Question

In a fraction, the numerator is 12 less than the denominator. If the numerator is decreased by 2, the fraction becomes\(\frac{5}{7}\). What is the value of the sum of numerator and denominator in the original fraction?

• A. 73
• B. 91
• C. 86
• D. 39

Answer 1

The Correct Option is C

Sum of the Numerator and Denominator 

- Let the numerator of the fraction be x and the denominator be y.

Step 1: Establish the First Condition

- According to the first condition, the numerator is 12 less than the denominator:

x = y - 12

Step 2: Establish the Second Condition

- According to the second condition, when the numerator is decreased by 2, the fraction becomes \( \frac{5}{7} \).

\[ \frac{x - 2}{y} = \frac{5}{7} \] - Substituting \( x = y - 12 \) into the equation:

\[ \frac{(y - 12) - 2}{y} = \frac{5}{7} \] Simplifying: \[ \frac{y - 14}{y} = \frac{5}{7} \]

Step 3: Solve for \( y \)

Cross-multiplying: \[ 7(y - 14) = 5y \] \[ 7y - 98 = 5y \] \[ 2y = 98 \] \[ y = 49 \]

Step 4: Solve for \( x \)

Now, substituting \( y = 49 \) into \( x = y - 12 \): \[ x = 49 - 12 = 37 \]

Step 5: Find the Sum of Numerator and Denominator

The sum is: \[ x + y = 37 + 49 = 86 \]

Conclusion:

The sum of the numerator and denominator is 86.