Introduction
A ratio compares the relationship between two values. It tells us how much there is of one thing compared to another.
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 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. 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.
Examples
Example 1
Write 30 to 70 as a fully simplified ratio.
Two ratios are equivalent if one of the ratios can be increased or decreased by some multiple to be equal to the other ratio.
A ratio is a simplified ratio if the values are integers and have no common factors other than 1.
Divide a quantity by 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.
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
Examples
Example 2
A salad dressing is supposed to have a 5\text{:}16 ratio of vinegar to oil. If there are 13\text{ mL} of vinegar, how many \text{mL} of oil should be added? Round your answer to the nearest whole \text{mL}.
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
Triple ratio
We can also use ratios to relate three quantities in the form 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.
Examples
Example 3
A piece of rope is cut into three lengths in the ratio 3\text{:}4\text{:}8. The shortest length of rope is measured to be 18\text{ m} long.
Find the middle length of the rope.
Find the longest length of the rope.
We can use ratios to relate three quantities in the form a:b:c.