9.3Dividing a Quantity in a Given Ratio

To share a quantity in a given ratio, find the total number of units, then find the value of one unit. Multiply to get each share.

The example below, before any working: the total and the ratio, and nothing else.

Sharing pocket money in a three-way ratio

Three siblings share $\displaystyle \$240$ of birthday money in the ratio $\displaystyle 1 : 3 : 4$ . How much does each sibling receive?

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\displaystyle \text{Total units}

\displaystyle = 1 + 3 + 4

\displaystyle = 8

\displaystyle \text{One unit}

\displaystyle = \dfrac{\$240}{8}

\displaystyle = \$30

First sibling:

1 \times \$30 = \$30

Second sibling:

3 \times \$30 = \$90

Third sibling:

4 \times \$30 = \$120

Check:

\$30 + \$90 + \$120 = \$240

✓

Combining two ratios into one

In a school orchestra, the ratio of strings to woodwind players is $\displaystyle 5 : 3$ , and the ratio of woodwind to brass players is $\displaystyle 4 : 1$ . Find the ratio of strings to woodwind to brass.

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Woodwind is the shared quantity:

3

units in

5 : 3

, and

4

units in

4 : 1

Scale each ratio so woodwind reads the same in both. The LCM of

3

and

4

12

	Strings	Woodwind	Brass
given	5	3
× 4	20	12
given		4	1
× 3		12	3
together	20	12	3

both ratios now say 12 units of woodwind, so they can be read as one

Strings : Woodwind : Brass

= 20 : 12 : 3

Reverse ratio: finding the total from the difference

Ali and Ben share some money in the ratio $\displaystyle 3 : 5$ . Ben receives $\displaystyle \$40$ more than Ali. Find the total amount shared.

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\displaystyle \text{Difference in units}

\displaystyle = 5 - 3

\displaystyle = 2 \text{ units}

\displaystyle \text{Two units}

\displaystyle = \$40

\displaystyle \text{One unit}

\displaystyle = \$40 \div 2

\displaystyle = \$20

\displaystyle \text{Total units}

\displaystyle = 3 + 5

\displaystyle = 8

\displaystyle \text{Total amount}

\displaystyle = 8 \times \$20

\displaystyle = \$160

Now you try

Share $\$160$ in the ratio $3 : 5$ . Find the larger share.

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\displaystyle \text{Total parts}

\displaystyle = 3 + 5 = 8

\displaystyle \text{1 part}

\displaystyle = \frac{160}{8} = 20

\displaystyle \text{Larger share}

\displaystyle = 5 \times 20

\displaystyle = 100

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