If two is added to the denominator of a rational number it becomes 1/3 and if…

Concept

A statement "a fraction becomes 1/k when something is changed" is an equation between two fractions. Cross-multiply each such statement to turn it into a linear equation, then solve the resulting system (here by elimination) for the unknown numerator and denominator.

Application

Let the rational number be x/y, where x is the numerator and y the denominator.

1. First condition: adding 2 to the denominator gives 1/3, so x/(y+2) = 1/3. Cross-multiplying: 3x = y + 2, i.e. 3x − y = 2.

2. Second condition: adding 4 to the numerator gives 1/2, so (x+4)/y = 1/2. Cross-multiplying: 2(x+4) = y, i.e. 2x − y = −8.

3. Eliminate y by subtracting the second equation from the first: (3x − y) − (2x − y) = 2 − (−8), which gives x = 10.

4. Substitute x = 10 into 3x − y = 2: 30 − y = 2, so y = 28.

5. The required sum is x + y = 10 + 28 = 38.

Cross-check

Verify both conditions with x = 10, y = 28: x/(y+2) = 10/30 = 1/3 ✓ and (x+4)/y = 14/28 = 1/2 ✓. Both hold, so the numerator-plus-denominator sum is 38.