You have learned how to multiply fractions using both grid models and the standard algorithm. As you worked through the problems in this lesson, you may have noticed that sometimes the product of two fractions was smaller than the fractions that were multiplied together. It is also possible for the product of two fractions to be larger than the fractions that were multiplied together. The product’s value depends on what kinds of fractions were multiplied together.
Look at the table shown.
- If two proper fractions are multiplied, their product is smaller than either fraction.
- If a proper fraction is multiplied by an improper fraction, mixed number, or whole number other than 0 or 1, then the product is always larger than the proper fraction.
- If an improper fraction is multiplied by another improper fraction, mixed number, or whole number other than 0 or 1, then the product is always larger than the improper fraction.
Let's Watch
It is important to understand these relationships so that you can check your multiplication as you work. It is also important for estimating products. In the video below, the instructor demonstrates each of these relationships. Pay close attention to the types of fractions she multiplies and what their products are.
You may want to use the study guide to follow along. If so, click below to download the study guide.
Let's work through a few examples of predicting products when multiplying fractions.
1. Is the product of \( \frac{18}{7} \times \frac{1}{5} \) smaller or larger than \( \frac{18}{7} \)?
I first notice that the value we are comparing the product to is one of the factors of the multiplication expression. We will determine the type of fractions we are working with, and then determine how the product will compare to that first factor. \( \frac{18}{7} \) is an improper fraction, and \( \frac{1}{5} \) is a proper fraction. When we multiply an improper fraction by a proper fraction, the value is always less than the improper fraction. In this case, the product will be smaller than \( \frac{18}{7} \).
We can calculate the product to check our work \( \frac{18}{7} \) times \( \frac{1}{5} \). 18 \\( \times \) 1 is 18. 7 \( \times \) 5 is 35. \( \frac{18}{35} \) is smaller than \( \frac{18}{7} \), because \( \frac{18}{35} \) is proper fraction while \( \frac{18}{7} \) is an improper fraction. The product will be smaller than \( \frac{18}{7} \).
2. Is the product of \( \frac{15}{4} \times \frac{2}{3} \) smaller or larger than \( \frac{15}{4} \)?
Again, the value we are comparing our product to is one of the factors in the multiplication expression. Let’s determine what type of fractions we are multiplying. \( \frac{15}{4} \) is an improper and \( \frac{2}{3} \) is a proper fraction. When an improper fraction is multiplied by a proper fraction, the product is always smaller than the improper fraction.
We can calculate the product to check our work. \( \frac{15}{4} \) times \( \frac{2}{3} \). 15 \\( \times \) 2 is 30 and 4 \( \times \) 3 is 12. \( \frac{30}{12} \) is equivalent to \( 2\frac{6}{12} \), while \( \frac{15}{4} \) is equivalent to \( 3\frac{3}{4} \). The product that we got is smaller than \( \frac{15}{4} \). The product will be smaller than \( \frac{15}{4} \).
3. Is the product of \( \frac{12}{5} \times \frac{10}{3} \) smaller or larger than \( \frac{12}{5} \)?
In this problem we have two improper fractions. When two improper fractions are multiplied, the product is greater than both the improper fractions. This means the product will be greater than \( \frac{12}{5} \).
We can calculate the product to check our work. 12 \( \times \)10 is 120, and 5 \( \times \) 3 is 15. \( \frac{12}{5} \) as a mixed number is \( 2\frac{2}{5} \), and \( \frac{120}{15} \) is equivalent to 8. The product that we got is greater than \( \frac{12}{5} \) The product will be greater than \( \frac{12}{5} \).
Together we have worked through examples of how to predict products when multiplying fractions. Refer to this video if you need help when working on these types of problem.
Question
Suppose \( \frac{18}{7} \) is multiplied by its reciprocal, \( \frac{7}{18} \).
Predict if their product will be larger or smaller than \( \frac{18}{7} \). Then multiply the factors to show you are correct.
The product will be smaller than \( \frac{18}{7} \).
Multiplying: \( \frac{18}{7} \times \frac{7}{18} = \frac{18 \times 7}{7 \times 18} = \frac {126}{126} = 1 \). The product is smaller than \( \frac{18}{7} \). Another way to write \( \frac{18}{7} \) is \( 2 \frac{4}{7} \).
