Question

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

• A. $\frac{15}{56}$
• B. $\frac{56}{15}$
• C. $\frac{7}{8}$
• D. $\frac{8}{7}$

Hint:

When you know the sum and product of two numbers, the sum of their reciprocals is found directly using $\frac{x+y}{xy}$ without solving for the numbers individually.

Answer 1

The Correct Option is A

- Step 1: Understanding the problem - We are told two numbers have a sum $x + y = 15$ and a product $xy = 56$. 

We need the sum of their reciprocals $\frac{1}{x} + \frac{1}{y}$. 

- Step 2: Using the identity for reciprocals - Recall: \[ \frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy} \] 

This formula comes from taking a common denominator: \[ \frac{1}{x} + \frac{1}{y} = \frac{y}{xy} + \frac{x}{xy} = \frac{x + y}{xy} \] 

- Step 3: Substituting known values - We know $x + y = 15$ and $xy = 56$, 

so: \[ \frac{1}{x} + \frac{1}{y} = \frac{15}{56} \] 

- Step 4: Verifying with actual numbers - The numbers satisfy $t^2 - 15t + 56 = 0$. 

Solving: \[ t^2 - 15t + 56 = 0 \implies (t - 7)(t - 8) = 0 \implies t = 7, 8 \] 

Reciprocals: $\frac{1}{7} + \frac{1}{8} = \frac{8 + 7}{56} = \frac{15}{56}$. 

This confirms the calculation. 

- Step 5: Conclusion - The sum of their reciprocals is exactly $\frac{15}{56}$.