Discussion

If the sum of two numbers is 9 and the sum of their reciprocals is \(\dfrac{1}{2}\), then those numbers will be

Answer: C

The numbers are 6 and 3.

Let the numbers be a and b, with a + b = 9. The sum of reciprocals is

\[ \begin{aligned} \frac{1}{a} + \frac{1}{b} &= \frac{a + b}{ab} = \frac{1}{2} \\ ab &= 2 \times 9 = 18 \end{aligned} \]

Two numbers with sum 9 and product 18 are 6 and 3, and indeed 1/6 + 1/3 = 1/2.

Each other pair also adds up to 9 but has the wrong product: 1 × 8 = 8, 2 × 7 = 14 and 4 × 5 = 20, so their reciprocals sum to 9/8, 9/14 and 9/20.

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