If 2 is added to the denominator of a rational number, it becomes 1/2 and if…

Concept

A rational number is written as a fraction numerator/denominator. When a condition says the fraction equals a target value after a change, translate it into an equation by cross-multiplication: a/b = p/q means a*q = b*p. Two independent conditions give two linear equations, which are solved simultaneously for the numerator and the denominator.

Application

Let the numerator be N and the denominator be D, so the number is N/D.

1. Condition 1 - adding 2 to the denominator gives 1/2: N/(D + 2) = 1/2. Cross-multiply: 2N = D + 2, so D = 2N - 2.

2. Condition 2 - adding 34 to the numerator gives 1/4: (N + 34)/D = 1/4. Cross-multiply: 4(N + 34) = D, so D = 4N + 136.

3. Equate the two expressions for D: 2N - 2 = 4N + 136, which gives -2N = 138, so N = -69.

4. Substitute back: D = 2(-69) - 2 = -140.

5. Required sum = N + D = -69 + (-140) = -209.

Cross-check

The number is -69/-140 = 69/140. Add 2 to the denominator: -69/(-138) = 69/138 = 1/2. Add 34 to the numerator: (-69 + 34)/-140 = -35/-140 = 35/140 = 1/4. Both conditions hold, so the sum is -209.