Lesson 62: Writing Mixed Numbers as Improper Fractions

New Concept

To convert a mixed number to an improper fraction, multiply the denominator by the whole number, add the numerator, and keep the original denominator.

What’s next

This is a foundational skill for working with fractions. Next, you'll apply this technique in worked examples and practice problems, including multiplication.

Property

To convert a mixed number like $3\frac{5}{6}$ to an improper fraction, change the whole number into a fraction with the same denominator. Then, add this to the fraction part.

Examples

$2\frac{3}{4} = (2 \times \frac{4}{4}) + \frac{3}{4} = \frac{8}{4} + \frac{3}{4} = \frac{11}{4}$
$1\frac{2}{5} = (1 \times \frac{5}{5}) + \frac{2}{5} = \frac{5}{5} + \frac{2}{5} = \frac{7}{5}$
$4\frac{1}{3} = (4 \times \frac{3}{3}) + \frac{1}{3} = \frac{12}{3} + \frac{1}{3} = \frac{13}{3}$

Explanation

Imagine you have pies! To change $3\frac{5}{6}$ into just slices, first convert the whole pies. Each whole pie has 6 sixths, so three pies give you $3 \times 6 = 18$ slices. Now, add the 5 extra slices from the fractional part. Altogether, that's $18 + 5 = 23$ yummy slices, giving you the improper fraction $\frac{23}{6}$.

Property

To find the numerator of an improper fraction, multiply the denominator by the whole number and then add the original numerator. The denominator remains the same.

Examples

$3\frac{1}{4} = \frac{(4 \times 3) + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}$
$2\frac{4}{5} = \frac{(5 \times 2) + 4}{5} = \frac{10 + 4}{5} = \frac{14}{5}$
$6\frac{1}{2} = \frac{(2 \times 6) + 1}{2} = \frac{12 + 1}{2} = \frac{13}{2}$

Explanation

Ready for a math shortcut? To quickly turn a mixed number into an improper fraction, use this simple trick. Multiply the denominator (bottom number) by the whole number, then add the numerator (top number) to get your new top number. The denominator stays the same—it’s the magic ingredient that doesn't change! It’s a fast-pass to the answer.

Property

To multiply mixed numbers, you must first convert each mixed number into an improper fraction. After converting, multiply the numerators together and the denominators together.

Examples

$1\frac{2}{3} \times 2\frac{2}{5} = \frac{5}{3} \times \frac{12}{5} = \frac{60}{15} = 4$
$1\frac{1}{2} \times 2\frac{1}{3} = \frac{3}{2} \times \frac{7}{3} = \frac{21}{6} = \frac{7}{2}$
$2\frac{1}{4} \times 1\frac{1}{5} = \frac{9}{4} \times \frac{6}{5} = \frac{54}{20} = \frac{27}{10}$

Explanation

Multiplying mixed numbers is a two-step dance! First, you must change your partners by converting each mixed number into its improper fraction costume. Only then can they dance. Multiply the numerators together and the denominators together. Trying to multiply them before they change costumes will lead to a dance disaster and a completely wrong answer!