Question

The ratio of two numbers is 7 to 3 and the sum of their squares is 232. Find the numbers.

Answer 1

Let's call the two numbers in this problem "x" and "y". We know that the ratio of x to y is 7 to 3, so we can write:

x/y = 7/3

We can use this information to solve for one of the variables in terms of the other. For example, we can solve for y in terms of x:

y = (3/7) x

Now we know that one of the numbers is equal to (3/7) times the other. We also know that the sum of their squares is 232, so we can write:

x^2 + y^2 = 232

Substituting y = (3/7) x, we get:

x^2 + (3/7)^2 x^2 = 232

Simplifying, we get:

(1 + (3/7)^2) x^2 = 232

(1 + 9/49) x^2 = 232

(58/49) x^2 = 232

x^2 = (232 * 49) / 58

x^2 = 196

Taking the square root of both sides, we get:

x = 14

Now we can use the equation y = (3/7) x to find y:

y = (3/7) * 14 = 6

Therefore, the two numbers are 14 and 6.

Answer 2

Let the two numbers be 7x and 3x.

We know that (7x)^2 + (3x)^2 = 232.

Expanding this equation, we get 58x^2 = 232.

Dividing both sides by 58, we get x^2 = 4.

Therefore, x = +/- 2.

Plugging this value of x into the expression for the two numbers, we get that the numbers are 14 and 6, or -14 and -6.