4.6 Add and Subtract Mixed Numbers

Mixed numbers combine a whole number with a fraction, like $3\frac{2}{5}$. Adding and subtracting them means handling the whole-number part and the fraction part carefully, especially when you need to borrow or deal with different denominators.

Adding and Subtracting Mixed Numbers

Addition of Mixed Numbers

The core idea: add the whole numbers together, then add the fractions together.

1. Add the whole-number parts: $3 + 2 = 5$
2. Add the fractional parts: $\frac{1}{5} + \frac{2}{5} = \frac{3}{5}$
3. Combine them: $5\frac{3}{5}$

What if the fractions add up to more than 1? Convert the improper fraction to a mixed number and add the extra whole number to your total.

For example, $3\frac{3}{4} + 1\frac{3}{4}$:

1. Add the whole numbers: $3 + 1 = 4$
2. Add the fractions: $\frac{3}{4} + \frac{3}{4} = \frac{6}{4}$
3. Convert $\frac{6}{4}$ to a mixed number: $1\frac{2}{4} = 1\frac{1}{2}$
4. Add that back to the whole number: $4 + 1\frac{1}{2} = 5\frac{1}{2}$

Visual models can make this click. Picture each whole number as a fully shaded square and each fraction as a partially shaded square. When you combine the shaded parts and they fill up a whole square, that's your improper fraction turning into an extra whole number.

Subtraction of Mixed Numbers

Subtraction works the same way in reverse, but there's a catch: sometimes the fraction you're subtracting is bigger than the fraction you're subtracting from. When that happens, you need to borrow.

When borrowing isn't needed (the top fraction is already bigger):

$5\frac{3}{4} - 2\frac{1}{4}$

1. Subtract the fractions: $\frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}$

2. Subtract the whole numbers: $5 - 2 = 3$

3. Result: $3\frac{1}{2}$

When borrowing is needed (the top fraction is smaller):

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

You can't subtract $\frac{3}{4}$ from $\frac{1}{4}$, so you borrow 1 from the whole number:

1. Borrow 1 from 4, making it 3. Convert that 1 into a fraction with the same denominator: $1 = \frac{4}{4}$

2. Add the borrowed fraction to the existing fraction: $\frac{4}{4} + \frac{1}{4} = \frac{5}{4}$. Now you have $3\frac{5}{4}$

3. Subtract the fractions: $\frac{5}{4} - \frac{3}{4} = \frac{2}{4} = \frac{1}{2}$

4. Subtract the whole numbers: $3 - 1 = 2$

5. Result: $2\frac{1}{2}$

Working with Different Denominators

When the fractional parts have different denominators, you need a common denominator before you can add or subtract. The least common denominator (LCD) is the smallest number both denominators divide into evenly.

Example: $2\frac{1}{2} + 1\frac{1}{3}$

1. Find the LCD of 2 and 3. The LCD is 6.

2. Convert each fraction to an equivalent fraction with a denominator of 6:

   • $\frac{1}{2} = \frac{3}{6}$ (multiply numerator and denominator by 3)
   • $\frac{1}{3} = \frac{2}{6}$ (multiply numerator and denominator by 2)

3. Rewrite the problem: $2\frac{3}{6} + 1\frac{2}{6}$

4. Add whole numbers: $2 + 1 = 3$

5. Add fractions: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$

6. Result: $3\frac{5}{6}$

The same process applies to subtraction with different denominators. Find the LCD, convert, then subtract (borrowing if needed). Always simplify your final answer by reducing the fraction to lowest terms.

Key Vocabulary

• Equivalent fractions: Fractions that represent the same value with different numerators and denominators ($\frac{1}{2} = \frac{3}{6}$)
• Least common denominator (LCD): The smallest denominator two fractions can share, used to rewrite fractions before adding or subtracting
• Improper fraction: A fraction where the numerator is greater than or equal to the denominator ($\frac{7}{4}$), which can be converted to a mixed number ($1\frac{3}{4}$)
• Simplify: Divide the numerator and denominator by their greatest common factor to write the fraction in lowest terms ($\frac{4}{6} = \frac{2}{3}$)