How to Solve Ratios and Proportions?

Ratios and Proportions

        A ratio is a comparison between quantities. A proportion is a set of ratios that are equal. Ratios in a proportion are related to one another by multiplication by some constant.

Ratios

        There are a few different ways to express a ratio. For example, the ratio of boys to girls in a class can be expressed as:

• 2 boys for every 3 girls
• 2 to 3
• 2:3
• 2/3

        Ratios can also be expressed as part-to-part or part-to-whole ratios. The above example (boys to girls) is an example of a part-to-part ratio. Assuming there are only 5 students in the class, an example of a part-to-whole ratio is the ratio of girls to students in the class, or 3:5. For boys, the ratio would be 2:5.

            

        All ratios can be scaled to form equivalent ratios by multiplying both parts of the ratio by the same constant. This is useful in everyday applications such as cooking, were scaling a recipe up or down may be necessary. For example, a pasta recipe that calls for 2 cups of pasta to 3 cups of water that feeds 2 people could be doubled to an equivalent ratio of 4:6 to feed 4 people.

Proportions

        Proportions are equations made up of two equivalent ratios. The following are all proportions:

• 2:3 = 6:9

• $\frac{2}{3} = \frac{6}{9}$

        A ratio of 2:3 is said to be proportional to a ratio of 4:6 (or 6:9, 8:12, etc.). Proportions indicate that the relative sizes of the objects being compared are the same. This means that given two objects that are proportional, it is possible to determine certain attributes of either object given information about the other; this is done by solving the proportion using cross multiplication.

Here are some examples:

        1. At PJ’s Pet Daycare, the ratio of cats to dogs is 3:5. If the total number of pets enrolled is 80, then how many of them are cats?

Calculating Ratio and Proportion

        With a ratio of 3:5, the daycare cares for three cats for every five dogs. Simplified this means that if the daycare only cared for eight animals, you would know that three would be cats and the rest would be five dogs. That is how we get the fraction, ⅜, to work the problem. Multiply that by the total number of pets to suss out how many of the total would be cats.

        Sometimes you need to do a little calculating to determine your second ratio. Set up the problem normally, but use x in place of your unknown number. The first fraction will be your known ratio. After setting up your two fractions, cross-multiply to determine what x is. See the question below as an example.

        2. We have a recipe for dog biscuits that requires two cups of flour. This makes 25 dog biscuits. Liesel’s birthday party is next week so we need to make triple the regular amount. How many cups of flour will the recipe need in order to make 75 dog biscuits?

Solve for x to Find the Proportion

        Problems dealing with finding the proportion are pre-algebraic equations so not only did you learn how to find the proportions, you also now have a foundation for algebra!