Two positive numbers differ by 12 and their reciprocals

swerve wrote:Two positive numbers differ by \(12\) and their reciprocals differ by \(\frac{4}{5}\). What is their product?

A. \(\frac{2}{15}\)
B. \(\frac{48}{5}\)
C. \(15\)
D. \(42\)
E. \(60\)

The OA is C

Source: Manhattan Prep

Two positive numbers differ by 12
Let x = the smaller number
So x + 12 = the larger number
NOTE: our goal is to find the value of x(x + 12)

Their reciprocals differ by 4/5
We get: 1/x - 1/(x+12) = 4/5
Multiply both sides by x to get: 1 - x/(x + 12) = 4x/5
Multiply both sides by 5 to get: 5 - 5x/(x + 12) = 4x
Multiply both sides by (x + 12) to get: 5(x + 12) - 5x = 4x(x +12)
Expand left side to get: 5x + 60 - 5x = 4x(x +12)
Simplify left side to get: 60 = 4x(x +12)
Divide both sides by 4 to get: 15 = x(x +12)

Answer: C

Cheers,
Brent

swerve wrote:Two positive numbers differ by \(12\) and their reciprocals differ by \(\frac{4}{5}\). What is their product?

A. \(\frac{2}{15}\)
B. \(\frac{48}{5}\)
C. \(15\)
D. \(42\)
E. \(60\)

The OA is C

Source: Manhattan Prep

We can create the equations:

x - y = 12

and

1/y - 1/x = 4/5 (notice that x > y, so 1/y > 1/x)

Let’s multiply the second equation by xy, and we have:

x - y = (4/5)xy

Since x - y = 12, we have:

12 = (4/5)xy

Multiplying both sides by 5/4, we have:

15 = xy

Answer: C