Improper fractions and mixed numbers

An improper fraction like 11/4 is more than one whole. As a mixed number it is 2 3/4: two wholes and three quarters.

What improper fractions and mixed numbers are

An improper fraction has a numerator as big as, or bigger than, its denominator: 5/4, 11/4, 9/3. It is one whole or more. (“Improper” is just a name; there is nothing wrong with these fractions, and in algebra they are often the most useful form.)

A mixed number writes the same amount as whole numbers plus a proper fraction: 11/4 = 2 3/4. People usually say mixed numbers in everyday life: “two and three quarter cups”, “one and a half hours”.

See it: pictures to draw with your child

Copy these onto paper or a whiteboard. A quick sketch makes the idea concrete before any written method.

11/4 = 2 3/4
11 quarters fill 2 whole circles with 3 quarters left: 2 3/4.
012311/4
On the number line, 11/4 is three quarters past 2.
2 1/3 = 7/3
2 wholes are 6 thirds; one more third makes 7/3.

Improper fraction to mixed number

Divide the numerator by the denominator. The answer is the whole number, and the remainder is the new numerator.

  1. 11/4: 11 ÷ 4 = 2 remainder 3.
  2. 2 wholes, and 3 quarters left over.
  3. So 11/4 = 2 3/4.

Pictures make this clear: draw 11 quarters, fill whole circles 4 quarters at a time, and count 2 full circles and 3 quarters left. This is the same idea as fractions as division.

Mixed number to improper fraction

Multiply the whole number by the denominator, then add the numerator. Keep the same denominator.

  1. 2 3/4: 2 wholes are 2 × 4 = 8 quarters.
  2. Add the 3 quarters: 8 + 3 = 11.
  3. So 2 3/4 = 11/4.

Children need this direction for multiplying and dividing mixed numbers, which are much easier as improper fractions. The fraction calculator shows this step first.

Mixed numbers on a number line

On a line marked in quarters, 4/4 sits at 1, 8/4 at 2 and 12/4 at 3. So 11/4 is three quarters past 2, the same point as 2 3/4. Children who place a few improper fractions on a number line rarely forget how the two forms match. See fractions on a number line.

Whole numbers as fractions

Some improper fractions are whole numbers: 8/4 = 2, 15/5 = 3, 6/6 = 1. The division leaves no remainder. Any whole number can be written as a fraction with any denominator: 3 = 3/1 = 6/2 = 12/4. This is useful for adding a whole number to a fraction, and for multiplying fractions by whole numbers.

Try it the other way too: ask “how many quarters make 3 wholes?” (12) or “how many thirds make 5?” (15). Quick questions like these make converting between the two forms automatic.

Quiz: convert both ways

Write improper fractions as mixed numbers and mixed numbers as improper fractions.

Where this fits: fractions step by step

  1. Halves and quarters
  2. What are fractions
  3. Unit fractions
  4. Numerator and denominator
  5. Fractions on a number line
  6. Fractions of a set
  7. Equivalent fractions
  8. Simplifying fractions
  9. Comparing fractions
  10. Ordering fractions
  11. Improper fractions and mixed numbers (this page)
  12. Adding fractions
  13. Subtracting fractions
  14. Finding a common denominator
  15. Adding mixed numbers
  16. Subtracting mixed numbers
  17. Fraction of a number
  18. Fractions × whole numbers
  19. Multiplying fractions
  20. Fractions as division
  21. Dividing fractions by whole numbers
  22. Dividing fractions
  23. Fractions to decimals
  24. Fraction word problems

What to say to your child

A short script for teaching it the first time. Pause after each line and let your child do the next step.

  1. “Here are 7 half-sandwiches. How many whole sandwiches can we make?” (3.)
  2. “And any half left over?” (1.) “So 7 halves is 3 and a half.”
  3. “We write 7/2 = 3 1/2.”
  4. “Now go back. 3 1/2: how many halves in 3 wholes?” (6.) “Plus 1 more half?” (7 halves!)
  5. “Multiply, then add.”

More worked examples

Example 1: 13/5 to a mixed number

13 ÷ 5 = 2 r 3. So 13/5 = 2 3/5.

Example 2: 4 2/3 to an improper fraction

4 × 3 = 12, 12 + 2 = 14. So 4 2/3 = 14/3.

Example 3: a whole number

20/4: 20 ÷ 4 = 5 with no remainder, so 20/4 = 5.

Example 4: a measurement

9/2 cups of flour is 4 1/2 cups.

Practice questions (answers below)

Ask these out loud or write them down. Check each answer, and ask “how did you get that?” for one or two.

  1. 7/2 as a mixed number
  2. 9/4 as a mixed number
  3. 17/5 as a mixed number
  4. 12/3 as a whole number
  5. 1 3/4 as an improper fraction
  6. 2 2/5 as an improper fraction
  7. 5 1/3 as an improper fraction
  8. Which is bigger: 11/4 or 2 1/2?
  9. Write 3 as a fraction with denominator 5
  10. Stretch: 47/6 as a mixed number
Show the answers
  1. 3 1/2
  2. 2 1/4
  3. 3 2/5
  4. 4
  5. 7/4
  6. 12/5
  7. 16/3
  8. 11/4 (2 3/4)
  9. 15/5
  10. 7 5/6

Common mistakes

Using the quotient as the numeratorWrites 11/4 = 3 2/4.Fix: the quotient is the whole number and the remainder is the numerator: 2 3/4.
Changing the denominatorWrites 2 3/4 = 11/8.Fix: the denominator stays the same: 11/4.
Adding instead of multiplyingWrites 2 3/4 = 9/4 (2 + 3 + 4).Fix: multiply the whole number by the denominator first: 2 × 4 + 3 = 11.

How to teach it at home

Snack halves

Count halves of sandwiches or apples and say them both ways.

Recipe check

Read 1 1/2 cups as 3/2 cups.

Number line

Mark a ruler or tape at quarters and find 5/4, 9/4 and 11/4.

By grade

More for each grade: Grade 4 math and the CBSE class 4 maths guide; Grade 5 math and the CBSE class 5 maths guide.

Worksheets and practice

Improper fractions and mixed numbers worksheet, page 1

Improper fractions and mixed numbers worksheet: printable PDF with an answer key. · Make a fresh test

Fraction practice · fraction calculator · fraction wall · fractions quiz · fraction levels · fraction games · fractions for kids

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Questions parents ask

What is an improper fraction?

A fraction whose numerator is as big as or bigger than its denominator, like 7/4. It is one whole or more.

What is a mixed number?

A whole number and a proper fraction together, like 1 3/4. It is another way to write an improper fraction.

How do you change an improper fraction to a mixed number?

Divide the numerator by the denominator. The answer is the whole number and the remainder goes over the same denominator: 11/4 = 2 3/4.

How do you change a mixed number to an improper fraction?

Multiply the whole number by the denominator, add the numerator, and keep the denominator: 2 3/4 = (2 × 4 + 3)/4 = 11/4.

What grade learns mixed numbers?

Grade 4 starts mixed numbers, and grades 5 and 6 use them in every fraction operation.