Mixed Numbers & Improper Fractions

A mixed number has an integer part and a fraction part, e.g. $3\frac{3}{4}$ means "three and three quarters".

An improper fraction (top-heavy fraction) has a numerator greater than or equal to the denominator, e.g. $\frac{15}{4}$.

Any fraction greater than or equal to 1 can be written as either a mixed number or an improper fraction.

$\frac{15}{4}$ means "fifteen quarters" and is equivalent to $3\frac{3}{4}$.

A mixed number is a sum: $3\frac{3}{4} = 3 + \frac{3}{4}$.

An improper fraction represents more than one whole: $\frac{15}{4} = 15$ quarters.

Conversion preserves value; the form changes for convenience in different operations.

Mixed to improper: Multiply the whole number by the denominator, add the numerator, write over the same denominator. $4\frac{6}{7} = \frac{4 \times 7 + 6}{7} = \frac{34}{7}$.

Improper to mixed: Divide the numerator by the denominator. The quotient is the whole part; the remainder over the denominator is the fraction part. $\frac{22}{3} = 7 \text{ remainder } 1 \Rightarrow 7\frac{1}{3}$.

Always convert mixed numbers to improper fractions before multiplying or dividing.

When a question asks for a mixed number, convert your improper fraction at the end.

For multiplication or division, convert mixed numbers to improper fractions first.

Check: multiply the whole by the denominator and add the numerator to verify your improper fraction.

Forgetting to add the numerator: $4\frac{6}{7}$ is $\frac{28+6}{7} = \frac{34}{7}$, not $\frac{28}{7}$.

Wrong remainder: When converting $\frac{17}{5}$, $17 \div 5 = 3$ remainder $2$, so the answer is $3\frac{2}{5}$, not $3\frac{1}{5}$.

Operating on mixed numbers directly: Do not multiply whole parts and fraction parts separately; convert first.

Mixed numbers appear when adding or subtracting fractions that give a result greater than 1.

Improper fractions are required for multiplying and dividing fractions.

Conversion links to division with remainder.