Introduction to Mixed Numbers

Learn what mixed numbers are, how to read and write them, how to draw them and where to find them on the number line.

Elementary20 min18 problems
Practice Problems

Definition

A mixed number is a number made of a whole number and a proper fraction written side by side.
234=2+342\frac{3}{4} = 2 + \frac{3}{4}
  • The whole number part (22) tells how many complete wholes you have
  • The fraction part (34\frac{3}{4}) tells what part of one more whole you have
  • The fraction part is always proper: its numerator is smaller than its denominator
We read 2342\frac{3}{4} as two and three quarters (or two and three fourths). The word and stands for the plus sign that is hidden between the two parts.
A mixed number is always greater than 1, because it has at least one whole. In the lesson on improper fractions you saw that the same amount can also be written as one fraction: 234=1142\frac{3}{4} = \frac{11}{4}. The next lessons show how to switch between the two forms.

Try it now

Which of these is a mixed number?

Worked Examples

What does 3123\frac{1}{2} mean, and how do you read it?
1

Find the whole number part

The big number in front is 33 → 3 complete wholes
2

Find the fraction part

The fraction next to it is 12\frac{1}{2}, a proper fraction because 1<21 < 2 → Half of one more whole
3

Put the two parts together

312=3+123\frac{1}{2} = 3 + \frac{1}{2} → Read: three and one half

Common Mistakes

Thinking 2132\frac{1}{3} means 2×132 \times \frac{1}{3}

Why it's wrong: In algebra, writing things side by side often means multiplying. In a mixed number it does not: the hidden sign is a plus.

Correct: 213=2+132\frac{1}{3} = 2 + \frac{1}{3}. Read the "and" in "two and one third" as plus.

Writing a fraction part that is not proper, like 1541\frac{5}{4}

Why it's wrong: The fraction part must be less than one whole. 54\frac{5}{4} already contains a whole (44\frac{4}{4}), so it belongs in the whole number part.

Correct: Move the extra whole over: 154=2141\frac{5}{4} = 2\frac{1}{4}. In a mixed number the numerator is always smaller than the denominator.

Placing 2142\frac{1}{4} near 14\frac{1}{4} on the number line

Why it's wrong: Looking only at the fraction part forgets the 2 wholes that come first.

Correct: First jump to the whole number (2), then move the fraction part (14\frac{1}{4}) further to the right.

Using the wrong denominator when reading a picture of several shapes

Why it's wrong: With 2 full circles and 3 of 4 parts of a third circle, some students count all the parts and write 1112\frac{11}{12}.

Correct: The denominator is the number of parts in ONE whole (4). The answer is 2342\frac{3}{4}.

Interactive Visual

5
5/3=123= 167%

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Fraction Number Line

02/411 2/422 2/433 2/44

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Practice Problems

18 problems
Problem 1 of 18
Easy
Which of these is a mixed number?

Why It Matters

Mixed numbers are the way people naturally talk about amounts that are more than a whole:
  • Cooking: A recipe asks for 1121\frac{1}{2} cups of milk or 2142\frac{1}{4} teaspoons of salt
  • Time: A film lasts 1341\frac{3}{4} hours, a road trip takes 2122\frac{1}{2} hours
  • Length: A shelf is 1141\frac{1}{4} meters long, a room is 3123\frac{1}{2} meters wide
  • Sports: A runner completes 4124\frac{1}{2} laps of the track
Saying "two and a half hours" is much easier to picture than "five halves of an hour". Mixed numbers show at a glance how many wholes you have and how much is left over.

Real World Applications

Cooking and Baking

Recipes use mixed numbers for amounts bigger than one cup, spoon or kilogram.

Example:

A pancake recipe needs 1121\frac{1}{2} cups of milk: fill the cup once completely and then once halfway.

1Try It Yourself

A bread recipe needs 2122\frac{1}{2} cups of flour, but you only have a half cup measure.

How many half cups of flour do you need?

Step 1: Write the mathematical expression

Two halves in each whole cup, plus the extra half:

Measuring Lengths

Builders, tailors and crafters measure lengths that are a few whole meters plus a part of a meter.

Example:

A curtain that is 1341\frac{3}{4} meters long is 1 meter plus three quarters of another meter, so 175 centimeters.

2Try It Yourself

A ribbon is 2342\frac{3}{4} meters long.

How many centimeters long is the ribbon?

Step 1: Write the mathematical expression

Whole meters in centimeters, plus three quarters of 100 centimeters:

Time and Schedules

Timetables and everyday speech use mixed numbers of hours: an hour and a half, two and a quarter hours.

Example:

A train trip of 1121\frac{1}{2} hours takes 60 + 30 = 90 minutes.

3Try It Yourself

Football training lasts 1141\frac{1}{4} hours.

How many minutes does the training last?

Step 1: Write the mathematical expression

One hour in minutes, plus a quarter of 60 minutes:

Key Takeaways

  • 1A mixed number is a whole number and a proper fraction written together, like 2342\frac{3}{4}
  • 2It means the whole number plus the fraction: 234=2+342\frac{3}{4} = 2 + \frac{3}{4}, read "two and three quarters"
  • 3The fraction part is always proper: its numerator is smaller than its denominator
  • 4To draw a mixed number, shade the full wholes, then shade the fraction part of one more whole
  • 5On the number line, a mixed number sits between its whole number and the next one: 2142\frac{1}{4} is between 2 and 3
  • 6The same amount can also be written as an improper fraction, like 234=1142\frac{3}{4} = \frac{11}{4}

Frequently Asked Questions

Yes. A mixed number is a short way of writing a sum of a whole number and a fraction. The plus sign is just left out.
Yes. A mixed number is a short way of writing a sum of a whole number and a fraction. The plus sign is just left out.
No. 44\frac{4}{4} is one whole, so it should be added to the whole number part. 3443\frac{4}{4} is simply 44, and 3543\frac{5}{4} is 4144\frac{1}{4}.
Mixed numbers are easier to picture in daily life: 2122\frac{1}{2} hours tells you right away it is a bit more than 2 hours. Improper fractions like 52\frac{5}{2} are often easier for calculations. Both name the same amount.
Compare the whole number parts first, and only if they are equal compare the fraction parts. The lesson Comparing Mixed Numbers explains this in detail.

Glossary

Mixed number
A number made of a whole number and a proper fraction, like 3123\frac{1}{2}
Whole number part
The number in front of the fraction; it tells how many complete wholes there are
Fraction part
The proper fraction in a mixed number; it shows the part of one more whole
Proper fraction
A fraction whose numerator is smaller than its denominator, like 34\frac{3}{4}; it is less than 1
Improper fraction
A fraction whose numerator is equal to or bigger than its denominator, like 74\frac{7}{4}; it is 1 or more

Formula Card

Meaning of a mixed number

abc=a+bca\frac{b}{c} = a + \frac{b}{c}

A mixed number is the whole number $a$ plus the fraction $\frac{b}{c}$

Rule for the fraction part

b<cb < c

The numerator of the fraction part is always smaller than the denominator

Position on the number line

a<abc<a+1a < a\frac{b}{c} < a + 1

A mixed number lies between its whole number part and the next whole number

Today's challengeOne problem, three tries

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