Solving a Fraction Word Problem

Video Explanation

Question

If the numerator of a fraction is multiplied by 3 and the denominator is reduced by 4, the fraction becomes \( \frac{3}{2} \). If the numerator is increased by 5 and the denominator is doubled, the fraction becomes \( \frac{2}{3} \). Find the fraction.

Solution

Step 1: Let the Variables

Let the numerator = \(x\)

Let the denominator = \(y\)

Step 2: Form the First Equation

\[ \frac{3x}{y – 4} = \frac{3}{2} \]

Cross multiply:

\[ 2(3x) = 3(y – 4) \]

\[ 6x = 3y – 12 \]

\[ 6x – 3y = -12 \]

Divide by 3:

\[ 2x – y = -4 \quad (1) \]

Step 3: Form the Second Equation

\[ \frac{x + 5}{2y} = \frac{2}{3} \]

Cross multiply:

\[ 3(x + 5) = 2(2y) \]

\[ 3x + 15 = 4y \]

\[ 3x – 4y = -15 \quad (2) \]

Step 4: Solve the Equations

Multiply equation (1) by 3:

\[ 6x – 3y = -12 \times 3 \]

\[ 6x – 3y = -36 \quad (3) \]

Multiply equation (2) by 2:

\[ 6x – 8y = -30 \quad (4) \]

Subtract (3) from (4):

\[ (6x – 8y) – (6x – 3y) = -30 – (-36) \]

\[ -5y = 6 \]

\[ y = -\frac{6}{5} \]

Conclusion

Since the denominator cannot be negative for a standard positive fraction, this system gives no valid positive fraction solution.

\[ \boxed{\text{No positive fraction satisfies the given conditions}} \]