Dividing a Quantity in a Ratio
Learn how to share an amount in a given ratio for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-05). Whether you are splitting money, sweets or a load of concrete, the method never changes: add the ratio numbers to get the total number of equal parts.
You will work out the value of one part, multiply to get each share, split amounts in two-part and three-part ratios, work backwards when you know one share rather than the total, and check the shares add to the total — a fair way to split bills, prizes and costs.
Theory
To divide a quantity in a given ratio, add the ratio numbers to get the total number of equal parts, find the value of one part, then multiply by each share’s number of parts. This Year 12 Standard 1 (NSW) guide also shows what to do when one share is known.
To share a total in the ratio \(a:b\) (or \(a:b:c\)), first find the total number of parts: \(a+b\) (or \(a+b+c\)). Each part is the same size.
Divide the total amount by the total number of parts to get the value of one part. Each share is its number of parts times one part. The shares should add back to the total.
Sometimes one share is known instead of the total. Then one part is the known share divided by its own number of parts. Multiply to get any other share, or the total.
Sharing a total in the ratio \(a:b\):
When one share is known:
How to divide a quantity in a ratio
- Add the ratio numbers to get the total number of parts.
- Divide the total amount by the total parts to find one part (or divide a known share by its own parts).
- Multiply one part by each share’s number of parts.
- Check the shares add to the total, and give money to the cent.
Add the parts, find one part, then multiply.
| \(\text{Total parts}\) | \(=\) | \(4+5=9\) |
| \(\text{One part}\) | \(=\) | \(\dfrac{\$135}{9}=\$15\) |
| \(\text{Tom}\) | \(=\) | \(4\times\$15=\$60\) |
| \(\text{Lily}\) | \(=\) | \(5\times\$15=\$75\) |
Check: \(\$60+\$75=\$135\). Tom gets \(\$60\) and Lily gets \(\$75\).
Add the three parts, find one part, then multiply for each trough.
| \(\text{Total parts}\) | \(=\) | \(2+3+7=12\) |
| \(\text{One part}\) | \(=\) | \(\dfrac{156}{12}=13\text{ L}\) |
| \(\text{Shares}\) | \(=\) | \(2\times13=26,\ 3\times13=39,\ 7\times13=91\text{ L}\) |
Check: \(26+39+91=156\). The troughs get \(26\text{ L}\), \(39\text{ L}\) and \(91\text{ L}\).
The larger share is \(5\) parts, so divide it by \(5\) to get one part.
| \(5\text{ parts}\) | \(=\) | \(\$65\) |
| \(\text{One part}\) | \(=\) | \(\dfrac{\$65}{5}=\$13\) |
| \(\text{Smaller share}\) | \(=\) | \(3\times\$13=\$39\) |
| \(\text{Total parts}\) | \(=\) | \(5+3=8\) |
| \(\text{Total}\) | \(=\) | \(8\times\$13=\$104\) |
The smaller share is \(\$39\) and the total is \(\$104\).
Add the parts, find one part to the cent, then multiply.
| \(\text{Total parts}\) | \(=\) | \(2+2+1=5\) |
| \(\text{One part}\) | \(=\) | \(\dfrac{\$84.50}{5}=\$16.90\) |
| \(2\text{ parts}\) | \(=\) | \(2\times\$16.90=\$33.80\) |
Check: \(\$33.80+\$33.80+\$16.90=\$84.50\). They pay \(\$33.80\), \(\$33.80\) and \(\$16.90\).
Common pitfalls
Frequently asked questions
How do you divide an amount in a ratio?
Add the numbers in the ratio to get the total number of parts. Divide the amount by that total to find one part, then multiply one part by each number in the ratio.
How do you share 135 dollars in the ratio 4 to 5?
There are 4 plus 5, which is 9 parts. One part is 135 divided by 9, which is 15 dollars. The shares are 4 times 15, which is 60 dollars, and 5 times 15, which is 75 dollars.
How do you divide a quantity in a three-part ratio?
Add all three numbers to get the total parts, find one part by dividing the quantity by that total, then multiply one part by each of the three numbers.
How do you find the total when you only know one share?
Divide the known share by its own number of parts to find one part. Then multiply one part by the total number of parts.
How can I check my answer when sharing in a ratio?
Add the shares together. They should equal the original total. Also check the larger share goes with the larger number in the ratio.