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?

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Answer 1 · 2017-01-07T22:26:04

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

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

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

We know from the first sentence:

$3 l = 4 s$ $3l = 4s$

We know from the second sentence:

$l + s = 21$ $l + s = 21$

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

$l - l + s = 21 - l$ $l - l + s = 21 - l$

$0 + s = 21 - l$ $0 + s = 21 - l$

$s = 21 - l$ $s = 21 - l$

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

$3 l = 4 (21 - l)$ $3l = 4(21 - l)$

$3 l = 84 - 4 l$ $3l = 84 - 4l$

$3 l + 4 l = 84 - 4 l + 4 l$ $3l + color(red)(4l) = 84 - 4l + color(red)(4l)$

$7 l = 84 - 0$ $7l = 84 - 0$

$7 l = 84$ $7l = 84$

$(7 l) / 7 = \frac{84}{7}$ $(7l)//color(red)(7) = 84/color(red)(7)$

$(color(red)(cancel(color(back)(7)))l)/cancel(color(red)(7)) = 12$

$l = 12$

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