Question

A fraction becomes \(\frac{1}{2}\) when 1 is added to its numerator and denominator, and becomes \(\frac{1}{4}\) when 1 is subtracted from its numerator and denominator. Find the fraction.

• A. \(\frac{4}{9}\)
• B. \(\frac{3}{5}\)
• C. \(\frac{4}{13}\)
• D. \(\frac{2}{5}\)

Hint:

Write the fraction as \(\frac{x}{y}\) and turn each condition into a linear equation in \(x\) and \(y\).

Answer 1

The Correct Option is D

Step 1: Set up the first equation.
Let the fraction be \(\frac{x}{y}\), where \(x\) is the numerator and \(y\) is the denominator.
Adding 1 to both the numerator and the denominator gives \(\frac{1}{2}\), so
\[ \frac{x+1}{y+1} = \frac{1}{2} \]
Cross multiply:
\[ 2(x+1) = y+1 \]
\[ y = 2x+1 \]

Step 2: Set up the second equation.
Subtracting 1 from both the numerator and the denominator gives \(\frac{1}{4}\), so
\[ \frac{x-1}{y-1} = \frac{1}{4} \]
Cross multiply:
\[ 4(x-1) = y-1 \]
\[ y = 4x-3 \]

Step 3: Solve the two equations together.
Both expressions equal \(y\), so equate them:
\[ 2x+1 = 4x-3 \]
\[ 4 = 2x \]
\[ x = 2 \]
Put \(x=2\) into \(y = 2x+1\):
\[ y = 2(2)+1 = 5 \]

Step 4: Check the answer and rule out the other options.
The fraction is \(\frac{2}{5}\). Add 1 to both parts: \(\frac{3}{6}=\frac{1}{2}\), correct. Subtract 1 from both parts: \(\frac{1}{4}\), correct.
Option (a) \(\frac{4}{9}\): adding 1 gives \(\frac{5}{10}=\frac{1}{2}\), which looks fine, but subtracting 1 gives \(\frac{3}{8}\), not \(\frac{1}{4}\), so it fails the second condition.
Option (b) \(\frac{3}{5}\): adding 1 gives \(\frac{4}{6}=\frac{2}{3}\), not \(\frac{1}{2}\), so it fails right away.
Option (c) \(\frac{4}{13}\): adding 1 gives \(\frac{5}{14}\), not \(\frac{1}{2}\), so it also fails.

Final Answer:
The fraction is \(\frac{2}{5}\), which is option (d).
\[ \boxed{\dfrac{2}{5}} \]