Using Sum and Product — Sum of Reciprocals Two numbers have sum 10 and product 20. What is the sum of their reciprocals?

Introduction / Context:This classic identity-based question rewards recognizing a known expression. When you know the sum and product of two numbers, you can compute the sum of their reciprocals without finding the numbers individually.

Given Data / Assumptions:

• a + b = 10.
• a * b = 20.
• We seek (1/a) + (1/b).

Concept / Approach:Use the identity: (1/a) + (1/b) = (a + b) / (a * b). Substituting the given sum and product yields the result directly. This avoids solving a quadratic to find a and b separately.

Step-by-Step Solution:Apply the identity: (1/a) + (1/b) = (a + b) / (ab).Plug in values: = 10 / 20.Simplify: 10 / 20 = 1/2.Hence, the sum of reciprocals is 1/2.

Verification / Alternative check:As a check, solve a and b as roots of t^2 − 10t + 20 = 0. Compute reciprocals and confirm their sum is 1/2; the identity guarantees consistency.

Why Other Options Are Wrong:1, 3/5, 11/6, and 2/5 arise from misapplying formulas or arithmetic slips in simplification.

Common Pitfalls:Attempting to find a and b explicitly (extra work); inverting the identity incorrectly as (ab)/(a + b).

Final Answer:1/2