Addition and Subtraction of Mixed Numbers

Master adding and subtracting fractions and mixed numbers with unlike denominators

1. Recall: Adding and Subtracting Fractions

Key Rule

To add or subtract fractions with unlike denominators, first make the denominators the same by finding a common denominator. The lowest common multiple of the denominators is the easiest one to use.

Example 1: Add $\frac{2}{3}$ and $\frac{1}{5}$

Step 1: Find the lowest common multiple of 3 and 5.

Multiples of 3: 3, 6, 9, 12, 15, 18...
Multiples of 5: 5, 10, 15, 20...
The common denominator is 15.

Step 2: Convert both fractions to equivalent fractions with denominator 15.

\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}

\frac{1}{5} = \frac{1 \times 3}{5 \times 3} = \frac{3}{15}

Step 3: Add the numerators.

$\frac{10}{15} + \frac{3}{15} = \frac{13}{15}$

Example 2: Subtract $\frac{3}{4}$ from $\frac{5}{6}$

$\frac{5}{6} - \frac{3}{4}$

The common denominator of 4 and 6 is 12.

$\frac{5}{6} = \frac{10}{12}$ and $\frac{3}{4} = \frac{9}{12}$

$\frac{10}{12} - \frac{9}{12} = \frac{1}{12}$

Practice: Find $\frac{1}{8}$ + $\frac{9}{10}$

ShowHide solution

The common denominator of 8 and 10 is 40.

$\frac{1}{8} = \frac{5}{40}$ and $\frac{9}{10} = \frac{36}{40}$

$\frac{5}{40} + \frac{36}{40} = \frac{41}{40} = 1\frac{1}{40}$

2. Adding Mixed Numbers

Strategy

Add the whole numbers first
Add the fractional parts (find a common denominator if needed)
If the fraction sum is improper, convert and add to the whole

Example: Add $1\frac{1}{2}$ and $1\frac{1}{3}$

Visual: Adding the mixed numbers

$1\frac{1}{2}$

$1\frac{1}{3}$

Step 1: Add whole numbers: 1 + 1 = 2

Step 2: Add fractional parts (the common denominator of 2 and 3 is 6)

$\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

Step 3: Combine: $2\frac{5}{6}$

Example: When fraction sum exceeds 1

Add $2\frac{3}{4}$ and $1\frac{1}{2}$

Whole numbers: 2 + 1 = 3

Fractions: $\frac{3}{4} + \frac{1}{2} = \frac{3}{4} + \frac{2}{4} = \frac{5}{4} = 1\frac{1}{4}$

Final: 3 + $1\frac{1}{4} = 4\frac{1}{4}$

Practice: Add $3\frac{2}{5}$ + $2\frac{3}{4}$

ShowHide solution

Whole numbers: 3 + 2 = 5

Fractions (common denominator 20): $\frac{8}{20} + \frac{15}{20} = \frac{23}{20} = 1\frac{3}{20}$

Answer: 5 + $1\frac{3}{20} = 6\frac{3}{20}$

3. Subtracting Mixed Numbers (with Regrouping)

When to Regroup (Rename)

When the fractional part being subtracted is larger than the fractional part you have, you need to borrow 1 whole and convert it to a fraction.

Example 1 (Subtracting from a whole number): $5 - 1\frac{3}{7}$

5 has no fraction part to subtract from, so regroup: take 1 whole from 5 and rename it as sevenths. Think of 5 as $4\frac{7}{7}$

$4\frac{7}{7} - 1\frac{3}{7}$

Whole numbers: 4 − 1 = 3

Fractions: $\frac{7}{7} - \frac{3}{7} = \frac{4}{7}$

Answer: $3\frac{4}{7}$

Example 2 (With Regrouping): Subtract $1\frac{5}{8}$ from $3\frac{1}{4}$

Problem: $3\frac{1}{4} - 1\frac{5}{8}$

$\frac{1}{4} = \frac{2}{8}$ . We cannot take $\frac{5}{8}$ away from $\frac{2}{8}$ , because $\frac{2}{8} < \frac{5}{8}$ .

