Lesson Objectives

- Demonstrate an understanding of how to convert between a mixed number and an improper fraction
- Learn how to add mixed numbers
- Learn how to subtract mixed numbers
- Learn how to multiply mixed numbers
- Learn how to divide mixed numbers

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

In this lesson, we will discuss methods to perform operations with mixed numbers: addition, subtraction, multiplication, and division.
When we perform operations with mixed numbers, we have two choices: work with the mixed numbers or convert the mixed numbers into improper fractions.

Example 1: Perform each indicated operation: $$5\frac{1}{7} + 2\frac{1}{3}$$ Add the fraction parts first: $$\frac{1}{7}+\frac{1}{3} = \frac{10}{21}$$ Next, we add the whole numbers: $$5 + 2 = 7$$ Our answer: $$5\frac{1}{7} + 2\frac{1}{3} = 7\frac{10}{21}$$ Example 2: Perform each indicated operation: $$9\frac{4}{5} - 7\frac{3}{8}$$ Subtract the fraction parts first: $$\frac{4}{5} - \frac{3}{8} = \frac{17}{40}$$ Next, we subtract the whole numbers: $$9 - 7 = 2$$ Our answer: $$9\frac{4}{5} - 7\frac{3}{8} = 2\frac{17}{40}$$ Example 3: Perform each indicated operation: $$11\frac{1}{4} - 6\frac{3}{8} + 1\frac{1}{2}$$ With addition and subtraction, we work left to right. Here we will begin with subtraction: $$11\frac{1}{4} - 6\frac{3}{8}$$ Subtract the fraction parts first: $$\frac{1}{4} - \frac{3}{8}$$ 1/4 is too small to subtract 3/8 away. We can use a little trick here. Let's borrow 1 from 11: $$11\frac{1}{4} = 10 + 1 + \frac{1}{4}$$ Now let's write 1 as the fraction 4/4 and add this to our fraction 1/4. $$\frac{4}{4} + \frac{1}{4} = \frac{5}{4}$$ Now we can perform our operation with the fractions: $$\frac{5}{4} - \frac{3}{8} = \frac{7}{8}$$ Now we subtract our whole number parts. Remember we borrowed a 1 from 11, now we have 10 - 6. $$10 - 6 = 4$$ Our answer for the subtraction problem: $$11\frac{1}{4} - 6\frac{3}{8} = 4\frac{7}{8}$$ Now we move into addition. Our new problem becomes: $$4\frac{7}{8} + 1\frac{1}{2}$$ Add the fraction parts first: $$\frac{7}{8} + \frac{1}{2} = \frac{11}{8} = 1\frac{3}{8}$$ Add the whole numbers, make sure to include the 1 from the addition of fractions in the last step. $$4 + 1 + 1 = 6$$ Our answer: $$11\frac{1}{4} - 6\frac{3}{8} + 1\frac{1}{2} = 6\frac{3}{8}$$

Example 4: Perform each indicated operation: $$3\frac{1}{5} \cdot 7\frac{2}{3}$$ Convert each mixed number into an improper fraction: $$3\frac{1}{5} = \frac{(5 \cdot 3) + 1}{5} = \frac{16}{5}$$ $$7\frac{2}{3} = \frac{(3 \cdot 7) + 2}{3} = \frac{23}{3}$$ Now we can multiply the improper fractions: $$\frac{16}{5} \cdot \frac{23}{3} = \frac{368}{15}$$ We can convert our answer into a mixed number if required: $$\frac{368}{15} = 24\frac{8}{15}$$ Our answer: $$3\frac{1}{5} \cdot 7\frac{2}{3} = 24\frac{8}{15}$$ Example 5: Perform each indicated operation: $$-9\frac{4}{5} \div 3\frac{2}{15}$$ Convert each mixed number into an improper fraction: $$-9\frac{4}{5} = -\frac{(5 \cdot 9) + 4}{5} = -\frac{49}{5}$$ Be very careful when converting a negative mixed number into an improper fraction. The easiest way is to treat the number as positive, perform the conversion and then apply the negative to the result. $$3\frac{2}{15} = \frac{(15 \cdot 3) + 2}{15} = \frac{47}{15}$$ Now we can divide the improper fractions: $$-\frac{49}{5} \div \frac{47}{15} = -\frac{49}{5} \cdot \frac{15}{47}$$ Now we multiply: $$-\frac{49}{5} \cdot \frac{15}{47} = -\frac{147}{47}$$ We can convert our answer into a mixed number if required: $$-\frac{147}{47} = -3\frac{6}{47}$$ Our answer: $$-9\frac{4}{5} \div 3\frac{2}{15} = -3\frac{6}{47}$$

