Question

If the numerator of a fraction is increased by 25% and the denominator decreased by 20% the new value is $\dfrac{5}{4}$. What was the original fraction?
[a] $\dfrac{3}{5}$
[b] $\dfrac{4}{5}$
[c] $\dfrac{7}{8}$
[d] $\dfrac{3}{7}$

Answer 1

Hint: Assume that the numerator of the original fraction is x and the denominator of the original fraction is y. Hence find the numerator and denominator of the new fraction in terms of x and y. Compare the new numerator to be equal to 5k and the new denominator to be equal to 4k. Hence find the value of x and y in k. Hence find the original fraction.

Complete step-by-step answer:

Let the numerator of the original fraction be x, and the denominator of the original fraction be y.
Since the numerator of the new fraction is increased by 25%
Hence, we have
The numerator of the new fraction $=x+\dfrac{25}{100}\times x=1.25x$
Since the denominator of the new fraction is decreased by 20%
Hence, we have
The denominator of the new fraction $=y-\dfrac{20}{100}y=0.8x$
Hence, we have $1.25x:08y::5:4$
Here extremes are 1.25x and 4, and the means are 0.8y and 5
Since the product of means is equal to the product of extremes, we have
$\left( 1.25x \right)\times 4=0.8y\times 5\Rightarrow 5x=4y$
Dividing both sides by 5, we get
$x=\dfrac{4y}{5}$
Dividing both sides by y, we get
$\dfrac{x}{y}=\dfrac{4}{5}$
Hence the original fraction is $\dfrac{4}{5}$
Hence option [b] is correct.

Note: Verification:
The numerator of the original fraction = 4
Hence the numerator of the new fraction $=4+\dfrac{25}{100}\times 4=4+1=5$
The denominator of the original fraction = 5
Hence, the denominator of the new fraction $=5-\dfrac{20}{100}\times 5=5-1=4$
Hence, the fraction $=\dfrac{5}{4}$
Hence our answer is verified to be correct.