Middle Years

# 3.01 Ratios

Lesson

A ratio compares the relationship between two values. It tells us how much there is of one thing compared to another.

If we want to describe the relationship between the number of blue dots and the number of green dots, we could say that there is $1$1 blue dot for every $3$3 green dots. We could also express this as a ratio, which we would write as $1:3$1:3.

It is important that both quantities are in the same units, otherwise we cannot compare them with a ratio. In the example above, we are comparing one blue dot to three green dots, so a single dot can be thought of as our unit.

### Equivalent and simplified ratios

We can make equivalent ratios written in the form $a:b$a:b by multiplying or dividing each value in the ratio by the same number to preserve the proportions.

A ratio is a simplified ratio if there is no equivalent ratio with smaller integer values. This is the same as saying that the two integers in the ratio have a highest common factor of $1$1. We simplify ratios by dividing both sides of a ratio expression by a common factor. A simplified ratio uses only integers. A ratio that includes fractions or decimals is not yet fully simplified and can be increased or decreased by an appropriate multiple to simplify it.

#### Worked example

##### Example 1

A recipe for a salad dressing includes $20$20 ml of vinegar and $60$60 ml of olive oil.

a) Fill in the table shown below to make equivalent ratios for a larger amount of dressing.

Vinegar to Olive Oil
$20$20 : $60$60
$30$30 : $90$90
$40$40 : $\editable{}$
$\editable{}$ : $300$300

Think: What do you need to multiply one side of the ratio by to get the known value? Multiply both sides by this value to get an equivalent ratio.

Do:

 Vinegar to Olive oil Vinegar to Olive oil $20$20 : $60$60 $20$20 : $60$60 $\times2$×2 $\times2$×2 $\times5$×5 $\times5$×5 $40$40 : $\editable{}$ $\editable{}$ : $300$300

So the completed table is:

Vinegar to Olive Oil
$20$20 : $60$60
$30$30 : $90$90
$40$40 : $120$120
$100$100 : $300$300

b) What is the simplified ratio of vinegar to olive oil in the dressing?

Think: What is the highest common factor of $20$20 and $60$60? Divide both sides of the ratio by this number.

Do:
 Vinegar to Olive oil $20$20 : $60$60 $\div20$÷20 $\div20$÷20 $1$1 : $3$3

The simplified ratio of vinegar to olive oil is $1:3$1:3. A simplified ratio is great for simple recipes, this one tells us we need three times as much olive oil than vinegar. So rather than measuring in millilitres, if we have $1$1 tablespoon on vinegar we know we need to put $3$3 tablespoons of olive oil.

#### Practice question

##### question 1

Write $30$30 to $70$70 as a fully simplified ratio.

### Dividing using a ratio

We may also want to divide a quantity into a given ratio. This can be done using two main methods. The unitary method and the fraction method.

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. The fraction method finds what fraction of the whole each side of the ratio is.

Dividing using a ratio

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)

Fraction method

• Divide each side of the ratio by the total number of parts (by adding all the numbers in the ratio)
• Each share is worth the corresponding fraction multiplied by the total amount

#### Worked examples

##### Example 1

Amir and Keira shared $\$720$$720 in the ratio 4:54:5. How much did each person get? Method 1. The unitary method Think: There are 4+5=94+5=9 parts in total, so we can find one part by dividing \720$$720 by $9$9 parts to get $\$80$$80. We can now use the knowledge that Amir gets 44 parts and Keira gets 55 parts to find each share of the money. Do:  Amir's share == 4\times\804×80 == \320$$320 Keira's share $=$= $5\times\$80$5×$80 $=$= $\$400$$400 Reflect: The total of Amir's share and Keira's share should sum to the total amount: \320+\400=\720$$320+$400=$720

Method 2. The fraction method

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.

 Amir's share $=$= $\frac{4}{9}\times\$720$49​×$720 $=$= $\$320$$320 Keira's share == \frac{5}{9}\times\72059​×720 == \400$$400

Notice that multiplying $720$720 by $\frac{5}{9}$59 is effectively the same as dividing it by $9$9 (the total number of parts) and then multiplying it by $5$5 (the number of parts we want to find).

#### Practice question

##### question 2

A salad dressing is supposed to have a $5:16$5:16 ratio of vinegar to oil.

If there are $13$13 mL of vinegar, how many mL of oil should be added?

### Triple ratio

We can also use ratios to relate three quantities in the form $a:b:c$a:b:c.

This works using the same rules as before, however all three quantities must be multiplied or divided by the same number when simplifying or finding equivalent ratios.

#### Worked example

##### example 2

A certain paint colour has a ratio of blue, yellow and red given by $2:3:6$2:3:6. If $20$20 ml of blue paint is added how much red paint will be added?

Think: We can consider just the relevant parts of the ratio for this question, the blue to red ratio will be $2:6$2:6.

Do: We can multiply both sides by $20$20 to find an equivalent ratio:

 $2:6$2:6 $=$= $1:3$1:3 Simplifying the ratio $=$= $20:60$20:60 Multiplying both sides by $20$20 Amount of red paint $=$= $60$60 mL

Reflect: Notice that the ratio of just two parts can be simplified, however the original ratio can not be simplified as not all three parts can simplify.

#### Practice question

##### question 3

A piece of rope is cut into three lengths in the ratio $3:4:8$3:4:8. The shortest length of rope is measured to be $18$18 m long.

1. Find the middle length of the rope.

2. Find the longest length of the rope.

3. Now calculate the total length of the rope.