Question

The income of A is $20\% $ higher than that of B. the income of B is $25\% $ less than that of C. What percent less is A’s income from C’s income?
A) $7\% $
B) $8\% $
C) $10\% $
D) $12.5\% $

Answer 1

Hint:
We can take the incomes of the 3 persons as three variables. Then we can form equations with the given relation between the salaries. Then we can find the relation between the salaries of A and C. Then we can see how less the income of A is from income of C. Then we can find the required percentage by converting the fraction.

Complete step by step solution:
Let income of A be a, income of B be b and income of C be c.
It is given that income of A is $20\% $ higher than that of B. So, the income A is income of B added with $20\% $ income of B. so we can write,
 $ \Rightarrow a = b + 20\% b$
We can write the percentage as fraction. So, we get,
 $ \Rightarrow a = b + \dfrac{{20}}{{100}}b$
On taking the LCM, we get,
 $ \Rightarrow a = \dfrac{{100 + 20}}{{100}}b$
So, we have,
 $ \Rightarrow a = \dfrac{{120}}{{100}}b$
On cancelling the zeros in the numerator and denominator, we get,
 $ \Rightarrow a = \dfrac{{12}}{{10}}b$ … (1)
It is given that income of B is $25\% $ less than that of C. So, the income B is income of C minus $25\% $ of the income of C. so we can write,
 $ \Rightarrow b = c - 25\% c$
We can write the percentage as fraction. So, we get,
 $ \Rightarrow b = c - \dfrac{{25}}{{100}}c$
On taking the LCM, we get,
 $ \Rightarrow b = \dfrac{{100 - 25}}{{100}}c$
So, we have,
 $ \Rightarrow b = \dfrac{{75}}{{100}}c$ … (2)
Now we can substitute equation (2) in (1).
 $ \Rightarrow a = \dfrac{{12}}{{10}} \times \dfrac{{75}}{{100}}c$
On simplification, we get,
 $ \Rightarrow a = \dfrac{{900}}{{1000}}c$
We can make the denominator to 100.
 $ \Rightarrow a = \dfrac{{90}}{{100}}c$
As we need to find how much percentage less is a from c, we can write the equation as,
 $ \Rightarrow a = \dfrac{{100 - 10}}{{100}}c$
 $ \Rightarrow a = c - \dfrac{{10}}{{100}}c$
Now we write the fraction as percentage,
 $ \Rightarrow a = c - 10\% c$
From the equation, we can say that a is $10\% $ less than c.
So, A’s income is $10\% $ less than C’s income.
Therefore, the required solution is $10\% $

So, the correct answer is option C.

Note:
Alternate solution to this problem is given by,
Let income of A be a, income of B be b and income of C be c.
It is given that income of A is $20\% $ higher than that of B. It will be $120\% $ of income of B.
 $ \Rightarrow a = 120\% b$ .. (a)
It is given that income of B is $25\% $ less than that of C. So, the income B is $75\% $ of income of B
 $ \Rightarrow b = 75\% c$ .. (b)
On substituting (b) in (a), we get,
  $ \Rightarrow a = 120\% \times 75\% c$
Now we can use one of the percentages as fraction.
 $ \Rightarrow a = \dfrac{{120 \times 75}}{{100}}\% c$
On multiplying the numerators, we get,
 $ \Rightarrow a = \dfrac{{9000}}{{100}}\% c$
On cancelling the zeros, we get,
 $ \Rightarrow a = 90\% c$
So, the income of A is $90\% $ of C.
So, A’s income is $10\% $ less than C’s income.
Therefore, the required solution is $10\% $