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⁞ Percentage

Aptitude ➤ Percentage ➤ 1

1. Concept of Percentage: By a certain percent, we mean that many hundredths.
Thus, x percent means x hundredths, written as x %.

To express x% as a fraction: We have, x % = x/100 .
Thus, 20 % = 20/100 = 1/5 .
To express a/b as a percent: We have, a/b = (a/b) x 100 %.
Thus, 1/4 = (1/4) x 100 % = 25 %.

2. Percentage Increase/Decrease:

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

[R/(100 + R)] x 100 %

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

[R/(100 - R)] x 100 %

3. Results on Population:
Let the population of a town be P now and suppose it increases at the rate of R % per annum, then:
1. Population after n years = P [1 + (R/100 )]n
2. Population n years ago = P / [1 + (R/100 )]n

4. Results on Depreciation:
Let the present value of a machine be P. Suppose it depreciates at the rate of R % per annum. Then:
1. Value of the machine after n years = P [1 - (R/100 )]n
2. Value of the machine n years ago = P / [1 - (R/100 )]n
3. If A is R % more than B, then B is less than A by [R/(100 + R)] x 100 %.
4. If A is R % less than B, then B is more than A by [R/(100 - R)] x 100 %.

Question 6
Q6.  The population of a certain town increased at a certain rate per cent per annum. Now it is 456976. Four years ago, it was 390625. What will it be 2 years hence ?
 

Answer :  494265


Suppose the population increases at r % per annum. Then,
390625 [1+(r/100)]4 = 456976
∴ [1+(r/100)]2 = √456976/390625 = 676/625

Therefore, Population 2 years hence = 456976 [1+(r/100)]2
= 456976 x (676/625) = 494265 approximately.
⚔ DISCUSS

Question 7
Q7.  A number is increased by 25 % and then it is decreased by 25 %. Find the net increase or decrease per cent.
 

Answer :  6.25 %


% change = (25)2/100 = 625/100 = 6.25 % decrease

♥ If the value is first increased by x % and then decreased by x % then there is only decrease which is equal to (x2/100)


⚔ DISCUSS
Question 8
Q8.  What percentage of numbers from 1 to 70 have 1 or 9 in the unit's digit ?
 

Answer :  20 %


Clearly, the numbers which have 1 or 9 in the unit's digit, have squares that end in the digit 1. Such numbers from 1 to 70 are 1, 9, 11, 19, 21, 29, 31, 39, 41, 49, 51, 59, 61, 69.
⇒ Number of such number = 14

∴ Required percentage = (14/70) x 100 % = 20 %.
⚔ DISCUSS

Question 9
Q9.  The price of a headphone is decreased by 10 % and 20 % in two successive years. What percent of price of a headphone is decreased after 2 years ?
 

Answer :  28 %


♥ If the value is decreased successively by x % and y % then the final decrease is given by

[-x-y+(xy/100)] %

Put x = - 10 and y = - 20, then
- 10 - 20 + (-10 x -20)/100 = - 30 + 2 = - 28 %

∴ The price of the headphone decreases by 28 %.

♥ If the value is increased successively by x % and y % then the final increase is given by

[x+y+(xy/100)] %
⚔ DISCUSS

Question 10
Q10.  In a certain school, 20% of students are below 8 years of age. The number of students above 8 years of age is 2/3 of the number of students of 8 years of age which is 48. What is the total number of students in the school ?
 

Answer :  100


Let the number of students be x. Then,
Number of students above 8 years of age = (100 - 20)% of x = 80% of x.

∴ 80% of x = 48 + (2/3) of 48
⇒ (80/100)x = 80
⇒ x = 100.
⚔ DISCUSS

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