Mixed-Improper Conversion Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Mixed-Improper Conversion.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The process of converting between mixed-number form and improper-fraction form, which represent the same value.

Mixed to improper: multiply the whole number by the denominator, add the numerator, keep the denominator. Improper to mixed: divide numerator by denominator to get the whole part and remainder.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Conversion changes the name, not the amount.

Common stuck point: The procedure for mixed-improper conversion is the easy part; the trap is multiplying the whole number by the numerator. Asking "Am I changing notation while keeping the same point on the number line?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I changing notation while keeping the same point on the number line?

Worked Examples

Example 1

easy

Convert

3\frac{2}{7}

to an improper fraction.

Answer

\frac{23}{7}

First step

Multiply the whole number by the denominator:

3 \times 7 = 21

Full solution

2
Add the numerator: $21 + 2 = 23$ .
3
Place over the original denominator: $\frac{23}{7}$ .

The formula for converting a mixed number to an improper fraction is

w\frac{a}{b} = \frac{wb + a}{b}

. Multiply the whole part by the denominator, add the numerator, and keep the denominator unchanged.

Example 2

medium

Convert

\frac{41}{6}

to a mixed number, then verify by converting back to an improper fraction.

Example 3

medium

Convert

5\frac{7}{12}

to an improper fraction, then check by converting back.

Example 4

medium

Without dividing, explain why

\dfrac{50}{7}

is between

7

and

8

, then give its mixed form.

Example 5

hard

A board is

7\frac{1}{4}

feet long, and we cut pieces of

\dfrac{3}{4}

ft each. How many full pieces, and what fraction left over?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Convert

5\frac{3}{8}

to an improper fraction.

Example 2

hard

Subtract

\frac{29}{5} - 3\frac{2}{5}

by first converting to a common form.

Example 3

easy

Convert

2\frac{1}{3}

to an improper fraction.

Example 4

easy

Convert

1\frac{3}{4}

to an improper fraction.

Example 5

easy

Convert

3\frac{2}{5}

to an improper fraction.

Example 6

easy

Convert

\frac{9}{2}

to a mixed number.

Example 7

easy

Convert

\frac{7}{3}

to a mixed number.

Example 8

easy

Convert

\frac{11}{4}

to a mixed number.

Example 9

easy

Convert

4\frac{1}{2}

to an improper fraction.

Example 10

easy

Convert

\frac{8}{4}

to a whole or mixed number.

Example 11

medium

Convert

5\frac{3}{8}

to an improper fraction, then state the numerator.

Example 12

medium

Convert

\frac{23}{5}

to a mixed number.

Example 13

medium

Which is larger,

3\frac{1}{4}

\frac{15}{4}

Example 14

medium

Convert

\frac{30}{7}

to a mixed number.

Example 15

medium

A board is

\frac{17}{4}

feet long. Write its length as a mixed number.

Example 16

medium

Convert

6\frac{5}{6}

to an improper fraction.

Example 17

medium

Convert

\frac{45}{6}

to a mixed number and simplify the fraction part.

Example 18

medium

Convert

\frac{19}{3}

to a mixed number.

Example 19

medium

Convert

\frac{50}{8}

to a mixed number and simplify the fraction part.

Example 20

challenge

A recipe needs

\frac{23}{4}

cups of flour. You only have a

\frac{1}{4}

-cup scoop. How many scoops, and what does that equal as a mixed number of cups?

Example 21

challenge

Find a whole number

n

so that

n\frac{2}{5}=\frac{32}{5}

Example 22

challenge

Without full division, explain why

\frac{100}{9}

is between 11 and 12, then give its mixed form.

Example 23

easy

Convert

4\frac{1}{5}

to an improper fraction.

Example 24

easy

Convert

2\frac{3}{8}

to an improper fraction.

Example 25

easy

Convert

\dfrac{13}{5}

to a mixed number.

Example 26

easy

Convert

\dfrac{25}{4}

to a mixed number.

Example 27

easy

A ribbon is

\dfrac{9}{4}

meters long. Write it as a mixed number.

Example 28

medium

Convert

\dfrac{47}{8}

to a mixed number.

Example 29

medium

Which is larger,

4\frac{2}{3}

\dfrac{15}{3}

Example 30

medium

Add

2\frac{1}{6} + 1\frac{2}{3}

by first converting to improper fractions.

Example 31

medium

Subtract

\dfrac{27}{4} - 3\frac{1}{2}

Example 32

medium

Convert

\dfrac{72}{10}

to a mixed number and simplify the fraction part.

Example 33

medium

A pizza is cut into

8

slices and

\dfrac{19}{8}

pizzas are eaten. How many whole pizzas and what fraction of a pizza?

Example 34

medium

Convert

\dfrac{100}{8}

to a mixed number and simplify.

Example 35

medium

Order from smallest to largest:

\dfrac{17}{5},\ 3\frac{1}{2},\ \dfrac{18}{5}

Example 36

hard

Find a whole number

n

so that

n\frac{3}{7} = \dfrac{45}{7}

Example 37

hard

Express

\dfrac{200}{9}

as a mixed number.

Example 38

hard

A recipe needs

2\frac{2}{3}

cups of sugar; you make

4

batches. How many cups total?

Example 39

hard

Find the value:

3\frac{1}{2} \times \dfrac{4}{7}

Example 40

challenge

For what positive integers

n

does

\dfrac{n}{6}

equal a mixed number with whole part exactly

4

Example 41

challenge

Find all positive integers

a < 10

such that

a\frac{1}{2}

equals an improper fraction with numerator a perfect square.

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Related Concepts

Mixed Numbers Improper Fractions Adding Fractions with Unlike Denominators Multiplying Fractions

Background Knowledge

These ideas may be useful before you work through the harder examples.

mixed numbersimproper fractions