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

Options

A. 1400
B. 1600
C. 1800
D. 2000

Answer

To solve the problem step by step, we can follow these instructions:

1. **Define the Variables**: Let the two positive integers be \( a \) and \( b \).

2. **Set Up the Equations**: According to the problem, we have two equations:
- The sum of the two integers:
\[ a + b = 80 \]
- The difference between the two integers:
\[ a - b = 20 \]

3. **Solve for \( a \) and \( b \)**: We can solve these two equations simultaneously.
- First, add the two equations:
\[ (a + b) + (a - b) = 80 + 20 \]
This simplifies to:
\[ 2a = 100 \implies a = \frac{100}{2} = 50 \]
- Now, substitute \( a = 50 \) back into the first equation to find \( b \):
\[ 50 + b = 80 \implies b = 80 - 50 = 30 \]

4. **Calculate the Difference of Squares**: Now, we need to find the difference between the squares of \( a \) and \( b \):
\[ a^2 - b^2 \]
We can use the values we found:
\[ a^2 = 50^2 = 2500 \]
\[ b^2 = 30^2 = 900 \]
Therefore,
\[ a^2 - b^2 = 2500 - 900 = 1600 \]

5. **Final Answer**: The difference between the squares of the two numbers is:
\[ \boxed{1600} \]