Step 1: Convert to the same denominator (common denominator 8)

$3\frac{1}{4} = 3\frac{2}{8}$

Step 2: Regroup (rename) by borrowing 1 whole

$3\frac{2}{8} = 2 + 1 + \frac{2}{8} = 2 + \frac{8}{8} + \frac{2}{8} = 2\frac{10}{8}$

Step 3: Now subtract

$2\frac{10}{8} - 1\frac{5}{8}$

Whole numbers: 2 − 1 = 1

Fractions: $\frac{10}{8} - \frac{5}{8} = \frac{5}{8}$

Answer: $1\frac{5}{8}$

Practice 1: $3\frac{4}{9} - 1\frac{5}{6}$

ShowHide solution

The common denominator of 9 and 6 is 18.

$3\frac{4}{9} = 3\frac{8}{18}$ and $1\frac{5}{6} = 1\frac{15}{18}$

Since $\frac{8}{18} < \frac{15}{18}$ , regroup:

$3\frac{8}{18} = 2\frac{26}{18}$

$2\frac{26}{18} - 1\frac{15}{18} = 1\frac{11}{18}$

Answer: $1\frac{11}{18}$

Practice 2: $4\frac{5}{12} - 1\frac{1}{6}$

ShowHide solution

The common denominator of 12 and 6 is 12.

$1\frac{1}{6} = 1\frac{2}{12}$

Since $\frac{5}{12} > \frac{2}{12}$ , no regrouping needed!

Whole numbers: 4 − 1 = 3

Fractions: $\frac{5}{12} - \frac{2}{12} = \frac{3}{12} = \frac{1}{4}$

Answer: $3\frac{1}{4}$

Practice 3: $7\frac{1}{10} - 3\frac{1}{2}$

ShowHide solution

The common denominator of 10 and 2 is 10.

$3\frac{1}{2} = 3\frac{5}{10}$

Since $\frac{1}{10} < \frac{5}{10}$ , regroup:

$7\frac{1}{10} = 6\frac{11}{10}$

$6\frac{11}{10} - 3\frac{5}{10} = 3\frac{6}{10} = 3\frac{3}{5}$

Answer: $3\frac{3}{5}$

Key Takeaways

Unlike denominators: Find a common denominator first before adding or subtracting
Adding mixed numbers: Add wholes first, then add fractions
Improper fraction sum: If fractions add to more than 1, convert and add to whole number
Subtracting with regrouping: If the fraction being subtracted is larger, borrow 1 whole
Regrouping (renaming 1 whole): $3\frac{2}{8} = 2 + \frac{8}{8} + \frac{2}{8} = 2\frac{10}{8}$
Always simplify: Express final answers in simplest form

Addition and Subtraction of Mixed Numbers

1. Recall: Adding and Subtracting Fractions

Key Rule

Example 1: Add 23\frac{2}{3}32​ and 15\frac{1}{5}51​

Example 2: Subtract 34\frac{3}{4}43​ from 56\frac{5}{6}65​

Practice: Find 18\frac{1}{8}81​ + 910\frac{9}{10}109​

2. Adding Mixed Numbers

Strategy

Example: Add 1121\frac{1}{2}121​ and 1131\frac{1}{3}131​

Example: When fraction sum exceeds 1

Practice: Add 3253\frac{2}{5}352​ + 2342\frac{3}{4}243​

3. Subtracting Mixed Numbers (with Regrouping)

When to Regroup (Rename)

Example 1 (Subtracting from a whole number): 5−1375 - 1\frac{3}{7}5−173​

Example 2 (With Regrouping): Subtract 1581\frac{5}{8}185​ from 3143\frac{1}{4}341​

Practice 1: 349−1563\frac{4}{9} - 1\frac{5}{6}394​−165​

Practice 2: 4512−1164\frac{5}{12} - 1\frac{1}{6}4125​−161​

Practice 3: 7110−3127\frac{1}{10} - 3\frac{1}{2}7101​−321​

Key Takeaways

Example 1: Add $\frac{2}{3}$ and $\frac{1}{5}$

Example 2: Subtract $\frac{3}{4}$ from $\frac{5}{6}$

Practice: Find $\frac{1}{8}$ + $\frac{9}{10}$

Example: Add $1\frac{1}{2}$ and $1\frac{1}{3}$

Practice: Add $3\frac{2}{5}$ + $2\frac{3}{4}$

Example 1 (Subtracting from a whole number): $5 - 1\frac{3}{7}$

Example 2 (With Regrouping): Subtract $1\frac{5}{8}$ from $3\frac{1}{4}$

Practice 1: $3\frac{4}{9} - 1\frac{5}{6}$

Practice 2: $4\frac{5}{12} - 1\frac{1}{6}$

Practice 3: $7\frac{1}{10} - 3\frac{1}{2}$