Using identities — The difference of two numbers is 5 and the difference of their squares is 135. What is the sum of the numbers?

Introduction / Context:
Another direct application of the difference-of-squares identity. When you know a - b and a^2 - b^2, you can immediately compute a + b by rearranging the identity without solving for a and b individually.

Given Data / Assumptions:

• a - b = 5.
• a^2 - b^2 = 135.
• We need a + b.

Concept / Approach:
Use a^2 - b^2 = (a - b)(a + b). Solve for a + b by dividing the known difference of squares by the known difference of the numbers. This is a fast and reliable technique in timed exams.

Step-by-Step Solution:
Identity: a^2 - b^2 = (a - b)(a + b).Substitute: 135 = 5 * (a + b).Compute a + b = 135 / 5 = 27.Hence, the sum is 27.

Verification / Alternative check:
Optionally solve: Let a + b = 27 and a - b = 5 → a = 16, b = 11. Then a^2 - b^2 = 256 - 121 = 135, confirming the result.

Why Other Options Are Wrong:

• 25/30/32/35: None equals 135 / 5; these values do not satisfy the identity with the given numbers.

Common Pitfalls:
Multiplying instead of dividing by the known difference; mixing up (a + b) with (a - b); making arithmetic slips when dividing 135 by 5.

Final Answer:
27