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 . 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.

A proportion is represented as two ratios that are set equal to one another.

For example:

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.

The ratios must be set up where the relationship between the values makes sense. The man’s height creates the length of his shadow, as does the building’s height. Thus, the height of the man and the building can be placed in the numerator of each ratio, while their shadow lengths can be placed in the denominators of each ratio. Once the ratios are set up such that the same type of values are in the numerators and the denominators, the proportion works and you can solve for the unknown value.

Therefore, the proportion *does not* make sense if you place the height of the building over the man’s shadow length or the man’s height over the building shadow length. This is because there is no longer a correlation within the individual ratios, as the man’s height does not create the building’s shadow, and vice versa.

**Practice Problems**

- The ratio of boys to girls in a class is 2:3. If there are 15 girls in the class, how many total students are in the class?
- Yolanda is 62 inches tall. If the ratio of Yolanda to Mary’s height is 4:3, how tall is Mary?
- At a pet store, there are 8 dogs, 18 cats, 36 hamsters, and 15 birds. What is the ratio of hamsters to cats?
- Jimmy can run 2 miles in 15 minutes. How long does it take him to run 3 miles?
- On Monday, Julie ate 1/3 of a pie. On Tuesday, she ate half of what was left. What is the ratio of the amount of pie she ate on Monday to the amount of pie she ate on Tuesday?

**Answers**

- 25
- 46.5
- 2:1
- 22.5 min
- 1:1

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