This lesson we want to divide a quantity into a given ratio. This can be done using two main methods. The unitary method and the fraction method.
Method 1. The unitary method
- Calculate the total number of parts (by adding all the numbers in the ratio).
- Calculate what one part is worth (by dividing the given value by the total number of parts)
- Calculate what each share of the ratio is worth (by multiplying what one part is worth with each number in the ratio)
The unitary method is named for the key step in which we find one part (one unit) of the whole amount. From there we can find the value of any number of parts.
Method 2. The fraction method
Use our knowledge of writing ratios as fractions and then multiply each fraction by the given value.
Recall, if we wanted to describe the ratio of green dots to blue dots in the image above, we would write it as ratio $2:3$2:3. We can divide the ratio by the total number of dots to see fraction each colour makes up of the group, $\frac{2}{5}:\frac{3}{5}$25:35 that is two-fifths of the total number of dots are green and three-fifths are blue. The denominator represents the total number of parts in the ratio.
Worked examples
Example 1
Amir and Keira shared $\$720$$720 in the ratio $4:5$4:5. How much did each person get?
Method 1. The unitary method
Think: There are $4+5=9$4+5=9 parts in total, so we can find one part by dividing $\$720$$720 by $9$9 parts to get $\$80$$80. We can now use the knowledge that Amir gets $4$4 parts and Keira gets $5$5 parts to find each share of the money.
Do:
| Amir's share | $=$= | $4\times\$80$4×$80 |
| $=$= | $\$320$$320 | |
| Keira's share | $=$= | $5\times\$80$5×$80 |
| $=$= | $\$400$$400 |
Check: The total of Amir's share and Keira's share should sum to the total amount:
$\$320+\$400=\$720$$320+$400=$720
Method 2. The fraction method
Think: Since we know there are $9$9 parts in total, and Amir gets $4$4 parts and Keira gets $5$5 parts, then Amir will get $\frac{4}{9}$49 of the total and Keira will get $\frac{5}{9}$59 of the total.
Do:
| Amir's share | $=$= | $\frac{4}{9}\times\$720$49×$720 |
| $=$= | $\$320$$320 | |
| Keira's share | $=$= | $\frac{5}{9}\times\$720$59×$720 |
| $=$= | $\$400$$400 |
Example 2
Divide $60$60 cm in the ratio $1:2:3$1:2:3.
Think: First we find the length of one part. There are $1+2+3=6$1+2+3=6 parts in total, and $60\div6$60÷6 is $10$10. This means that one part is $10$10 cm in length.
Do: $60$60 cm divided in the ratio $1:2:3$1:2:3 would give one length of $10$10 cm, another length of $20$20 cm, and a final length of $30$30 cm. The total of these three lengths is $10+20+30=60$10+20+30=60 cm, as expected.
Practice questions
Question 1
$56$56 building blocks are shared between Mohamad and Isabelle in the ratio $2:5$2:5.
What fraction of the blocks does Mohamad receive?
$\frac{\editable{}}{\editable{}}$
How many blocks does Mohamad receive?
How many blocks does Isabelle have?
Question 2
The ratio of adults to children on a train is $9:4$9:4. If the train is carrying $130$130 passengers:
Find the number of adults on the train.
Find the number of children on the train.
Question 3
The length of a garden bed is split into three sections for beans, turnips and parsley respectively in the ratio $6:7:63$6:7:63.
If the total length of the garden bed is $19$19 metres:
What is the length of the side for beans?
What is the length of the side for turnips?
What is the length of the side for parsley?