The sum and difference of two numbers are 20 and 8 respectively. What is the difference of the squares of these two numbers?

Introduction / Context:
This question tests your understanding of algebraic identities, especially the identity for the difference of squares. When you know the sum and difference of two numbers, you can quickly compute the difference of their squares without finding the numbers individually. This is a time-saving technique that appears often in aptitude and quantitative exams.

Given Data / Assumptions:

• The sum of two numbers is 20.
• The difference between the two numbers is 8 (larger minus smaller).
• We are asked for the difference of their squares.
• Let the numbers be a and b with a > b.

Concept / Approach:
We use the important algebraic identity: a^2 − b^2 = (a − b) * (a + b). Instead of first solving for a and b, we can directly apply this identity because the question gives us both (a + b) and (a − b). This greatly simplifies the calculation. This approach demonstrates how understanding identities helps solve problems quickly and accurately without unnecessary steps.

Step-by-Step Solution:
Step 1: Let the two numbers be a and b such that a + b = 20 and a − b = 8. Step 2: Recall the identity: a^2 − b^2 = (a + b) * (a − b). Step 3: Substitute the given values of a + b and a − b into the identity. Step 4: Compute a^2 − b^2 = 20 * 8. Step 5: Multiply 20 * 8 = 160. Step 6: Therefore, the difference of the squares of the two numbers is 160.

Verification / Alternative check:
As an alternative, you can find the actual numbers. Adding the two equations a + b = 20 and a − b = 8 gives 2a = 28, so a = 14. Subtracting the second from the first gives 2b = 12, so b = 6. Now compute a^2 − b^2 = 14^2 − 6^2 = 196 − 36 = 160. This confirms the result obtained directly using the identity and shows that both methods are consistent.

Why Other Options Are Wrong:
Option 120 and 140: These values could arise from incorrect intermediate multiplications or misreading the given sum or difference, but they do not match the identity.
Option 180 and 100: These are typical distractors that might result from incorrectly using 20^2 − 8^2 or other mistaken combinations. They do not match the algebraic identity with the given values.

Common Pitfalls:
A common error is to square the sum and difference separately, for example doing 20^2 − 8^2 instead of using the correct identity. Another mistake is to attempt full expansion (a + b)^2 − (a − b)^2 without simplifying properly, which takes longer and is prone to algebra mistakes. Remember that a^2 − b^2 factors directly into (a + b) * (a − b), which is simple to compute when both terms are given.

Final Answer:
The difference of the squares of the two numbers is 160.