Introduction
We previously learned how to write ratios that match the information given to us. 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.
Equivalent ratios
Consider a cake recipe that uses 1 cup of milk and 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.
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 cups of milk and 8 cups of flour. Now the ratio of milk to flour is 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 and 2:8 are equivalent ratios.
Now consider if we wanted to make enough cakes to use up 4 cups of milk. How many cakes would this make, and how much flour would we need?
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.
Two cakes require twice the ingredients of one cake, but in the same proportion.
And since this is an equivalence relation, we can also say the same for the reverse:
One cake requires half the ingredients of two cakes, but in the same proportion.
Two ratios are equivalent if one of the ratios can be increased or decreased by some multiple to be equal to the other ratio.
Examples
Example 1
The ratio of students to teachers competing in a charity race is 10 : 3. If 70 students take part in the race, how many teachers are there?
Example 2
Complete the table of equivalent ratios and use it to answer the following questions.
| Dogs | to | Cats |
|---|---|---|
| 9 | : | 5 |
| 18 | : | 10 |
| 27 | : | |
| 45 | : | |
| : | 50 |
If there are 270 dogs, how many cats are there expected to be?
Which of the following is the fully simplified ratio for 270 : 150?
Two ratios are equivalent if one of the ratios can be increased or decreased by some multiple to be equal to the other ratio.
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 greatest common factor of 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.
A ratio is a simplified ratio if there is no equivalent ratio with smaller integer values.
The simplified ratio uses only integers. A ratio that uses fractions or decimals is not yet fully simplified and can be increased or decreased by the appropriate multiple to simplify it.
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.
The ratio of the length of your hand to your height is approximately 1:10. Try measuring your height using the length of your hand. How accurate is this ratio?
Examples
Example 3
Write 540 cents to \$3.00 as a fully simplified ratio.
A ratio is a simplified ratio if there is no equivalent ratio with smaller integer values.