One number is 55 more than another. The difference between their squares is 6565. What are the numbers?

Let the two numbers be \( x \) and \( y \), where \( x \) is 55 more than \( y \). Thus, we can write the relationship between the two numbers as:

\[
x = y + 55
\]

We are also given that the difference between their squares is 6565, which leads to the equation:

\[
x^2 - y^2 = 6565
\]

Using the difference of squares formula:

\[
x^2 - y^2 = (x - y)(x + y)
\]

Substituting \( x = y + 55 \) into the equation:

\[
(y + 55 - y)(y + 55 + y) = 6565
\]

This simplifies to:

\[
55(2y + 55) = 6565
\]

Now, divide both sides by 55:

\[
2y + 55 = \frac{6565}{55} = 119
\]

Solve for \( y \):

\[
2y = 119 - 55
\]
\[
2y = 64
\]
\[
y = \frac{64}{2} = 32
\]

Now, substitute \( y = 32 \) into the equation \( x = y + 55 \):

\[
x = 32 + 55 = 87
\]

Thus, the two numbers are:

\[
x = 87 \quad \text{and} \quad y = 32
\]

Would you like more details on any part of the solution? Here are 5 related questions:

1. How does the difference of squares formula work in other examples?
2. What happens if the two numbers differ by a different amount, say 100?
3. Can you derive the same result using another method, such as factoring?
4. How can you generalize this problem for different numbers?
5. How can you verify the solution by substituting values back into the original equation?

**Tip**: Always check your solution by substituting the values back into the original problem to ensure accuracy.

One number is 55 more than another. The difference between their squares is 6565. What are the numbers?

Solve the problem of finding two numbers where one is 55 more than the other, and the difference between their squares is 6565. This problem involves algebraic concepts like difference of squares and linear equations, and is suitable for Grades 8-10.

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