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1. If the numerator of a fraction is increased by $\frac{1}{4}$ and the denominator is decreased byy $\frac{1}{3}$ , the new fraction obtained is $\frac{33}{64}$ . What was the original fraction ?

A. $\frac{3}{7}$

B. $\frac{5}{7}$

C. $\frac{7}{9}$

D. Cannot be determined

Solution:

Let the fraction be

\frac{x}{y}

Then,

\begin{aligned} \Leftrightarrow \frac{x + \frac{1}{4}}{y - \frac{1}{3}} = \frac{33}{64} \\ \Leftrightarrow \frac{3 (4 x + 1)}{4 (3 y - 1)} = \frac{33}{64} \\ \Leftrightarrow \frac{4 x + 1}{3 y - 1} = \frac{33}{64} \times \frac{4}{3} \\ \Leftrightarrow \frac{4 x + 1}{3 y - 1} = \frac{11}{16} \\ \Leftrightarrow 16 (4 x + 1) = 11 (3 y - 1) \\ \Leftrightarrow 64 x + 16 = 33 y - 11 \\ \Leftrightarrow 64 x - 33 y = - 27 \end{aligned}

Which cannot be solved to find

\frac{x}{y}

Hence, the original fraction cannot be determined from the given data.

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