Reading a Mixed Number

What does $3\frac{1}{2}$ mean, and how do you read it?

Find the whole number part: The big number in front is $3$ = 3 complete wholes

Find the fraction part: The fraction next to it is $\frac{1}{2}$, a proper fraction because $1 < 2$ = Half of one more whole

Put the two parts together: $3\frac{1}{2} = 3 + \frac{1}{2}$ = Read: three and one half

Answer: $3\frac{1}{2}$ means 3 wholes plus one half. We read it as "three and one half" or "three and a half".

Writing a Mixed Number from a Picture

There are 2 whole pizzas on the table and a third pizza with 3 of its 8 equal slices left. Write the amount of pizza as a mixed number.

Count the complete wholes: Two pizzas are complete = Whole number part = 2

Find the denominator of the fraction part: The third pizza is cut into 8 equal slices = Denominator = 8

Find the numerator of the fraction part: 3 of those 8 slices are left = Fraction part = $\frac{3}{8}$

Write the mixed number: $2 + \frac{3}{8} = 2\frac{3}{8}$ = $2\frac{3}{8}$

Answer: There are $2\frac{3}{8}$ pizzas on the table.

Drawing a Mixed Number with Fraction Bars

Draw $1\frac{2}{3}$ using fraction bars.

Draw the wholes: The whole number part is $1$, so draw 1 bar and shade all of it = 1 fully shaded bar

Draw one more bar for the fraction part: The denominator is $3$, so split a second bar of the same size into 3 equal parts = A bar with 3 equal parts

Shade the fraction part: The numerator is $2$, so shade 2 of the 3 parts = 2 of 3 parts shaded

Check the drawing: 1 full bar plus $\frac{2}{3}$ of a bar = $1\frac{2}{3}$

Answer: One fully shaded bar and a second bar with 2 of its 3 equal parts shaded show $1\frac{2}{3}$.

Placing a Mixed Number on the Number Line

Mark $2\frac{1}{4}$ on a number line from 0 to 3.

Find the two whole numbers around it: $2\frac{1}{4}$ is more than 2 but less than 3 = Between 2 and 3

Split that space into equal parts: The denominator is $4$, so divide the space from 2 to 3 into 4 equal parts = Marks at $2\frac{1}{4}$, $2\frac{2}{4}$, $2\frac{3}{4}$

Count from the whole number: Start at 2 and move 1 part to the right = The first mark after 2

Answer: $2\frac{1}{4}$ is the first of the four marks after 2, a quarter of the way from 2 to 3.

Mixed Numbers in Time

A bike tour lasts $2\frac{1}{3}$ hours. How many minutes is that?

Change the whole hours to minutes: $2 \times 60 = 120$ = 120 minutes

Change the fraction part to minutes: $\frac{1}{3}$ of 60 minutes is $60 \div 3 = 20$ = 20 minutes

Add the two parts: $120 + 20 = 140$ = 140 minutes

Answer: $2\frac{1}{3}$ hours is 140 minutes, which is 2 hours and 20 minutes.

Mistake: Thinking $2\frac{1}{3}$ means $2 \times \frac{1}{3}$

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

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

Mistake: Writing a fraction part that is not proper, like $1\frac{5}{4}$

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

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

Mistake: Placing $2\frac{1}{4}$ near $\frac{1}{4}$ on the number line

Why: 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 ($\frac{1}{4}$) further to the right.

Mistake: Using the wrong denominator when reading a picture of several shapes

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

Correct: The denominator is the number of parts in ONE whole (4). The answer is $2\frac{3}{4}$.

Cooking and Baking

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

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

Measuring Lengths

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

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

Time and Schedules

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

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

Q: Is $2\frac{1}{4}$ really the same as $2 + \frac{1}{4}$?

A: 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.

Q: Can the fraction part be $\frac{4}{4}$ or bigger?

A: No. $\frac{4}{4}$ is one whole, so it should be added to the whole number part. $3\frac{4}{4}$ is simply $4$, and $3\frac{5}{4}$ is $4\frac{1}{4}$.

Q: Why use mixed numbers when improper fractions exist?

A: Mixed numbers are easier to picture in daily life: $2\frac{1}{2}$ hours tells you right away it is a bit more than 2 hours. Improper fractions like $\frac{5}{2}$ are often easier for calculations. Both name the same amount.

Q: How can I tell which of two mixed numbers is bigger?

A: 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.