The sum of two numbers is 45. The sum of their quotient and its reciprocal is…
2024
The sum of two numbers is 45. The sum of their quotient and its reciprocal is 2.5 (i.e. if the quotient of the two numbers is a/b, then a/b + b/a = 2.5). Find the product of the numbers.
- A.
600
- B.
780
- C.
450
- D.
500
Attempted by 244 students.
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Correct answer: C
Solution:
Let the two numbers be a and b. Given
a + b = 45
a/b + b/a = 2.5, which implies (a^2 + b^2)/ab = 2.5
Compute (a + b)^2:
(a + b)^2 = a^2 + b^2 + 2ab = 45^2 = 2025, so a^2 + b^2 = 2025 - 2ab.
Substitute into (a^2 + b^2)/ab = 2.5 to get (2025 - 2ab)/ab = 2.5.
So 2025/ab - 2 = 2.5, hence 2025/ab = 4.5 and 4.5·ab = 2025.
Therefore ab = 2025/4.5 = 450.
Answer: 450