14.02 Equivalent and simplified ratios

Writing a ratio can let us compare things mathematically but has only limited use in solving problems. We can build upon this with the use of equivalent ratios and simplified ratios.

 

Exploration

Consider a cake recipe that uses $1$1 cup of milk and $4$4 cups of flour. What is the ratio of milk to flour used in the cake?

Letting the unit be "the number of cups", we can express the information given as the ratio $1:4$1:4.

What if we want to make two cakes? We would need to double the amount of milk and flour we use. This means we will need $2$2 cups of milk and $8$8 cups of flour. Now the ratio of milk to flour is $2:8$2:8.

But how do we get two different ratios from the same recipe? The secret is that the two ratios actually represent the same proportion of milk to flour. We say that $1:4$1:4 and $2:8$2:8 are equivalent ratios.

Now consider if we wanted to make enough cakes to use up $4$4 cups of milk. How many cakes would this make, and how much flour would we need?

 

Equivalent ratios

Equivalent ratios are useful for when we want to change the value of one quantity but also keep it in the same proportion to another quantity. After calculating how much the value of the first quantity has increased, we can increase the value of the second quantity by the same multiple to preserve the ratio.

We saw in the cake example that increasing both the amount of milk and the amount of flour by the same multiple preserved the ratio. That's because this is the same as having multiple sets of the same ratio.

 

And since this is an equivalence relation, we can also say the same for the reverse:

 

 

Worked examples

Example 1

The ratio of tables to chairs is $1:2$1:2. If there are $14$14 chairs, how many tables are there?

Think: The ratio $1:2$1:2 says that each table has two chairs. We want to increase both sides of this ratio by some multiple to get the equivalent ratio that looks like $\editable{?}:14$?:14. The missing number in this ratio will be the number of tables needed for $14$14 chairs.

Do: The first thing we can figure out is by what multiple the chairs have been increased. We can find this by dividing $14$14 by $2$2. This tells us how many sets of two chairs there are:

Number of pairs of chairs	$=$=	$14\div2$14÷​2
	$=$=	$7$7

If there are $7$7 sets of two chairs, and each table has two chairs, then we will need $7\times1=7$7×1=7 tables.

 

Example 2

The ratio of players to teams is $60:10$60:10. If there are only $12$12 students present, how many teams can be made?

Think: We only have $12$12 out of $60$60 students. What fraction of the students are present? To preserve the ratio, we want to take the same fraction of the usual $10$10 teams.

Do: The fraction of students present is $12$12 out of $60$60 which we can write as:

Fraction of students present	$=$=	$12$12 out of $60$60
	$=$=	$\frac{12}{60}$1260​
	$=$=	$\frac{1}{5}$15​

If only one fifth of the students are present, they can only make one fifth of the usual number of teams. So we can make $10\times\frac{1}{5}=2$10×15​=2 teams with $12$12 students.

In both of these worked examples we were able to preserve the ratio by either increasing or decreasing the initial ratio by some multiple. It is important to note that whenever we performed an operation we applied it to both sides of the ratio.

Tables	to	Chairs				Students	to	Teams
$1$1	:	$2$2				$60$60	:	$10$10
$\times7$×7		$\times7$×7				$\div5$÷​5		$\div5$÷​5
$7$7	:	$14$14				$12$12	:	$2$2

 

Simplified ratios

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.

Since the simplified ratio is the smallest integer valued ratio, this also means that all the ratios equivalent to it are multiples of it. This makes the simplified ratio very useful for solving equivalent ratio questions that don't have very nice numbers.

 

 

Worked example

example 3

The ratio of sultanas to nuts in a bag of trail mix is always $32:56$32:56. If there are $12$12 sultanas left, how many nuts are left?

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.

$32:56$32:56	$=$=	$\frac{32}{8}:\frac{56}{8}$328​:568​
	$=$=	$4:7$4:7

Since $4$4 and $7$7 have no common factors except for $1$1, this is a simplified ratio. 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. But we know the ratio of sultanas and nuts is always $4:7$4:7, which means there must be $7\times3=21$7×3=21 nuts left.

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.

 

 

The application of equivalent and simplified ratios is useful for when we want to keep things in proper proportion while changing their size, or when we want to measure large objects by considering their ratio with smaller objects.