Question

The fraction $\dfrac{x}{y}$ is altered by decreasing x by 25 percent and increasing y by 25 percent. The new fraction is what percent less than the original?
A. 35
B. 40
C. 42
D. 45

Answer 1

Hint: First we’ll find the $25\% $of x and y, then according to the given conditions we’ll form a new fraction. Now using this new fraction we will find that by what percentage this new fraction is lesser than the original fraction.

Complete step by step answer:

Given data: fraction$ = \dfrac{x}{y}$
$\dfrac{x}{y}$ is altered by decreasing x by 25 percent and increasing y by 25 percent
Now, $25\% $ of x$ = \dfrac{{25}}{{100}}x$
$ = \dfrac{x}{4}$
And, $25\% $ of y$ = \dfrac{{25}}{{100}}y$
$ = \dfrac{y}{4}$
Therefore numerator of the new fraction$ = x - \dfrac{x}{4}$
$ = \dfrac{{3x}}{4}$
And the denominator of the new fraction$ = y + \dfrac{y}{4}$
$ = \dfrac{{5y}}{4}$
Therefore the new fraction$ = \dfrac{{\dfrac{{3x}}{4}}}{{\dfrac{{5y}}{4}}}$
Multiplying numerator and denominator by 4
$ = \dfrac{{3x}}{{5y}}$
Now for calculating the percentage change we use \[percentage = \dfrac{{old{\text{ }}fraction - new{\text{ }}fraction}}{{old{\text{ }}fraction}} \times 100\],
Therefore the percentage of the amount the new fraction is lesser from the original fraction
$ = \dfrac{{\dfrac{x}{y} - \dfrac{{3x}}{{5y}}}}{{\dfrac{x}{y}}} \times 100$
Multiplying the numerator and the denominator by $\dfrac{y}{x}$
$ = \left( {1 - \dfrac{3}{5}} \right) \times 100$
On simplification we get,
$ = \dfrac{2}{5} \times 100$
$ = 40\% $
Hence, The new fraction is $40\% $ less than the original fraction.
Hence, Option(B) is correct.

Note: Remember always that during finding the percentage of any value that value is measured with-respect-to the original value. Similarly, in this case, when we find the percentage do not divide by the new fraction we have mentioned it measured with-respect-to original value i.e. the original fraction.