4.02 Equivalent and simplified ratios
Introduction
We previously learned how to write ratios that represent the information given to us. Writing a ratio can help us compare things mathematically. We can build upon this with the use of equivalent ratios and simplified ratios to help us solve problems.
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?
We can express the information given as the ratio 1:4.
What if we want to make two cakes? We will need 2 cups of milk and 8 cups of flour which is double the amount that we used. Now the ratio of milk to flour is 2:8.
We say that 1:4 and 2:8 are 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.
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 tables to chairs is 1:2. If there are 14 chairs, how many tables are there?
Example 2
The ratio of players to teams is 60:10 . If there are only 12 students present, how many teams can be made?
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. Therefore, all ratios equivalent to the simplified ratio have to be multiples of it. This makes the simplified ratio very useful for solving equivalent ratio questions that don't have very nice numbers.
Let's say we have a recipe for 5 cakes using 25 cups of flour. The ratio of the number of cakes to the number of cups of flour is 5:25.
To find the number of cups of flour to make 1 cake, we can simplify the equation by dividing 5:25 by 5.
\dfrac{5}{5}: \dfrac{25}{5}= 1:5.
For this cake recipe, one cake requires 5 cups of flour. We know the ratio is simplified because there is no other whole number that can further divide into both parts of the ratio and there are not any equivalent ratios with smaller integer values than 1:5 .
Examples
Example 3
Simplify the ratio 10:24.
Example 4
Write 54 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.
-
The simplified ratio uses only integers.
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.