11.03 Find ratios in context

Ratios tell us about the relative sizes of two or more values. They are often used in everyday life, whether it's for dividing up money, mixing paint colours, cooking, or mixing cement.

For example, if a recipe called for the ratio of milk to flour to be $3:4$3:4 and I used six cups of milk, how much flour should I use? What if I used five cups of milk, how much flour should I use now? Sometimes we can quickly identify the equivalent ratio and sometimes it may require a few steps to ensure we obtain the correct value.

 

Keeping it in proportion

When we say that $2$2 quantities in a ratio are in proportion, we mean that they are equal, or equivalent. It's the same idea as equivalent fractions. 

The following ratios are in proportion

$3:3$3:3

$\frac{1}{2}:\frac{1}{2}$12​:12​

$\frac{1}{2}:\frac{3}{6}$12​:36​

 

Worked examples

Example 1

If the following two quantities are in proportion. Find the missing value. $\frac{1}{2}:\frac{\editable{}}{4}$12​:4​.

Think: The fraction one-half is in the same proportion as what other fraction with a denominator of $4$4? We can think of this as finding an equivalent fraction, or how many quarters are equivalent to one-half? 

Do: The answer is $2$2. We could also multiply both sides of the ratio by $4$4 to find the unknown value.

Example 2

The ratio of sultanas to nuts in a brand of trail mix is always $32:56$32:56. If a snack size pack has $12$12 sultanas, how many nuts are there?

Think: Since $12$12 is not a factor of $32$32, we can first turn $32:56$32:56 into a simplified ratio, then find the equivalent ratio that looks like $12:\editable{?}$12:?.

Do: We can simplify the ratio $32:56$32:56 by decreasing both sides of the ratio until the highest common factor of the two numbers is $1$1. Use the fact that both $32$32 and $56$56 are divisible by $8$8.

$32$32	$:$:	$56$56
$\frac{32}{8}$328​	$:$:	$\frac{56}{8}$568​	Divide both sides by $8$8
$4$4	$:$:	$7$7
$12$12	$:$:	?

 

So we now know that for every $4$4 sultanas there are $7$7 nuts. To get from $4$4 sultanas to $12$12 sultanas we multiply by $3$3 which means there must be $7\times3=21$7×3=21 nuts in the pack.

Reflect: We started with the ratio $32:56$32:56, then found the simplified ratio $4:7$4:7, and used this to find the equivalent ratio $12:21$12:21 that was relevant to our problem. Notice that the three ratios are all equivalent, but only $4:7$4:7 is a simplified ratio.

Example 3

The ratio of milk to flour in a recipe is $3:4$3:4. If we use $5$5 cups of milk, how much flour is required?

Think: We have a ratio of $3:4$3:4 and want an equivalent ratio of $5:\editable{}$5:. However, $5$5 is not a nice multiple of $3$3. We can approach this problem in two ways:

Method 1. Solve step by step as we did for exchange rates, using a unitary method as follows:

$3$3	$:$:	$4$4	Write the given ratio
$1$1	$:$:	$\frac{4}{3}$43​	Divide by $3$3 to find the amount of flour per $1$1 cup of milk
$1\times5$1×5	$:$:	$\frac{4}{3}\times5$43​×5	Multiply by $5$5 to find the amount of flour for $5$5 cups of milk
$5$5	$:$:	$\frac{20}{3}$203​

Thus, we need $\frac{20}{3}=6\frac{2}{3}$203​=623​ cups of flour.

Method 2. Use our understanding of ratios to convert the ratio to a fraction 

Think: The ratio of milk to flour as $3:4$3:4 tells us:

• the amount of milk is $\frac{3}{4}$34​ times the amount of flour
• the amount of flour is $\frac{4}{3}$43​ times the amount of milk

Since we know the amount of milk we can simply multiply by the fraction $\frac{4}{3}$43​ to find the amount of flour.

Do:

Amount of flour	$=$=	$5\times\frac{4}{3}$5×43​
	$=$=	$\frac{20}{3}$203​

So again we get the result of $\frac{20}{3}$203​ cups of flour but in less steps.

 

Practice questions

Question 1

question 2

question 3

Question 4

Question 5