Question

When $2$ is added to the numerator of a fraction, $\frac{4}{5}$ is obtained. Whereas when $2$ is subtracted from the denominator of the fraction, $\frac{2}{3}$ is obtained. Comment on the type of fraction.

• A. The fraction is a proper fraction.
• B. The fraction is an improper fraction.
• C. There exist no such fractions.

Answer 1

The correct option is A The fraction is a proper fraction.

Let the fraction be $\frac{x}{y}.$

It is given that $\frac{x+2}{y}=\frac{4}{5}$
$\Rightarrow 5(x+2)=4y$
$\Rightarrow 5x+10=4y...\left(a\right)$

Also, $\frac{x}{y-2}=\frac{2}{3}$
$\Rightarrow 3x=2(y-2)$
$\Rightarrow 3x=2y-4...\left(b\right)$

Let's solve using the substitution method.
From $1^{st}$ equation, $x=\frac{4y-10}{5}$
Substituting this in the $1^{st}$ equation we get,
$3\left(\frac{4y-10}{5}\right)=2y-4\Rightarrow 3(4y-10)=5(2y-4)\Rightarrow 12y-30=10y-20$
$\Rightarrow 2y=10$
$\Rightarrow y=5$

Substituting this in $\left(a\right)$ we get,
$5x+10=4\left(5\right)$
$\Rightarrow 5x=20-10=10$
$\Rightarrow x=2$

$\therefore$ The required fraction is $\frac{2}{5}.$
It is a proper fraction becuase it's numerator is less than denominator.