Questionэ $a: b=2: 5$ @ь $b: c=3: 4$ , дьдпб $a: b: c=$

Studdy Solution

STEP 1

Assumptions
1. The ratio $a:b = 2:5$ is given.
2. The ratio $b:c = 3:4$ is given.
3. We need to find the combined ratio $a:b:c$ .

STEP 2

To find the combined ratio $a:b:c$ , we need to express the ratios $a:b$ and $b:c$ in terms of a common variable for $b$ .

STEP 3

For the ratio $a:b = 2:5$ , let $a = 2k$ and $b = 5k$ for some constant $k$ .

STEP 4

For the ratio $b:c = 3:4$ , let $b = 3m$ and $c = 4m$ for some constant $m$ .

STEP 5

Since both expressions for $b$ must be equal, set $5k = 3m$ .

STEP 6

Solve for $m$ in terms of $k$ by dividing both sides by 3.
$m = \frac{5k}{3}$

STEP 7

Substitute $m = \frac{5k}{3}$ into the expression for $c$ .
$c = 4m = 4 \left(\frac{5k}{3}\right) = \frac{20k}{3}$

STEP 8

Now, express $a$ , $b$ , and $c$ in terms of $k$ .
- $a = 2k$ - $b = 5k$ - $c = \frac{20k}{3}$

STEP 9

To simplify the ratio $a:b:c$ , find a common denominator for the terms. The common denominator is 3.

STEP 10

Convert each term to have the common denominator of 3.
- $a = 2k = \frac{6k}{3}$ - $b = 5k = \frac{15k}{3}$ - $c = \frac{20k}{3}$

STEP 11

The combined ratio $a:b:c$ is now:
$a:b:c = \frac{6k}{3} : \frac{15k}{3} : \frac{20k}{3}$

STEP 12

Remove the common factor $\frac{k}{3}$ from each term.
$a:b:c = 6 : 15 : 20$

STEP 13

Simplify the ratio by dividing each term by their greatest common divisor, which is 1.
$a:b:c = 6 : 15 : 20$
The combined ratio $a:b:c$ is $6:15:20$ .

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