Mixed Numbers and Improper Fractions — How to Convert

By MathPal Editorial Team·Published Apr 21, 2026

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

A mixed number like $3\frac{1}{2}$ combines a whole number and a fraction. An improper fraction like $\frac{7}{2}$ has a numerator larger than its denominator. Both represent the same value — you need to move fluently between the two forms depending on the operation you are performing.

Theory

What is a mixed number?

A mixed number has three parts:

$\underbrace{2}_{\text{whole}}\underbrace{\frac{3}{4}}_{\text{fraction}}$

It means $2 + \frac{3}{4}$. The fraction part is always proper (numerator smaller than denominator).

What is an improper fraction?

An improper fraction has a numerator greater than or equal to its denominator:

$\frac{11}{4}, \quad \frac{7}{3}, \quad \frac{5}{5}$

$\frac{5}{5} = 1$ is technically improper (numerator equals denominator), but is usually just written as 1.

Converting a mixed number to an improper fraction

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

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

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

Converting an improper fraction to a mixed number

Rule: Divide numerator by denominator. Quotient = whole number, remainder = new numerator.

$\frac{17}{5}: \quad 17 \div 5 = 3 \text{ remainder } 2 \quad \Rightarrow \quad 3\frac{2}{5}$

Why improper fractions are easier to compute

When multiplying or dividing fractions, improper fractions simplify the process — you don't need to deal with the whole-number part separately.

Adding with unlike denominators also becomes cleaner once everything is improper:

$1\frac{1}{2} + 2\frac{1}{3} = \frac{3}{2} + \frac{7}{3} = \frac{9}{6} + \frac{14}{6} = \frac{23}{6} = 3\frac{5}{6}$

Worked Examples

Example 1 — Mixed number to improper fraction

Convert $4\frac{3}{7}$ to an improper fraction.

Step 1: Multiply the whole number by the denominator: $4 \times 7 = 28$.

Step 2: Add the numerator: $28 + 3 = 31$.

Step 3: Keep the denominator: $\dfrac{31}{7}$.

Example 2 — Improper fraction to mixed number

Convert $\dfrac{29}{6}$ to a mixed number.

Step 1: Divide: $29 \div 6 = 4$ remainder $5$.

Step 2: The whole number is $4$, the new numerator is $5$, denominator stays $6$.

Answer: $4\dfrac{5}{6}$

Example 3 — Adding mixed numbers

Calculate $2\dfrac{3}{4} + 1\dfrac{1}{2}$.

Step 1: Convert to improper fractions: $\dfrac{11}{4}$ and $\dfrac{3}{2}$.

Step 2: Find a common denominator (LCD = 4): $\dfrac{11}{4} + \dfrac{6}{4} = \dfrac{17}{4}$.

Step 3: Convert back: $17 \div 4 = 4$ remainder $1$ → $4\dfrac{1}{4}$.

Example 4 — Multiplying mixed numbers

Calculate $1\dfrac{2}{3} \times 2\dfrac{1}{4}$.

Step 1: Convert: $\dfrac{5}{3} \times \dfrac{9}{4}$.

Step 2: Multiply numerators and denominators: $\dfrac{5 \times 9}{3 \times 4} = \dfrac{45}{12}$.

Step 3: Simplify: $\dfrac{45}{12} = \dfrac{15}{4} = 3\dfrac{3}{4}$.

Common Mistakes

Mistake 1 — Adding whole numbers and fractions separately without converting

❌ $2\frac{3}{4} + 1\frac{2}{3}$: add whole parts ($2+1=3$) and fraction parts ($\frac{3}{4}+\frac{2}{3}$) separately, then combine.

✅ This method works only when fraction parts don't exceed 1. When the fraction sum $\geq 1$, you must carry over to the whole number. Converting to improper fractions first avoids this trap.

Mistake 2 — Wrong multiplication in the conversion formula

❌ $3\frac{2}{5} = \frac{3+2}{5} = \frac{5}{5}$ (just adding instead of multiplying)

✅ $3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{17}{5}$ — multiply the whole number by the denominator first.

Mistake 3 — Forgetting to simplify after converting back

❌ Leaving an answer as $\frac{8}{6}$ instead of simplifying.

✅ Always simplify: $\frac{8}{6} = \frac{4}{3} = 1\frac{1}{3}$.

Practice Problems

Problem 1: Convert $5\dfrac{3}{8}$ to an improper fraction.

Show Answer

$5 \times 8 + 3 = 43$

$\dfrac{43}{8}$

Problem 2: Convert $\dfrac{37}{9}$ to a mixed number.

Show Answer

$37 \div 9 = 4$ remainder $1$

$4\dfrac{1}{9}$

Problem 3: Calculate $3\dfrac{1}{2} + 2\dfrac{2}{3}$.

Show Answer

$\dfrac{7}{2} + \dfrac{8}{3} = \dfrac{21}{6} + \dfrac{16}{6} = \dfrac{37}{6} = 6\dfrac{1}{6}$

Problem 4: Calculate $2\dfrac{1}{2} \times 1\dfrac{4}{5}$.

Show Answer

$\dfrac{5}{2} \times \dfrac{9}{5} = \dfrac{45}{10} = \dfrac{9}{2} = 4\dfrac{1}{2}$

Problem 5: Order from least to greatest: $2\dfrac{3}{4}$, $\dfrac{13}{5}$, $2\dfrac{1}{2}$.

Show Answer

Convert all to decimals: $2.75$, $2.6$, $2.5$.

Order: $2\dfrac{1}{2} < \dfrac{13}{5} < 2\dfrac{3}{4}$

Summary

• A mixed number combines a whole number and a proper fraction: $a\frac{b}{c}$.
• An improper fraction has numerator ≥ denominator: $\frac{n}{d}$ where $n \geq d$.
• Mixed → improper: multiply whole by denominator, add numerator, keep denominator.
• Improper → mixed: divide numerator by denominator; quotient is whole, remainder is new numerator.
• Use improper fractions when multiplying, dividing, or adding fractions with unlike denominators.

Related Topics

• Fractions — Complete Guide — full coverage of fraction operations
• Adding Fractions with Unlike Denominators — LCD method
• How to Multiply and Divide Fractions — fraction multiplication and division

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