# Four times the larger of two numbers is equal to seven times the smaller The sum of the numbers is 22. How do you find the numbers?

Dec 3, 2016

The two numbers are 8 and 14.

#### Explanation:

First, let's call our two numbers $n$ and $m$.

Then, we are told, in mathematical terms:

$4 n = 7 m$

We are also told:

$n + m = 22$

If we solve this equation for n we get:

$n + m - m = 22 - m$

$n + 0 = 22 - m$

$n = 22 - m$

We can now substitute $22 - m$ for $n$ in the first equation and solve for $m$:

$4 \left(22 - m\right) = 7 m$

$88 - 4 m = 7 m$

$88 - 4 m + 4 m = 7 m + 4 m$

$88 = 11 m$

$\frac{88}{11} = \frac{11 m}{11}$

$8 = m$

We can now substitute $8$ for $m$ in the solution above to calculate $n$:

$n = 22 - 8$

$n = 14$

Dec 3, 2016

The larger is 14 the smaller is 8

#### Explanation:

Let the larger number be L and the smaller be S
So we know that
1) L+S=22
And
2)4L=7S
From equation 1) we get L=22-S
Substitute this into equation 2) to get 4(22-S)=7S
88-4S=7S
88=11S
S=8. The smaller number is 8
As they total 22 the larger number is 14.
Check: 4x14=7x8. Yes