# The larger of two numbers is 8 more than four times the smaller. If the larger is increased by four times the smaller, the result is 40. What are the numbers?

Oct 14, 2015

$4 \text{ }$ and $\text{ } 24$

#### Explanation:

Let's say that $x$ is the smaller number and $y$ is the larger number.

You know that if you multiply the smaller number by $4$ and add $8$, you get the larger number. This means that you can write

$4 \cdot x + 8 = y$

Moreover, you know that if you multiply the smaller number by $4$ and add it to the larger number, the result will be $40$.

$4 x + y = 40$

You now have a system of two equations with two unknowns, $x$ and $y$.

Notice that if you substitute the value of $y$ from the first equation into the second equation, you get

$4 x + {\underbrace{4 x + 8}}_{\textcolor{b l u e}{= y}} = 40$

$8 x = 32 \implies x = \frac{32}{8} = \textcolor{g r e e n}{4}$

This means that $y$ is

$y = 4 \cdot \left(\textcolor{g r e e n}{4}\right) + 8 = \textcolor{g r e e n}{24}$

The two numbers are

$x = 4 \text{ }$ and $\text{ } y = 24$