The numerator of a fraction is 6 less than the denominator.
Question:

The numerator of a fraction is 6 less than the denominator. If 3 is added to the numerator, the fraction is equal to $\frac{2}{3}$. What is the original fraction equal to?

Solution:

Let the denominator of the fraction be $\mathrm{x}$.

Therefore, the numerator will be $(x-6)$.

$\therefore$ Fraction $=\frac{\mathrm{x}-6}{\mathrm{x}}$

According to the question,

$\frac{\mathrm{x}-6+3}{\mathrm{x}}=\frac{2}{3}$

or $\frac{\mathrm{x}-3}{\mathrm{x}}=\frac{2}{3}$

or $3 \mathrm{x}-9=2 \mathrm{x}[$ After cross multiplication $]$

or $3 \mathrm{x}-2 \mathrm{x}=9$

or $\mathrm{x}=9$

Thus, the original fraction $=\frac{9-6}{9}=\frac{1}{3}$