# The sum of two numbers is 14.And the sum of the squares of these numbers is 100.Find the ratio of the numbers?

Feb 15, 2016

$3 : 4$

#### Explanation:

Call the numbers $x$ and $y$.

We are given:

$x + y = 14$

${x}^{2} + {y}^{2} = 100$

From the first equation, $y = 14 - x$, which we can substitute in the second to get:

$100 = {x}^{2} + {\left(14 - x\right)}^{2} = 2 {x}^{2} - 28 x + 196$

Subtract $100$ from both ends to get:

$2 {x}^{2} - 28 x + 96 = 0$

Divide through by $2$ to get:

${x}^{2} - 14 x + 48 = 0$

Find a pair of factors of $48$ whose sum is $14$. The pair $6$, $8$ works and we find:

${x}^{2} - 14 x + 48 = \left(x - 6\right) \left(x - 8\right)$

So $x = 6$ or $x = 8$

Hence $\left(x , y\right) = \left(6 , 8\right)$ or $\left(8 , 6\right)$

The ratio of the two numbers is therefore $6 : 8$, i.e. $3 : 4$