Lesson Objectives
• Demonstrate an understanding of improper fractions & mixed numbers
• Learn how to convert a mixed number into an improper fraction
• Learn how to convert an improper fraction into a mixed number

## How to Convert between Mixed Numbers & Improper Fractions

In our introduction to fractions lesson, we learned about proper fractions, improper fractions, and mixed numbers. Recall that an improper fraction represents a value that is equal to or larger than 1. When we have an improper fraction, the numerator will be greater than or equal to the denominator. As an alternative to the improper fraction form, we also have mixed numbers. A mixed number represents the sum of a whole number and a proper fraction. In some cases, it will be easier to work with mixed numbers. In other cases, it will be easier to work with improper fractions. It is important to be able to freely change back and forth between the two forms.

### How to Change an Improper Fraction into a Mixed Number

• Divide the numerator by the denominator
• The quotient from the division is the whole number for the mixed number
• The remainder from the division is the numerator for the fraction part of the mixed number
• The denominator from the improper fraction is used as the denominator for the fraction part of the mixed number
Let's try a few examples.
Example 1: Convert each improper fraction into a mixed number. $$\frac{23}{9}$$
• Divide the numerator by the denominator
• 23 ÷ 9 = 2 R5
• The quotient from the division (2) is the whole number for the mixed number
• Our whole number part for the mixed number will be 2
• The remainder from the division (5) is the numerator for the fraction part of the mixed number
• The numerator for our fraction part will be 5
• The denominator from the improper fraction (9) is used as the denominator for the fraction part of the mixed number
• The denominator for our fraction part will be 9
$$\frac{23}{9}=2\frac{5}{9}$$ Example 2: Convert each improper fraction into a mixed number. $$\frac{51}{5}$$
• Divide the numerator by the denominator
• 51 ÷ 5 = 10 R1
• The quotient from the division (10) is the whole number for the mixed number
• Our whole number part for the mixed number will be 10
• The remainder from the division (1) is the numerator for the fraction part of the mixed number
• The numerator for our fraction part will be 1
• The denominator from the improper fraction (5) is used as the denominator for the fraction part of the mixed number
• The denominator for our fraction part will be 5
$$\frac{51}{5}=10\frac{1}{5}$$ Example 3: Convert each improper fraction into a mixed number. $$\frac{119}{13}$$
• Divide the numerator by the denominator
• 119 ÷ 13 = 9 R2
• The quotient from the division (9) is the whole number for the mixed number
• Our whole number part for the mixed number will be 9
• The remainder from the division (2) is the numerator for the fraction part of the mixed number
• The numerator for our fraction part will be 2
• The denominator from the improper fraction (13) is used as the denominator for the fraction part of the mixed number
• The denominator for our fraction part will be 13
$$\frac{119}{13}=9\frac{2}{13}$$

### How to Change a Mixed Number into an Improper Fraction

• Multiply the denominator of the fraction part by the whole number part, add the result to the numerator of the fraction part. This will give us our numerator for the improper fraction
• The denominator from the fraction part of the mixed number is used as the denominator for the improper fraction
Let's try a few examples.
Example 4: Convert each mixed number into an improper fraction. $$3\frac{5}{7}$$
• Multiply the denominator of the fraction part (7) by the whole number part (3), add the result (21) to the numerator of the fraction part (5). This will give us our numerator for the improper fraction (26)
• (7 x 3) + 5 = 21 + 5 = 26
• The denominator from the fraction part of the mixed number (7) is used as the denominator for the improper fraction
$$3\frac{5}{7}=\frac{26}{7}$$ Example 5: Convert each mixed number into an improper fraction. $$9\frac{19}{25}$$
• Multiply the denominator of the fraction part (25) by the whole number part (9), add the result (225) to the numerator of the fraction part (19). This will give us our numerator for the improper fraction (244)
• (25 x 9) + 19 = 225 + 19 = 244
• The denominator from the fraction part of the mixed number (25) is used as the denominator for the improper fraction
$$9\frac{19}{25}=\frac{244}{25}$$ Example 6: Convert each mixed number into an improper fraction. $$16\frac{12}{13}$$
• Multiply the denominator of the fraction part (13) by the whole number part (16), add the result (208) to the numerator of the fraction part (12). This will give us our numerator for the improper fraction (220)
• (13 x 16) + 12 = 208 + 12 = 220
• The denominator from the fraction part of the mixed number (13) is used as the denominator for the improper fraction
$$16\frac{12}{13}=\frac{220}{13}$$

#### Skills Check:

Example #1

Convert the Mixed Number into an Improper Fraction. $$5\frac{29}{92}$$

A
$$\frac{93}{92}$$
B
$$\frac{489}{92}$$
C
$$\frac{523}{92}$$
D
$$\frac{711}{92}$$
E
$$\frac{577}{92}$$

Example #2

Convert the Mixed Number into an Improper Fraction. $$29\frac{42}{53}$$

A
$$\frac{1579}{53}$$
B
$$\frac{799}{53}$$
C
$$\frac{424}{53}$$
D
$$\frac{924}{53}$$
E
$$\frac{813}{53}$$

Example #3

Convert each Improper Fraction into a Mixed Number. $$\frac{305}{253}$$

A
$$5\frac{3}{253}$$
B
$$3\frac{79}{253}$$
C
$$2\frac{199}{253}$$
D
$$1\frac{52}{253}$$
E
$$4\frac{79}{253}$$