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# PERCENTAGE

**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 + 10∗20100

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} |

**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.

**A. **total votes be ‘x’. votes cast = 3x/4. invalid votes = (2/100)*(3x/4)

so valid votes are (98/100)*(3x/4)

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. **Raman`s 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

Q.

A sells his goods 30% cheaper than B and 30% dearer than C. By what percentage is the cost of C’s goods cheaper than B’s goods.

46.15

53.2

38.9

26.3

Ans .46.15

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.)

Q.

The length and the breadth of a rectangle are changed by +20% and by –10% respectively. What is the percentage change in the area of the rectangle.

75

5

10

8

Ans .8

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

Q.

Due to a 25% price hike in the price of rice, a person is able to purchase 20 kg less of rice for ` 400. Find the initial price.

3

5

4

9

Ans .4

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

Q.

A’s salary is 20% lower than B’s salary, which is 15% lower than C’s salary. By how much percent is C’s salary more than A’s salary?

48

60

30

45

Ans .48

Explanation :

Here I would take C as 100, getting B as 85 and A as 68.

Hence, the answer is (32 × 100/68).

Q.

The cost of manufacture of an article is made up of four components A, B, C and D which have a ratio of 3 : 4 : 5 : 6 respectively. If there are respective changes in the cost of +10%, –20%, –30% and +40%, then what would be the percentage change in the cost.

3

5

4

2.22

Ans .2.22

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%.

Q.

Harsh receives an inheritance of a certain amount from his grandfather. Of this he loses 32.5% in his effort to produce a film. From the balance, a taxi driver stole the sum of ` 1,00,000 that he used to keep in his pocket. Of the rest, he donated 20% to a charity. Further he purchases a flat in Ganga Apartment for ` 7.5 lakh. He then realises that he is left with only ` 2.5 lakh cash of his inheritance. What was the value of his inheritance?

22.5

25

20

30

Ans .20

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.

Q1: If the price of a commodity increases by 100%, then the reduction in consumption so as not to increase the expenditure is

50

100

75

25

ANS.50

Q2: If the price of the commodity decreases by 50%,then the increase in consumption so as to decrease the expenditure is

200

100

50

75

ANS.100

Q3: Let the population of the town be 100 now and suppose it increases at the rate of 10% per annum, then after 2 years it shall be:

100

120

121

115

ANS.121

Q4: Let the population of the town be 100 now and suppose it increases at the rate of 10% per annum, then 2 years ago it was:

82.6

80

78

76

ANS.82.6

Q5: the present value of a machine be 100. Suppose it depreciates at the rate 10% per annum. then after 2 years it shall be

80

81

78

75

ANS.81