The sum of two positive integers is 80 and the difference between them is 20. What is the difference between the squares of these two numbers?

Introduction / Context:

This question is a direct application of the identity for the difference of squares. Using algebraic identities helps you compute expressions involving squares much faster than finding the individual numbers and squaring them manually.

Given Data / Assumptions:

• The two positive integers are let us say a and b.
• a + b = 80.
• a - b = 20.
• We need to find a^2 - b^2.

Concept / Approach:

Use the identity a^2 - b^2 = (a + b)(a - b). The problem directly gives you the sum and the difference, so you do not even need to find a and b individually.

Step-by-Step Solution:

Verification / Alternative check:

You can find the numbers explicitly. Add the two equations: (a + b) + (a - b) = 80 + 20 gives 2a = 100, so a = 50. Then b = 80 - 50 = 30. Now compute a^2 - b^2 = 50^2 - 30^2 = 2500 - 900 = 1600, which matches the identity method.

Why Other Options Are Wrong:

The values 1400, 1800, 2000, and 1200 arise from incorrect multiplication or using the wrong expressions, such as adding squares instead of subtracting or misusing the sum and difference values.

Common Pitfalls:

A common mistake is to compute only one part, for example 80^2 or 20^2, instead of multiplying 80 and 20. Another frequent error is to ignore the identity and attempt a long solution by guessing numbers.

Final Answer:

The difference of the squares of the two numbers is 1600.