A fraction is reduced form is such that when it is squared and then its numerator

Ques - A fraction is reduced form is such that when it is squared and then its numerator is increased by 25% and the denominator is reduced to 80%, it results in 5/8 of the original fraction. The product of the numerator and denominator is :

(a) 6

(b) 12

(c) 10

(d) 7

Solution -

Let,

The numerator  = x

The denominator = y

So,

The fraction = x/y

Now,

When it is squared = x²/y²

Then,

Numerator is increased by 25%,

= x² + x² × (25/100)

And,

Denominator is reduced to 80%,

= y² - y² × (80/100)

According to the question,

[x² + x² × (25/100)] / [y² - y² × (80/100)] = 5/8(x/y)

[x² + (x²/4)] / [y² - (4y²/5)] = 5x/8y

[(4x² + x²)/4] / [(5y² - 4y²)/5] = 5x/8y

(5x²/4) / (y²/5) = 5x/8y

25x²/4y² = 5x/8y

25x²/4y² - 5x/8y = 0 

5x/4y (5x/y) - (1/2) = 0

x/y ≠ 0,

(5x/y) - (1/2) = 0

5x/y = 1/2

x/y = 1/10

Therefore,

The product of the numerator and denominator = 1 × 10

= 10

Hence,

The correct answer is option (c) 10.

Thank You