Discussion
Both the numerator and the denominator of \(\frac{3}{4}\) are increased by a positive integer, \(x\), and those of \(\frac{15}{17}\) are decreased by the same integer. This operation results in the same value for both the fractions.
What is the value of \(x\)?
Answer: C
\(x = 3\).
After the change the two fractions are \(\frac{3 + x}{4 + x}\) and \(\frac{15 - x}{17 - x}\). Setting them equal and cross-multiplying:
\[ \begin{aligned} (3 + x)(17 - x) &= (15 - x)(4 + x) \\ 51 + 14x - x^{2} &= 60 + 11x - x^{2} \\ 3x &= 9,\quad x = 3 \end{aligned} \]
Check: \(6/7\) and \(12/14 = 6/7\). With 1, 2 or 4 the fractions differ, for example \(4/5\) against \(14/16 = 7/8\) for x = 1.
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