Mixed Numbers: How to Add, Subtract, Multiply & Divide

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  • 0:28 Mixed Numbers &…
  • 1:46 Multiplying Mixed Numbers
  • 2:54 Dividing Mixed Numbers
  • 3:56 Adding & Subtracting…
  • 4:50 Adding & Subtracting…
  • 5:40 Borrowing With Mixed Numbers
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Lesson Transcript
Instructor: Betty Bundly

Betty has a master's degree in mathematics and 10 years experience teaching college mathematics.

In this lesson, we will learn how to calculate with fractions, a topic many students struggle with. Although considered elementary, fractions are found at all levels of mathematics, and knowing how to work with them is an important skill.

What Are Mixed Numbers?

A mixed number contains a whole number part and a fraction part, so it is a mixture of both. Mixed numbers are always more than 1. An example of a mixed number is a number such as 3 1/2. The number 3 represents three whole parts, and the number 1/2 represents a part less than one whole. It may be helpful to think of there being an invisible '+' between the whole number and the fraction, so that 3 1/2 is really the same as 3 + 1/2.

Mixed Numbers and Improper Fractions

Sometimes, it is necessary to express a fraction with only a numerator and denominator and no whole number part, as in the number 7/2. In this number, 7 is the numerator and 2 is the denominator. When a mixed number is changed to a form where it has only a numerator and denominator, it is called an improper fraction.

An improper fraction represents the same values as its mixed number counterpart, but the whole number parts are shifted to the numerator. This makes the numerator of an improper fraction larger than the denominator. Here are the steps to change a mixed number to its improper form:

  1. Multiply the numerator by the whole number.
  2. Add the product to the numerator. This number will be the new numerator.
  3. The denominator of the improper fraction is the same as the denominator in the original mixed number.

For example, to change 3 1/2 to an improper fraction:

  1. First multiply the denominator 2 by the whole number 3, making a product of 6.
  2. Add the 6 to the numerator 1, which makes 7. This number will be the new numerator.
  3. The denominator will remain 2. So the improper fraction form for 3 1/2 is 7/2.

Multiplying Mixed Numbers

One reason you need to know how to change mixed numbers to improper form is because all mixed numbers must be changed to improper fractions before multiplying. For example, to multiply 3 2/3 * 1 2/5, first change both mixed numbers to improper form. Then, as with all fraction multiplication, multiply the numerator of one by the numerator of the other, and the denominator of one by the denominator of the other. Here are the steps:

Change 3 2/3 to an improper fraction. 3 2/3 = (3 * 3 + 2)/3 = 11/2. Change 1 2/5 to an improper fraction. 1 2/5 = (5 * 1 + 2)/5 = 7/5.
Multiply the numerators to make the numerator of the final product. Multiply the denominators to make the denominator of the new product. (11/3) * (7/5) = (11 * 7) / (3 * 5) = 77/15

Don't forget to simplify the answer if you can.

Dividing Mixed Numbers

Another reason you need to know how to change mixed numbers to improper form is because mixed numbers must be changed to improper fractions before dividing.

First, recall that to divide any fraction, you multiply by the reciprocal of the divisor, where the divisor is the second fraction in a division problem. For example, the problem 2/3 ÷ 1/5 = 2/3 * 5/1 = 10/3.

Now, let's try 4 3/4 ÷ 1 2/5:

  1. Change all mixed numbers to improper fractions: 4 3/4 ÷ 1 2/5 = 19/4 ÷ 7/5
  2. Change the division into multiplication and flip the divisor: 19/4 ÷ 7/5 = 19/4 * 5/7
  3. Multiply the two fractions: 19/4 * 5/7 = 95/28.

Adding and Subtracting Common Denominators

To add or subtract any fractions, they must have the same denominator called a common denominator. Then, we simply add or subtract the numerators. If there's a mixed number in the problem, addition and subtraction must be performed separately between the whole number parts and the fraction parts.

  • Addition Example: 2 1/5 + 3 2/5 = 2 + 3 + (1 + 2)/5 = 5 + 3/5 = 5 3/5
  • Subtraction Example: 10 7/9 - 6 2/9 = (10 - 6) + (7 - 2)/9 = 4 + 5/9 = 4 5/9

Notice that subtraction took place only between the whole numbers or between the numerators.

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