Mixed Numbers, Improper Fractions and Data Displays | Grade 7 Math | LevelKite

Some amounts are bigger than one whole, such as a run of more than $1$ mile. You can name an amount like that in two ways.

A mixed number shows a number of full wholes together with a fraction less than $1$ , such as $2\frac{3}{4}$ . It means $2$ wholes plus $\frac{3}{4}$ of one more whole.

An improper fraction is a fraction whose numerator (the top number) is equal to or greater than its denominator (the bottom number), such as $\frac{11}{4}$ . The denominator tells you how many equal parts one whole is cut into. The numerator counts how many of those parts you have.

Both names only make sense when you know what the whole is, and when every whole is the same size and cut into the same number of equal parts. No amount in this lesson is negative. The model below shows one amount named both ways.

Each rectangle is one whole cut into

4

equal parts.

11

parts are shaded:

2

full wholes and

3

parts of the third.

Read the model in two ways. The stated whole is one rectangle. Each rectangle is cut into $4$ equal parts, so each part is $\frac{1}{4}$ of a whole.

Count wholes first. Two rectangles are fully shaded, and $3$ of the $4$ parts of the third rectangle are shaded. That is $2\frac{3}{4}$ .
Count parts only. There are $4 + 4 + 3 = 11$ shaded parts, and each one is a fourth. That is $\frac{11}{4}$ .

The shading did not change and the whole did not change, so $2\frac{3}{4}$ and $\frac{11}{4}$ are the same amount. The mixed number shows the full wholes at a glance. The improper fraction counts every part with one denominator.

Key concept

Two names, one amount

A mixed number and an improper fraction name the same amount when both use the same stated whole, cut into the same number of equal parts.

Mixed to improper: cut each full whole into its equal parts, then count all the parts.
Improper to mixed: group the parts into full wholes. Divide the numerator by the denominator. The quotient (the number of full groups) is the number of full wholes. The remainder (what is left over) is the number of extra parts. A remainder of $0$ means there is no fraction part.

In the formula, the left side is the mixed number: $w$ full wholes plus $n$ extra parts, where one whole has $d$ equal parts. All three are whole numbers ( $0$ , $1$ , $2$ and so on), $d$ is not $0$ , and $n$ is less than $d$ .

w + \frac{n}{d} = \frac{w \times d + n}{d}

Worked example

From a mixed number to an improper fraction

Write $3\frac{2}{5}$ as an improper fraction.

Name the whole and the parts. The denominator 5 says each whole is cut into 5 equal parts. So the mixed number means 3 full wholes plus 2 fifths of one more whole of the same size.
$3\frac{2}{5} = 3 + \frac{2}{5}$
Cut each of the 3 full wholes into fifths. One whole holds 5 fifths, so 3 wholes hold 3 groups of 5 fifths, which is 15 fifths. Cutting a whole into equal parts does not change how much there is.
$3 = \frac{3 \times 5}{5} = \frac{15}{5}$
Add the 2 fifths from the fraction part. Every part is a fifth of the same whole, so you can count them together: 15 fifths and 2 fifths make 17 fifths. The denominator stays 5 because the parts are still fifths.
$\frac{15}{5} + \frac{2}{5} = \frac{15 + 2}{5} = \frac{17}{5}$
Check that the size makes sense. With the same whole cut into fifths, 15 fifths is exactly 3 wholes and 20 fifths is exactly 4 wholes. Since 17 is between 15 and 20, the answer lies between 3 and 4, just as the mixed number does.
$\frac{15}{5} < \frac{17}{5} < \frac{20}{5}$

Answer

The mixed number $3\frac{2}{5}$ equals the improper fraction $\frac{17}{5}$ .

Each rectangle is one whole cut into

6

equal parts.

17

parts are shaded.

Worked example

Read a model, then rename the amount

Each rectangle in the model above is one whole. Write the shaded amount as an improper fraction and as a mixed number.

Find the whole and the size of one part. The whole is one rectangle, and each rectangle is cut into 6 equal parts. So every part is one sixth of a whole, and the denominator is 6.
Count the shaded parts, one rectangle at a time: 6 in the first, 6 in the second and 5 in the third. The total number of shaded sixths is the numerator.
$6 + 6 + 5 = 17, \quad \text{so the amount is } \frac{17}{6}$
Group the sixths into full wholes. Every 6 sixths make 1 whole, so divide 17 by 6. Two groups use 12 sixths and leave 5 sixths. Three groups would need 18 sixths, which is too many.
$17 \div 6 = 2 \text{ remainder } 5$
Write the mixed number. The quotient 2 is the number of full wholes, and the remainder 5 is the number of sixths left over. Grouping equal parts into wholes does not change the amount. The picture agrees: 2 rectangles are fully shaded and 5 sixths of the third are shaded.
$\frac{17}{6} = \frac{12}{6} + \frac{5}{6} = 2 + \frac{5}{6} = 2\frac{5}{6}$
Check by going back the other way. Cut the 2 full wholes into sixths and add the 5 extra sixths of the same whole. You get the 17 sixths you counted.
$2 \times 6 + 5 = 12 + 5 = 17$

Answer

The model shows $\frac{17}{6}$ , which is the same amount as the mixed number $2\frac{5}{6}$ .

