# Concept of Base Change

Question types covered:

Percent Change Reversal – If x is increased/decreased to y, then by what percent should y be decreased/increased to take it back to the original value?

If B is p% more/less than A, then A is how much percent less/more than B?

If z = x × y, and x is increased/decreased by p%, then by what percentage y should be decreased/increased to maintain the same value of z?

## Traditional method

Using the variables.

###### Q. If salary of Mragank is 20% more than the salary of Aanya, then the salary of Aanya is how much percent less than the salary of Mragank?

Explanation:

Let Salary of Aanya be x. Mragank is 20% more than the salary of Aanya.

So, Salary of Mragank = x + (20/100) × x = 1.2 x

Difference is salaries remain the same, i.e. 0.2x.

Required percentage = (0.2x/1.2x) × 100 = (1/6) × 100 = 16.67%

## Formula Method

If a value x is increased by p%, then to get back to the original x, we have to decrease the increased value by $\frac{p}{100+p}$ × 100%.OR

If y is p% more than x, then x is $\frac{p}{100+p}$ × 100% less than y.

If a value x is decreased by p%, then to get back to the original x, we have to increase the decreased value by $\frac{p}{100-p}$ × 100%.

OR

If y is p% less than x, then x is $\frac{p}{100-p}$ × 100% more than y.

###### Q. If salary of Mragank is 20% more than the salary of Aanya, then the salary of Aanya is how much percent less than the salary of Mragank?

Explanation:

If y is p% more than x, then x is $\frac{p}{100+p}$ × 100 % less than y.

So, Required percentage = (20/120) × 100 = (1/6) × 100 = 16.67%

## Percentage Method

###### Q. If salary of Mragank is 20% more than the salary of Aanya, then the salary of Aanya is how much percent less than the salary of Mragank?

Explanation:

Salary of Mragank is 20% more than the salary of Aanya. - Base is the salary of Aanya.

If Aanya’s salary = 100, then Mragank’s salary = 120 Salary of Aanya is how much percent less than the salary of Mragank. - Now, Base is the salary of Mragank.

Difference is salaries remain the same, i.e. 20. But now base will be 120.

Required percentage = (20/120) × 100 = (1/6) × 100 = 16.67%

## Fraction Method

Increase: In terms of fraction, if a value x is first increased by p% (= n/d) then to get back to the original number x, we have to decrease the increased value by $\frac{d}{d+n}$. (where d is denominator and n is numerator)

Decrease: In terms of fraction, if a value x is first decreased by p% (= n/d) then to get back to the original number x, we have to increase the increased value by $\frac{d}{d-n}$. (where d is denominator and n is numerator)

For example, a number is increased by 25%
25% = 𝒏/𝒅 = 𝟏/𝟒 - It implies an increase of 1 on 4

So, the original number ≡ 4 (i.e. d)

And new number ≡ 4 + 1 = 5 (i.e. d + n)

Percent change required in bringing down 5 to 4 again = 1/5 ≡ 20%

On the other hand, if a number is decreased by 25%
25% = 𝒏/𝒅 = 𝟏/𝟒 - It implies an decrease of 1 on 4

So, the original number ≡ 4 (i.e. d)

And new number ≡ 4 - 1 = 3 (i.e. d - n)

Percent change required in bringing up 3 to 4 again = 1/3 ≡ 33.33%

Explanation:

20% is equivalent to 1/5

If Aanya’s salary = 5, then Mragank’s salary = 6 Difference is salaries remain the same, i.e. 1. But now base will be 6.

Required percentage = 1/6 × 100 = 16.67%

###### Q. If salary of Mragank is 20% less than the salary of Aanya, then the salary of Aanya is how much percent more than the salary of Mragank?

Explanations :

If y is p% less than x, then x is $\frac{p}{100−p}$ × 100% more than y.

So, Required percentage = (20/80) × 100 = (1/4) × 100 = 25%

Salary of Mragank is 20% less than the salary of Aanya. - Base is the salary of Aanya.

If Aanya’s salary = 100, then Mragank’s salary = 80 Salary of Aanya is how much percent more than the salary of Mragank. - Now, Base is the salary of Mragank.

Difference is salaries remain the same, i.e. 20. But now base will be 80.

Required percentage = (20/80) × 100 = 1/4 × 100 = 25%

20% is equivalent to 1/5

If Aanya’s salary = 5, then Mragank’s salary = 4 Difference is salaries remain the same, i.e. 1. But now base will be 4.

Required percentage = (1/4) × 100 = 25%

## Percentage Method or Fraction Method?

Should we use the Percentage Method or Fraction Method?

Let us try to understand it using a couple of examples.

###### Q. On increasing a number by 37.5%, the new number obtained is 63 more than the original number. What must be the original number?

Explanations :

37.5% is equivalent to 63

So, 37.5% ≡ 63

Then, 100% ≡ 63 × 100/37.5 = 63 × 8/3 = 21 × 8 = 168

Complex calculation involved!

37.5% is equivalent to 3/8

If Original number = 8, then increase = 3

So, 3 ≡ 63

Then, 8 ≡ 63 × 8/3 = 21 × 8 = 168

Easier calculation as compared to percentage method.

###### Q. On increasing a number by 37.5%, the new number obtained is 231. What must be the original number?

Explanations :

137.5% (i.e. 100 + 37.5%) is equivalent to 231

So, 137.5% ≡ 231

Then, 100% ≡ 231 × 100/(100+37.5) = 231 × 100/137.5 = 231 × 8/11 = 21 × 8 = 168

Complex calculation involved!

37.5% is equivalent to 3/8

If Original number = 8, then new number = 8 + 3 = 11

So, 11 ≡ 231

Then, 8 ≡ 231 × 8/11 = 21 × 8 = 168

Easier calculation as compared to percentage method.

As you will keep on practicing you will develop a better sense of which method to use based on the kind of data given in the question.