Question

If a's income is $60\% $ more than that of b. By what percentage b's income is less than a.

Answer 1

Hint:Percentage- It is a number or ratio that can be expressed as a fraction of 100, which means, a part per hundred.The word percent means per 100.It is represented by the symbol
1. Percentage formula \[ = \left( {Value/Total{\text{ }}value} \right) \times 100\]
Like,
Three apples out of fifteen are rotten then.
Percentage of rotten apples \[ = \left( {3/15} \right) \times 100\]
\[ = 20\% \]
2. Percentage change
Increase in value $ = $ New Number - Original Number.
Now,
Divide the increase by the original number and multiply the answer by 100
$\% $ increase $ = $ Increase \[ \div \] Original Number $ \times $ 100.
If the answer is a negative number, which means the percentage change is a decrease
And if the answer is a positive number, which means the percentage change is an increase.

Complete step-by-step solution:
Let income of $b = x$
Then according to question
Income of $a = x + 60\% $ of x
$a = x + \dfrac{{60}}{{100}}x$
$a = x\left( {1 + \dfrac{3}{5}} \right)$
$a = x\left( {\dfrac{8}{5}} \right)$ or $\boxed{a = \dfrac{{8x}}{5}}$
So, we have income of $a = \dfrac{{8x}}{5}$ and $b = x$ we need to find by what percentage b’s income is less than a’s income.
Difference between their income, I
I $ = $ income of a – income of b
$ = \dfrac{{8x}}{5} - x$
$\boxed{I = \dfrac{{3x}}{5}}$
Percentage change in b’s income, $I\% $
$\Rightarrow I\% = \dfrac{I}{{Income\,of\,a}} \times 100$
$\Rightarrow I\% = \dfrac{{3x \times 5}}{{5 \times 8x}} \times 100$
$\Rightarrow I\% = \dfrac{{300}}{8} = 37.5\% $
So, b’s income is $37.5\% $ less than a’s income

Note:We can easily find the solution of the above question by using the following short method.
For that, Let income of b = 100 Rs ….(1)
Then, according to question
Income of $a = 100 + 60\% $ of 100
$a = 100 + \dfrac{{60}}{{100}} \times 100$
a $ = $ 160 Rs …..(2)
Difference in income $ = 60 - 100$ (From equation (1) & (2))
$ = $ 60 Rs
$\% $ of difference with respect to a’s income $ = \dfrac{{60}}{{160}} \times 100$
$ = 37.5\% $
So, b’s income is $37.5\% $ less than a’s income