Math Background

Dividing Fractions and Mixed Numbers: Overview

Learning how to multiply and divide fractions and mixed numbers is necessary to solve real-world problems and equations.

The concept of multiplying two fractions can be explained by using a model. Students have used models to find the product of two whole numbers.

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2 x 3 = 6

Models can also be used to find the product of two fractions. In this model of one-fourth x one-third, the large square represents 1 whole. The whole is divided horizontally into fourths and vertically into thirds. The result is 12 equal parts. One part is shaded.

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One fourth of one-third is shaded. So one-fourth x one-third = one-twelfth. One twelfth of the whole is shaded. Point out that the 1 stands for the shaded part, and the 12 stands for the total number of parts.

Another way to multiply fractions is to multiply the numerators and multiply the denominators.

multiplication of fractions

Multiplying fractions and whole numbers can be done the same way. Any number can be expressed as a fraction. Any number divided by 1 is that number; therefore, whole numbers may be written with a denominator of 1.

multiplication of fractions

If a mixed number is a factor, simply change the mixed number to an improper fraction and multiply. You may need to remind students that a mixed number is the sum of a counting number and a fraction. An improper fraction has a numerator that is greater than or equal to the denominator.

multiplication of fractions

Some students may write 1one-half as three-halves by multiplying the whole number part, 1, by the denominator, 2, before adding the numerator, 1.

Models can also be used to explain the division of fractions. The model below shows how to divide a unit fraction—a fraction in which the numerator is 1. It shows 3 ÷ one-fourth, or 3 wholes divided in fourths. You need to find how many fourths are in 3 wholes.

model

To divide by a unit fraction, multiply the whole by the denominator of the fraction. So 3 ÷ one-fourth = 3 x 4 = 12. For 2one-fourth ÷ one-half, the steps are: nine-fourths x 2 = eighteen-fourths = 4 two-fourths or 4 one-half.

Another way to divide fractions is to multiply by the reciprocal of the divisor. The reciprocal of a fraction is the fraction inverted. The product of a fraction and its reciprocal is always 1.

multiplication of fractions

For one-half ÷ three-fourths:

multiplication of fractions

When dividing with mixed numbers, first write the mixed number as an improper fraction, and then multiply by the reciprocal of the divisor.

multiplication of fractions

Houghton Mifflin Math Grade 5