Question

The sum of the squares of two natural numbers is 100. The square of the smaller number is four and a half times the larger number. Find the numbers.

Answer 1

Given: The sum of the squares of two natural numbers is 100 and the square of the smaller number is four and a half times the larger number.

Step-wise calculation:

1. Let the smaller number = x and the larger number = y (x, y ∈ N, x < y).

2. From the statement:

x² + y² = 100 

`x^2 = (9/2)y`   ...(Four and a half = `9/2`)

3. Solve for y from the second equation:

`y = (2/9)x^2`

4. Substitute into the first equation:

`x^2 + [(2/9)x^2]^2 = 100`

`x^2 + (4/81)x^4 = 100`

Let t = x².

Then: `(4/81)t^2 + t - 100 = 0`

Multiply by 81: 4t² + 81t – 8100 = 0

5. Solve the quadratic in t:

Discriminant D = 81² + 4 × 4 × 8100 

= 6561 + 129600

= 136161

= 369²

`t = (-81 ± 369)/8` 

Positive root: `t = (288)/8` 

= 36   ...(The negative root is invalid for t = x²)

6. So, x² = 36

⇒ x = 6   ...(Natural number) 

Then `y = (2/9)x^2`

 = `(2/9) xx 36`

= 8

The two natural numbers are 6 and 8.