If two is added to the denominator of a rational number it becomes 1 and if 4…

Concept

A rational number is a fraction numerator/denominator. When a worded condition says a fraction "becomes" a value after changing its numerator or denominator, translate each statement into a linear equation and solve the resulting system by substitution.

Application

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

1. Adding 2 to the denominator makes the fraction equal to 1: N / (D + 2) = 1, so N = D + 2.

2. Adding 4 to the numerator makes the fraction equal to 1/2: (N + 4) / D = 1/2, so 2(N + 4) = D, that is 2N + 8 = D.

3. Substitute N = D + 2 into 2N + 8 = D: 2(D + 2) + 8 = D, giving 2D + 4 + 8 = D.

4. Simplify: 2D + 12 = D, so D = -12, and then N = D + 2 = -10.

5. Required sum = N + D = -10 + (-12) = -22.

Cross-check

With N = -10 and D = -12 the fraction is -10/-12 = 5/6. Verify: N/(D+2) = -10/-10 = 1 and (N+4)/D = -6/-12 = 1/2. Both conditions hold, so the sum -22 is correct.