Mixed Numbers and Improper Fractions

Learn how to convert between mixed numbers and improper fractions, and how to add and subtract mixed numbers.

What Are Mixed Numbers and Improper Fractions?

A mixed number has a whole-number part and a fraction part. For example: $2\frac{3}{4}$ means "2 whole and three quarters".

An improper fraction has a numerator (top) larger than or equal to its denominator (bottom). For example: $\frac{11}{4}$

These two things represent the same amount:

$2\frac{3}{4} = \frac{11}{4}$

Note

Mixed numbers are easier to picture (you can imagine two whole pizzas and three slices of a fourth). Improper fractions are easier to calculate with.

Converting: Mixed Number → Improper Fraction

Rule: Multiply the whole number by the denominator, then add the numerator.

$a\frac{b}{c} = \frac{a \times c + b}{c}$

Example: Convert $3\frac{2}{5}$ to an improper fraction.

$3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5}$

Converting: Improper Fraction → Mixed Number

Rule: Divide the numerator by the denominator. The quotient is the whole number; the remainder is the new numerator.

Example: Convert $\frac{23}{6}$ to a mixed number.

$23 \div 6 = 3$ remainder $5$

$\frac{23}{6} = 3\frac{5}{6}$

Tip

When adding or subtracting mixed numbers, convert both to improper fractions first, do the calculation, then convert back.

Adding Mixed Numbers

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

Convert: $\frac{3}{2} + \frac{7}{3}$

Common denominator is $6$ : $\frac{9}{6} + \frac{14}{6} = \frac{23}{6}$

Convert back: $3\frac{5}{6}$

Warning

Never add whole numbers and fractions separately without making sure the fractions share the same denominator. $1\frac{1}{2} + 2\frac{1}{3} \neq 3\frac{2}{5}$ !

Real world: cooking measurements

A recipe needs $2\frac{1}{2}$ cups of flour for one batch of cookies. To make 3 batches, multiply: $3 \times 2\frac{1}{2} = 3 \times \frac{5}{2} = \frac{15}{2} = 7\frac{1}{2}$ cups. Converting to an improper fraction first makes the multiplication straightforward.

Real world: wood-cutting in carpentry

A carpenter has a board $5\frac{3}{4}$ m long and cuts off $2\frac{1}{3}$ m. Remaining length $= \frac{23}{4} - \frac{7}{3} = \frac{69}{12} - \frac{28}{12} = \frac{41}{12} = 3\frac{5}{12}$ m. Working with improper fractions avoids errors when subtracting across the whole-number boundary.

Tip Convert to improper fractions before any arithmetic — it is much easier to add, subtract, multiply, and divide fractions than mixed numbers directly.

Practice Problems

Convert $2\frac{3}{7}$ to an improper fraction.

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$2 \times 7 + 3 = 17$. Answer: $\dfrac{17}{7}$.

Convert $\dfrac{19}{4}$ to a mixed number.

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$19 \div 4 = 4$ remainder $3$. Answer: $4\dfrac{3}{4}$.