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 and their reciprocals
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Two positive numbers differ by 12swerve 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
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
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Brent
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We can create the equations: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
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
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