Question

The difference between two numbers is 5 and the difference between their squares is 65. Find the numbers.

Answer 1

Given:

Let the two numbers be x and y with x > y.

x – y = 5 and x² – y² = 65

Step-wise calculation:

1. Use the difference of squares identity x² – y² = (x – y)(x + y).

2. Substitute x – y = 5 into (x – y)(x + y) = 65:

5(x + y) = 65

⇒ x + y = 13

3. Solve the system x – y = 5, x + y = 13.

Add: 2x = 18

⇒ x = 9

Then y = 13 – x

= 13 – 9

= 4

The two numbers are 9 and 4.