Question

If the sum of two numbers is 15 and their product is 56, what is the sum of their reciprocals?

• A. $\dfrac{15}{56}$
• B. $\dfrac{56}{15}$
• C. $\dfrac{1}{15}$
• D. $\dfrac{1}{56}$

Hint:

For problems involving sum and product of numbers, use the identity $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{x + y}{xy}$ to find the sum of reciprocals efficiently.

Answer 1

The Correct Option is A

- Step 1: Let the two numbers be $x$ and $y$. Given $x + y = 15$ and $xy = 56$.
- Step 2: The sum of their reciprocals is $\dfrac{1}{x} + \dfrac{1}{y}$.
- Step 3: Using the identity for reciprocals, $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{x + y}{xy}$.
- Step 4: Substitute the given values: $\dfrac{x + y}{xy} = \dfrac{15}{56}$.
- Step 5: Verify options: Option (a) is $\dfrac{15}{56}$, which matches the result.
- Step 6: Check for correctness:
The numbers satisfying $x + y = 15$ and $xy = 56$ are roots of the quadratic $t^2 - 15t + 56 = 0$, with roots $t = 7, 8$.
Their reciprocals are $\dfrac{1}{7}, \dfrac{1}{8}$, and $\dfrac{1}{7} + \dfrac{1}{8} = \dfrac{8 + 7}{56} = \dfrac{15}{56}$.