/3%, then what difference does it make to a person who purchases an article with market price of Rs. 8400 ?

A. difference in final rate  = (3 1/2% rate on Rs 8400 ) - (3 1/3 % rate on Rs. 8400)

= (3 1/2 - 3 1/3)% of Rs 8400

= (1/6)% of Rs 8400

Q. In an election between two candidates, 75% of the voters cast their votes, out of which 2% of the votes were declared invalid. A candidate got 9261 votes which were 75% of the total valid votes. Find the total number of votes enrolled in that election.

we know 9261 = (3/4) * (98/100)*(3x/4) = 16800

Q. Reema's test had 75 problems i.e. 10 arithmetic, 30 algebra and 35 geometry problems. Although she answered 70% of the arithmetic , 40% of the algebra, and 60% of the geometry problems correctly.  she did not pass the test because she got less than 60% of the problems right. How many more questions she would have to answer correctly to earn 60% of the passing grade?

A. Reema's correct answers = 7 arithmetic, 12 algebra, 21 geometry.

Total correct = 7+12+21 = 40

total = 75, passing = 45. so she needs =5.

Q. Mr.J gave 40% of the money he had to his wife. he also gave 20% of the
remaining amount to his 3 sons. half of the amount now left was spent on
miscellaneous items and the remaining amount of Rs.12000 was deposited in the bank. how much money did Mr.J have initially?

A. let 'x' be the initial amount. wife gets = 2x/5, remaining amount = 3x/5; sons get = (1/5)*(3x/5) = 3x/25; amount now left = 12x/25;

amount spent on miscellaneous = 6x/25.

remaining amount = 6x/25 which is 12000 so x = 12000*25 / 6 = 50000

Q. Ramans  salary  was  decreased  by  50%  and  subsequently  increased  by 50%. How much percent does he lose?

A. Raman's salary be Rs. 100 [assume]. Then it becomes Rs. 50 as its reduced by 50%. Then its increased by 50% so becomes Rs. 75 so overall decrease is Rs. 25 so 25%.

Q. Paul  spends 75% of his  income. His  income  is  increased by 20% and he increased his expenditure by 10%. Find the percentage increase in his savings

A. Assume Paul earn's Rs. 100. he spends Rs. 75. so he saves Rs. 25 Now he earns Rs. 120 and he spends Rs. 82.5 so new savings is Rs. 37.5.

increasing in savings = 12.5 which is 50%.

Q. How many kg of pure salt must be added to 30kg of 2% solution of salt and water to increase it to 10% solution ?

A. salt in 2% solution of 30 kg is 0.6 kg now we want to find 'x' which will satisfy equation (0.6 + x) / (30 + x) = 1/10 [new ratio].

Q. In  the  new  budget  ,  the  price  of  kerosene  oil  rose  by  25%.  By  how much percent must a person reduce his consumption so that his expenditure  on it does not increase ?

A. Assume old price is 100 and new price becomes 125 so we must find how much is reduction in percentage is needed to get 100 from 125.

so we calculate how much % is 25 i.e. (125-100) of 125 it is 20% so a 20% reduction is needed.

OR

Directly apply short cut formula (see above).

Q. The population  of  a  town  is  1,76,400  .  If  it  increases  at  the  rate  of  5% per annum , what will be its population 2 years hence ? What was it 2 years ago ?

A. Formula: Population n years after= P * [1+(R/100)]^n

Population n years ago   = P / [1+(R/100)]^n

Ans .

46.15 

1. Explanation :

Let B = 100. Then A = 70, which is 30% more than C. Hence C = 23.07% less than A  = approx. 53.84. Hence answer is 46.15% approximately.
(This calculation can be done mentally if you are able to work through the calculations by the use of
percentage rule. The students are advised to try to assume the value of 100 for each of the variables A, B
and C and see what happens to the calculations involved in the problem. Since the value of 100 is
assumed for a variable to minimise the requirements of calculations to solve the problems, we should
ensure that the variable assumed as 100 should have the maximum calculations associated with it.)

Ans .

8 

1. Explanation :

The area of a rectangle is given by: length × breadth. If we represent these by:
Area = L × B = LB so then we will get the changed area as
Area(NEW) = 1.2 L × 0.9 B = 1.08 LB
Hence, the change in area is 8% increase

Ans .

 

1. Explanation :

Since price is rising by 25%, consumption has to decrease by 20%. But there is an actual
reduction in the consumption by 20 kg. Thus, 20% decrease in consumption is equal to a 20 kg drop in
consumption.
Hence, original consumption is: 100 kg of rice.
Money spent being  400, the original price of rice is  4 per kg

Ans .

 

1. Explanation :

Here I would take C as 100, getting B as 85 and A as 68.
Hence, the answer is (32 × 100/68).


Ans .

2.22 

1. Explanation :

Assume the cost components to be valued at 30, 40, 50 and 60 as you read the question. Then we
can get changed costs by effecting the appropriate changes in each of the four components.
Thus we get the new cost as 33, 32, 35 and 84 respectively.
The original total cost was 180 the new one is 184. The percent change is 4/180 = 2.22%.

Ans .

20 

1. Explanation :

He is left with  2.5 lakh after spending  7.5 lakh on the apartment.
Therefore, before the apartment purchase he has  10 lakh. But this is after the 20% reduction in his net
value due to his donation to charity. Hence, he must have given  2.5 lakh to charity (20% decrease
corresponds to a 25% increase). As such, he had 12.5 lakh before the charity. Further, he must have had 
13.5 lakh before the taxi driver stole the sum. From 13.5 lakh you can reach the answer by trial and error
trying whole number values. You will get that if he had 20 lakh and lost 32.5% of it he would be left with
the required 13.5 lakh.
Hence, the answer is  20 lakh.`

### Quiz

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