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Ratios and Proportions

Introduction to Ratios

A ratio demonstrates a mathematical relationship between two quantities, and can be represented as a fraction, two numbers joined by a colon, or two numbers joined by the word “to”, i.e. 3:1 or 3 to 1.

If the ratio is a fraction, generally the first value mentioned goes in the numerator while the second value goes in the denominator, so 3 to 1 can be written as $\frac{3}{1}$ . If the ratio is in the other two forms, generally the first value mentioned is written first while the second value mentioned is written second.

A ratio can represent a relationship between values that are parts of a whole (part to part) or a relationship between a part and its whole (part to whole). Because the ratio can be represented as a fraction, you must reduce the ratio to its simplest form.

Introduction to Proportions

A proportion is two ratios set equal to one another.

Proportions can help solve questions about scales of size or distance, such as a map or model. It is important that the placement of the known quantities in both ratios makes sense in proportions.

For example, if a man stands 6 feet tall and has a 10 foot shadow, you can then determine the height of a nearby building with a 20 foot shadow by setting up a proportion comparing the ratio of the man’s height to his shadow’s length and the ratio of the building's height to its shadow’s length.

When finding a proportion, you must make sure to compare equivalent values. In the problem above, the man stands at a certain height, and casts a shadow that is dependent on his height. Given the height of the building, you can then use proportional reasoning to find the length of its shadow. Similarly, if you were given the length of the building’s shadow, you could figure out the height of the building based on the proportion to the man.

However, if you were told the size of the building’s width, or the weight of the building, or the number of people in the building, you would not have the required information to set up a proportion. So always be careful to note what values are given in the problem.

Practice Problems

If a bag contains 10 red marbles, 4 blue marbles, and 6 green marbles:
1. What is the ratio of blue marbles to green marbles?
2. What is the ratio of blue marbles to all the marbles in the bag?
If the ratio of cats to dogs at an animal shelter is 2 to 1, and there are 25 dogs, how many cats are at the shelter?
A rope is 40 feet long and weighs 15 pounds. If the same material was used to extend the rope to 140 feet, how much would it weigh?

Answers to Practice Problems