Fractions greater than $1$ also appear in data displays. A running club recorded how far each of its $18$ members ran on Saturday, to the nearest quarter mile. The line plot below shows the data as marks stacked above a scale of evenly spaced values. Each marked position on the scale is a tick, and the number it stands for is its tick value.

Before you read a fraction from the plot, find three things:

The reference unit. This is what counts as $1$ whole. Here it is $1$ mile.
The tick values. The scale starts at $0$ and each step is $\frac{1}{4}$ mile, so every mile is cut into $4$ equal parts. The tick $7$ steps from $0$ stands for $\frac{7}{4}$ miles, which is the same distance as $1\frac{3}{4}$ miles.
The counts. Each mark is one runner. The number of marks stacked above a tick is the count of runners who ran that distance.

A tick value is a distance. The height of a stack is a number of runners. Keep those two numbers apart.

Line plot

Each mark is one runner, and each step on the scale is

\frac{1}{4}

mile. Five marks sit

6

steps from

0

, at

\frac{6}{4}

miles.

The table below lists the same $18$ runs. It writes every distance in fourths of a mile, so each numerator is the number of $\frac{1}{4}$ -mile steps from $0$ on the plot. The row for $\frac{6}{4}$ matches the tick $6$ steps from $0$ , and its count of $5$ matches the $5$ marks stacked there. Ticks with no marks, such as the ticks for $\frac{2}{4}$ and $\frac{10}{4}$ , have no row.

Every row uses the same whole ( $1$ mile) and the same equal parts (fourths). That is why you can compare two rows by their numerators: more fourths of the same mile means a longer run.

A tick label may show an equal value in another form. The tick $6$ steps from $0$ is $\frac{6}{4}$ miles, which is the same distance as $1\frac{2}{4}$ miles. Since $\frac{2}{4}$ and $\frac{1}{2}$ are equivalent, meaning equal in value, that is also $1\frac{1}{2}$ miles. Also, $\frac{4}{4}$ miles is exactly $1$ mile, because $4 \div 4 = 1$ with remainder $0$ .

Distances run on Saturday (reference unit: 1 mile)

Distance in miles	Number of runners
$\frac{3}{4}$	1
$\frac{4}{4}$	3
$\frac{5}{4}$	2
$\frac{6}{4}$	5
$\frac{7}{4}$	4
$\frac{8}{4}$	2
$\frac{9}{4}$	1

Worked example

Read the line plot and the table

Use the running club's line plot and table. (a) How many runners ran $1\frac{1}{4}$ miles? (b) What is the longest distance anyone ran, written as a mixed number? (c) How many runners ran $2$ miles or more?

Name the whole, the ticks and the counts. The reference unit is 1 mile. Each step on the scale is one fourth of a mile, so every mile is cut into 4 equal parts. Each mark, and each number in the runners column, counts runners.
For (a), rename the distance in fourths so it matches the table. The 1 full mile is 4 fourths, and there is 1 more fourth of the same mile, so there are 5 fourths in all. Renaming does not change the distance.
$1\frac{1}{4} = \frac{1 \times 4 + 1}{4} = \frac{5}{4}$
Read the count for 5 fourths of a mile. On the plot, count 5 steps from 0 and look at the stack there: it has 2 marks. The table row for 5 fourths also shows 2 runners.
For (b), find the mark farthest to the right. It sits 9 steps from 0, and the last table row is 9 fourths of a mile. Group the fourths into full miles: 8 fourths fill 2 miles and 1 fourth is left.
$9 \div 4 = 2 \text{ remainder } 1, \quad \frac{9}{4} = 2\frac{1}{4}$
For (c), rename 2 miles in fourths: each mile is 4 fourths, so 2 miles is 8 fourths. Then add the counts at that tick and at every tick to its right: 2 runners at 8 fourths, 1 runner at 9 fourths and 0 runners at 10 fourths.
$2 = \frac{2 \times 4}{4} = \frac{8}{4}, \quad 2 + 1 + 0 = 3$

Answer

Two runners ran $1\frac{1}{4}$ miles. The longest run was $\frac{9}{4}$ miles, which is $2\frac{1}{4}$ miles. Three runners ran $2$ miles or more.

A mixed number and an improper fraction are two names for the same amount, as long as both use the same whole cut into the same equal parts. To get the improper fraction, cut the full wholes into parts and count every part. To get the mixed number, group the parts into full wholes.

When you read a model, a line plot or a table, first ask what counts as $1$ whole and how many equal parts it has. The denominator counts the equal parts in one whole, even when a picture shows several wholes. Then you can match the tick values, the counts and the fractions.