Three times the larger of two numbers is equal to four times the smaller. The sum of the numbers is 21. How do you find the numbers?

Jan 7, 2017

See full process for solving this word problem below in the Explanation section:

Explanation:

Let us first deal with the first sentence of this word problem.

Let's call the larger number $l$ and the smaller number $s$.

We know from the first sentence:

$3 l = 4 s$

We know from the second sentence:

$l + s = 21$

Let's solve this second equation for $s$:

$l - l + s = 21 - l$

$0 + s = 21 - l$

$s = 21 - l$

Now we can substitute $21 - l$ for $s$ in the first equation and solve for $l$:

$3 l = 4 \left(21 - l\right)$

$3 l = 84 - 4 l$

$3 l + \textcolor{red}{4 l} = 84 - 4 l + \textcolor{red}{4 l}$

$7 l = 84 - 0$

$7 l = 84$

$\left(7 l\right) / \textcolor{red}{7} = \frac{84}{\textcolor{red}{7}}$

$\frac{\textcolor{red}{\cancel{\textcolor{b a c k}{7}}} l}{\cancel{\textcolor{red}{7}}} = 12$

$l = 12$

Next we can substitute $12$ for $l$ in the solution to the second equation:

$s = 21 - 12$

$s = 9$

The larger number is 12 and the smaller number is 9