Lesson 10: Writing Division Answers as Mixed Numbers

New Concept

A fraction that is equal to 1 or is greater than 1 is called an improper fraction. To convert a mixed number to an improper fraction, multiply the denominator by the whole number, add the numerator, and place the result over the original denominator.

$5\frac{1}{3} = \frac{3 \times 5 + 1}{3} = \frac{16}{3}$

What’s next

This is just the foundation. Next, we'll walk through worked examples, visual breakdowns, and challenge problems using these conversion skills.

Property

When dividing, instead of writing the answer with a remainder, we write it as a mixed number. The remainder becomes the numerator of the fraction, and the divisor becomes the denominator. For a division like $25 \div 4$, the answer is $6\frac{1}{4}$.

Examples

An 25-inch ribbon cut into 4 equal lengths: $25 \div 4 = 6\frac{1}{4}$ inches long for each piece.
A 35-inch ribbon cut into 4 equal lengths: $35 \div 4 = 8\frac{3}{4}$ inches long for each piece.
Dividing 100% by 3 to find what percent a third of a circle is: 100\% \div 3 = 33\tfrac{1}{3}\%.

Explanation

Sometimes a remainder just doesn't make sense! If you're splitting a 25-inch ribbon for 4 friends, saying "6 R 1" isn't helpful. Instead, we divide that leftover inch among the 4 pieces, giving everyone a fair share of $6\frac{1}{4}$ inches. This way, the answer actually solves the real-world problem correctly and fairly for everyone involved.

Property

A fraction that is equal to 1 or is greater than 1 is called an improper fraction. These can be rewritten as whole numbers or mixed numbers by performing the division indicated by the fraction bar, such as converting $\frac{5}{3}$ to $1\frac{2}{3}$.

Examples

Convert the improper fraction $\frac{5}{3}$ to a mixed number: $\frac{5}{3} = \frac{3}{3} + \frac{2}{3} = 1\frac{2}{3}$.
Convert the improper fraction $\frac{6}{3}$ to a whole number: $\frac{6}{3} = \frac{3}{3} + \frac{3}{3} = 2$.
Simplify the sum of two fractions: $\frac{4}{5} + \frac{4}{5} = \frac{8}{5} = 1\frac{3}{5}$.

Explanation

An improper fraction is like having too many slices for one pizza! If a pizza has 4 slices, and you have 5 slices ($\frac{5}{4}$), you really have one whole pizza and one extra slice. So, we convert it to a mixed number, like $1\frac{1}{4}$, to make it easier to understand how many "whole" things you have.

Property

To convert a mixed number to an improper fraction, multiply the denominator by the whole number and then add the numerator. This sum becomes the new numerator, placed over the original denominator. For example: $12\frac{1}{2} = \frac{2 \times 12 + 1}{2} = \frac{25}{2}$.

Examples

Convert $3\frac{1}{3}$ to an improper fraction: $3\frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3}$.
Convert $2\frac{3}{4}$ to an improper fraction: $2\frac{3}{4} = \frac{4 \times 2 + 3}{4} = \frac{11}{4}$.
Convert $5\frac{1}{2}$ to an improper fraction: $5\frac{1}{2} = \frac{2 \times 5 + 1}{2} = \frac{11}{2}$.

Explanation

Let's go backwards! To turn a mixed number like $3\frac{1}{4}$ into just a fraction, think about cutting all the whole things into slices. Three whole pizzas cut into fourths gives you $3 \times 4 = 12$ slices. Add the extra $\frac{1}{4}$ slice that you already had, and you have a grand total of 13 slices, or $\frac{13}{4}$.