Adding and Subtracting Mixed Numbers — Definition, Formula & Examples

Adding and subtracting mixed numbers means combining or finding the difference between numbers that have both a whole part and a fraction part, like

3\frac{1}{4}

and

1\frac{2}{3}

. The standard method is to convert each mixed number to an improper fraction, find a common denominator, then add or subtract.

To perform addition or subtraction on mixed numbers, express each mixed number as an improper fraction $\frac{a}{b}$ , rewrite both fractions with their least common denominator, compute the sum or difference of the numerators over that denominator, and convert the result back to a mixed number in simplest form.

How It Works

Start by converting each mixed number into an improper fraction: multiply the whole number by the denominator and add the numerator. Next, find the least common denominator (LCD) of the two fractions and rewrite each fraction with that denominator. Add or subtract the numerators while keeping the denominator the same. Finally, simplify the result and convert it back to a mixed number if needed. An alternative approach is to work with the whole numbers and fractions separately, but this can get tricky when subtracting requires borrowing.

Worked Example

Problem: Calculate

3\frac{1}{4} + 1\frac{2}{3}

Convert to improper fractions: Multiply the whole number by the denominator, then add the numerator.

3\frac{1}{4} = \frac{13}{4}, \quad 1\frac{2}{3} = \frac{5}{3}

Find the LCD and rewrite: The least common denominator of 4 and 3 is 12.

\frac{13}{4} = \frac{39}{12}, \quad \frac{5}{3} = \frac{20}{12}

Add the numerators: Keep the denominator and add the tops.

\frac{39}{12} + \frac{20}{12} = \frac{59}{12}

Convert back to a mixed number: Divide 59 by 12: the quotient is 4 with a remainder of 11.

\frac{59}{12} = 4\frac{11}{12}

Answer:

4\frac{11}{12}

Why It Matters

Adding and subtracting mixed numbers appears constantly in cooking (doubling recipes), measuring materials in shop class, and solving word problems throughout middle school math. Mastering this skill also builds the foundation for working with algebraic fractions in algebra courses.

Common Mistakes

Mistake: Adding the whole numbers and the fractions separately without finding a common denominator first.

Correction: You still need a common denominator for the fraction parts. For example,

\frac{1}{4} + \frac{2}{3}

is not

\frac{3}{7}

. Rewrite them as

\frac{3}{12} + \frac{8}{12} = \frac{11}{12}