Sum and difference of squares — The sum of two numbers is 29 and the difference of their squares is 145. What is the difference between the numbers?

Introduction / Context:
This problem is tailor-made for the difference-of-squares identity. Instead of determining each number, you can infer the difference directly from the product of sum and difference equal to the difference of squares.

Given Data / Assumptions:

• a + b = 29.
• a^2 - b^2 = 145.
• We need a - b.

Concept / Approach:
Recall that a^2 - b^2 = (a - b)(a + b). With the sum a + b known, divide the given difference of squares by the sum to get a - b directly. This avoids extra algebra and speeds up the solution.

Step-by-Step Solution:
Use identity: a^2 - b^2 = (a - b)(a + b).Substitute numbers: 145 = (a - b) * 29.Compute (a - b) = 145 / 29 = 5.Therefore, the difference between the numbers is 5.

Verification / Alternative check:
If desired, solve for a and b: a = (29 + 5)/2 = 17, b = (29 - 5)/2 = 12. Then 17^2 - 12^2 = 289 - 144 = 145, confirming the calculation.

Why Other Options Are Wrong:

• 13/8/11/7: None equals 145 / 29; only 5 makes the identity hold.

Common Pitfalls:
Adding or subtracting the given numbers prematurely; forgetting to divide by the sum; arithmetic errors when dividing 145 by 29 (remember 29 * 5 = 145).

Final Answer:
5