QuestionGiven $3a = 4b = 6c$ , find the ratio $a:b:c$ and if $a-b+c=21$ , determine $b$ .

Studdy Solution

STEP 1

Assumptions1. $a$ , $b$ , and $c$ are non-zero numbers. . $3a=4b=6c$ .
3. $a-b+c=21$ .

STEP 2

First, we need to find the ratio $a:b:c$ . We can do this by setting $a=4b=6c$ equal to a common variable, say $k$ .
$a = k$ $4b = k$ $6c = k$

STEP 3

Now, we can express $a$ , $b$ , and $c$ in terms of $k$ .
$a = \frac{k}{3}$ $b = \frac{k}{}$ $c = \frac{k}{6}$

STEP 4

The ratio $a:b:c$ can now be obtained by substituting the values of $a$ , $b$ , and $c$ from3.
$a:b:c = \frac{k}{3}:\frac{k}{4}:\frac{k}{6}$

STEP 5

We can simplify the ratio by multiplying each term by the least common multiple of the denominators (12).
$a:b:c =4k3k2k$

STEP 6

Since $k$ is a common factor, we can simplify further to get the ratio.
$a:b:c =432$

STEP 7

Now, we need to find the value of $b$ . We can do this by using the equation $a-b+c=21$ .

STEP 8

Substitute the values of $a$ , $b$ , and $c$ from3 into the equation.
$\frac{k}{3} - \frac{k}{4} + \frac{k}{6} =21$

STEP 9

To simplify the equation, multiply each term by the least common multiple of the denominators (12).
$4k -3k +2k =252$

STEP 10

implify the left side of the equation to find the value of $k$ .
$3k =252$

STEP 11

olve for $k$ .
$k = \frac{252}{3} =84$

STEP 12

Substitute $k =84$ into $b = \frac{k}{4}$ to find the value of $b$ .
$b = \frac{84}{4}$

STEP 13

Calculate the value of $b$ .
$b = \frac{84}{} =21$ So, $a:b:c =32$ and $b =21$ .

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