### How to Add and Subtract Mixed Numbers

When we add or subtract mixed numbers, it is usually faster to work directly with the mixed numbers. This means we will add or subtract the fraction parts first and then the whole numbers. Let’s look at a few examples.Example 1: Perform each indicated operation: $$5\frac{1}{7} + 2\frac{1}{3}$$ Add the fraction parts first: $$\frac{1}{7}+\frac{1}{3} = \frac{10}{21}$$ Next, we add the whole numbers: $$5 + 2 = 7$$ Our answer: $$5\frac{1}{7} + 2\frac{1}{3} = 7\frac{10}{21}$$ Example 2: Perform each indicated operation: $$9\frac{4}{5} - 7\frac{3}{8}$$ Subtract the fraction parts first: $$\frac{4}{5} - \frac{3}{8} = \frac{17}{40}$$ Next, we subtract the whole numbers: $$9 - 7 = 2$$ Our answer: $$9\frac{4}{5} - 7\frac{3}{8} = 2\frac{17}{40}$$ Example 3: Perform each indicated operation: $$11\frac{1}{4} - 6\frac{3}{8} + 1\frac{1}{2}$$ With addition and subtraction, we work left to right. Here we will begin with subtraction: $$11\frac{1}{4} - 6\frac{3}{8}$$ Subtract the fraction parts first: $$\frac{1}{4} - \frac{3}{8}$$ 1/4 is too small to subtract 3/8 away. We can use a little trick here. Let's borrow 1 from 11: $$11\frac{1}{4} = 10 + 1 + \frac{1}{4}$$ Now let's write 1 as the fraction 4/4 and add this to our fraction 1/4. $$\frac{4}{4} + \frac{1}{4} = \frac{5}{4}$$ Now we can perform our operation with the fractions: $$\frac{5}{4} - \frac{3}{8} = \frac{7}{8}$$ Now we subtract our whole number parts. Remember we borrowed a 1 from 11, now we have 10 - 6. $$10 - 6 = 4$$ Our answer for the subtraction problem: $$11\frac{1}{4} - 6\frac{3}{8} = 4\frac{7}{8}$$ Now we move into addition. Our new problem becomes: $$4\frac{7}{8} + 1\frac{1}{2}$$ Add the fraction parts first: $$\frac{7}{8} + \frac{1}{2} = \frac{11}{8} = 1\frac{3}{8}$$ Add the whole numbers, make sure to include the 1 from the addition of fractions in the last step. $$4 + 1 + 1 = 6$$ Our answer: $$11\frac{1}{4} - 6\frac{3}{8} + 1\frac{1}{2} = 6\frac{3}{8}$$

### Multiplying and Dividing Mixed Numbers

When we multiply or divide mixed numbers, it is usually faster to convert each into an improper fraction. We can then perform our multiplication or division and convert back to a mixed number if required. Let's take a look at a few examples.Example 4: Perform each indicated operation: $$3\frac{1}{5} \cdot 7\frac{2}{3}$$ Convert each mixed number into an improper fraction: $$3\frac{1}{5} = \frac{(5 \cdot 3) + 1}{5} = \frac{16}{5}$$ $$7\frac{2}{3} = \frac{(3 \cdot 7) + 2}{3} = \frac{23}{3}$$ Now we can multiply the improper fractions: $$\frac{16}{5} \cdot \frac{23}{3} = \frac{368}{15}$$ We can convert our answer into a mixed number if required: $$\frac{368}{15} = 24\frac{8}{15}$$ Our answer: $$3\frac{1}{5} \cdot 7\frac{2}{3} = 24\frac{8}{15}$$ Example 5: Perform each indicated operation: $$-9\frac{4}{5} \div 3\frac{2}{15}$$ Convert each mixed number into an improper fraction: $$-9\frac{4}{5} = -\frac{(5 \cdot 9) + 4}{5} = -\frac{49}{5}$$ Be very careful when converting a negative mixed number into an improper fraction. The easiest way is to treat the number as positive, perform the conversion and then apply the negative to the result. $$3\frac{2}{15} = \frac{(15 \cdot 3) + 2}{15} = \frac{47}{15}$$ Now we can divide the improper fractions: $$-\frac{49}{5} \div \frac{47}{15} = -\frac{49}{5} \cdot \frac{15}{47}$$ Now we multiply: $$-\frac{49}{5} \cdot \frac{15}{47} = -\frac{147}{47}$$ We can convert our answer into a mixed number if required: $$-\frac{147}{47} = -3\frac{6}{47}$$ Our answer: $$-9\frac{4}{5} \div 3\frac{2}{15} = -3\frac{6}{47}$$

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