Mixed numbers

What is a mixed number?

A mixed number combines a whole number and a proper fraction: 2 3/4 means 2 wholes and 3/4 more. You see mixed numbers in recipes ("1 1/2 cups of flour"), measurements ("5 3/8 inches") and time ("2 1/2 hours").

Mixed numbers and improper fractions

Every mixed number can also be written as an improper fraction:

2 3/4 = 4/4 + 4/4 + 3/4 = 11/4

Both describe the same amount. Mixed numbers are easier to picture; improper fractions are easier to calculate with.

Converting a mixed number to an improper fraction

1. Multiply the whole number by the denominator.
2. Add the numerator.
3. Put the result over the same denominator.

Example: 3 2/5

• 3 × 5 = 15
• 15 + 2 = 17
• 17/5

Converting an improper fraction to a mixed number

1. Divide the numerator by the denominator.
2. The quotient is the whole number.
3. The remainder is the new numerator; keep the denominator.

Example: 23/6

• 23 ÷ 6 = 3 R5
• 3 5/6

A quick table

Mixed numbers on a number line

2 3/4 sits between 2 and 3, three quarters of the way to 3. Counting in quarters: 2, 2 1/4, 2 2/4, 2 3/4, 3.

Comparing mixed numbers

Compare the whole numbers first. If they’re the same, compare the fractions.

• 3 1/8 > 2 7/8 (3 wholes beats 2 wholes)
• 4 2/3 > 4 1/2 (same wholes; 2/3 > 1/2)

Worked examples

Example 1: Write 4 1/3 as an improper fraction.

4 × 3 + 1 = 13 → 13/3.

Example 2: Write 29/8 as a mixed number.

29 ÷ 8 = 3 R5 → 3 5/8.

Example 3: A recipe needs 9/4 cups of milk. How is that written as a mixed number?

9 ÷ 4 = 2 R1 → 2 1/4 cups.

Common mistakes

• Adding the whole number to the numerator without multiplying: writing 3 2/5 as 5/5.
• Changing the denominator when converting. It stays the same.
• Forgetting to simplify the fraction part, e.g. 2 4/8 is better as 2 1/2.

Try it yourself

1. Convert 2 5/6 to an improper fraction.
2. Convert 7 1/2 to an improper fraction.
3. Convert 17/3 to a mixed number.
4. Convert 40/9 to a mixed number.
5. Which is bigger: 3 3/5 or 18/5?

Answers: 1) 17/6 2) 15/2 3) 5 2/3 4) 4 4/9 5) They are equal (3 3/5 = 18/5).

Quick recap

• A mixed number is a whole number plus a proper fraction.
• Mixed → improper: multiply, add, keep the denominator.
• Improper → mixed: divide; the remainder is the new numerator.