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Two positive numbers differ by 12 and their reciprocals

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Two positive numbers differ by 12 and their reciprocals

by swerve » Thu Feb 28, 2019 8:36 am

00:00

A

B

C

D

E

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Difficult

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

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by Brent@GMATPrepNow » Thu Feb 28, 2019 9:18 am
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)

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Brent
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by Scott@TargetTestPrep » Mon Mar 04, 2019 7:00 pm
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