Question

In a fraction, if the numerator is increased by 40% and the denominator is decreased by 20%, then by what percent is the new fraction more or less than the original fraction.
(a) 75% more
(b) 60% less
(c) 50% more
(d) 175% more

Answer 1

Hint:Assume the original fraction which has x as its numerator and y as its denominator. It is given that in a fraction the numerator is increased by 40% and the denominator is decreased by 20%. The numerator of the new fraction will be \[\left( x+\text{ }40\%\text{ }of\text{ }x \right)\] and denominator of the new fraction will be\[\left( y-\text{ }20\%\text{ }of\text{ }y \right)\] . Now, get the new fraction. Compare the original fraction and new fraction. If the new fraction is more than that of the original fraction then we can say that the new fraction is more than that of the original fraction. Now, get the difference between the new fraction and the original function. Now, calculate the percentage of the difference with respect to the original fraction.

Complete step-by-step answer:
According to the question, it is given that in a fraction the numerator is increased by 40% and the denominator is decreased by 20%.
Let us assume the original fraction which has x as its numerator and y as its denominator.
The original fraction is \[\dfrac{x}{y}\] …………………….(1)
It is given that the numerator of the original fraction is increased by 40%.
The numerator of the new fraction is (x+ 40% of x) …………………..(2)
Now, solving equation (2), we get
\[\begin{align}
  & x+40\%\,of\,x \\
 & =x+\dfrac{40}{100}x \\
 & =\dfrac{140x}{100} \\
\end{align}\]
\[=\dfrac{7x}{5}\] ……………….(3)
It is also given that the denominator of the original fraction is decreased by 20%.
The numerator of the new fraction is (y- 20% of y) …………………..(4)
Now, solving equation (4), we get
\[\begin{align}
  & y-20\%\,of\,y \\
 & =y-\dfrac{20}{100}y \\
 & =\dfrac{80y}{100} \\
\end{align}\]
\[=\dfrac{4y}{5}\] ……………….(5)
We have got a new fraction that has a numerator equal to \[\dfrac{7x}{5}\] and the denominator is \[\dfrac{4y}{5}\] .
Therefore, our new fraction is = \[\dfrac{\text{Numerator}}{\text{Denominator}}=\dfrac{\dfrac{7x}{5}}{\dfrac{4y}{5}}=\dfrac{7x}{4y}\] ………………(6)
From equation (1), we have the original fraction which is equal to \[\dfrac{x}{y}\] .
Now, comparing equation (1) and equation (6), we get
\[\begin{align}
  & \dfrac{x}{y}<\dfrac{7x}{4y} \\
 & \text{Original fraction New fraction } \\
\end{align}\]
We can see that the original fraction is less than the new fraction.
Now, calculating how much is the original fraction more than the new fraction =
\[\dfrac{7x}{4y}-\dfrac{x}{y}=\dfrac{7x-4x}{4y}=\dfrac{3x}{4y}\] ………………..(7)
We have to calculate the percentage of the new fraction is more than the original fraction.
Percentage = \[\dfrac{\dfrac{3x}{4y}}{\dfrac{x}{y}}\times 100=\dfrac{3}{4}\times 100=3\times 25=75%\] .
Therefore, the new fraction is more than the original fraction by 75%.
Hence, the correct option is option (a).

Note: In this question, while calculating the percentage, one may divide \[\dfrac{3x}{4y}\] by the new fraction and then multiply it by 100. This is wrong. Here, we cannot divide \[\dfrac{3x}{4y}\] by the new fraction because in the question it is asked to find the percentage is by which the new fraction is more or less with respect to the original fraction. Therefore we have to divide \[\dfrac{3x}{4y}\] by the original fraction.