Compare and Order Fractions, Decimals and Mixed Numbers | Grade 5 Math | LevelKite

You can write the same number in more than one form. A form is a way of writing a number. This lesson uses three forms: fractions, decimals and mixed numbers.

A mixed number is a whole number and a fraction written together, such as $2\frac{3}{5}$ . It means $2$ wholes plus $\frac{3}{5}$ of one more whole. The fraction part is less than $1$ .

A decimal such as $2.6$ works the same way. The digits before the decimal point show the whole number part: $2$ wholes. The digits after the point show a part that is less than $1$ : here $0.6$ , which is $6$ tenths.

So each form has a whole number part and a part less than $1$ . When you change a mixed number into a decimal, or a decimal into a mixed number, the wholes stay the same. Only the part less than $1$ needs a new form. You already know how to change a fraction into a decimal and a decimal into a fraction, so you are ready.

Worked example

From a mixed number to a decimal

Write $2\frac{3}{5}$ as a decimal.

Split the mixed number into its two parts. You can do this because a mixed number means a whole number plus a fraction.
$2\frac{3}{5} = 2 + \frac{3}{5}$
Keep the whole number part, 2. Convert only the fraction part. Each fifth is the same amount as 2 tenths, so 3 fifths is 6 tenths. Six tenths is written 0.6.
$\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10} = 0.6$
Join the two parts again. The 2 wholes go in the ones place and the 6 tenths go in the tenths place.
$2 + 0.6 = 2.6$

Answer

As a decimal, $2\frac{3}{5}$ is $2.6$ . The whole number part, $2$ , stayed the same.

The same steps work in the other direction. Take the decimal $4.75$ .

Split it. $4.75 = 4 + 0.75$ .
Keep the whole number part. It stays $4$ .
Convert the part less than one. The decimal $0.75$ is $75$ hundredths, so $0.75 = \frac{75}{100}$ . Each group of $25$ hundredths makes one fourth, so $75$ hundredths make three fourths: $\frac{75}{100} = \frac{3}{4}$ .
Join the parts. $4 + \frac{3}{4} = 4\frac{3}{4}$ .

So $4.75 = 4\frac{3}{4}$ . In both directions the whole number part stays the same. This works because the part you convert is less than $1$ before and after, so it never adds a whole or takes one away.

Next, you will compare and order numbers in different forms. To compare two numbers is to decide which one is greater, or to find that they are equal. To order a list of numbers is to arrange them by size, for example from least to greatest.

Every fraction, decimal and mixed number is a point on the number line. On the number lines in this lesson, numbers get greater as you move to the right. So the greater number is the point farther to the right. Equal numbers, such as $2\frac{3}{5}$ and $2.6$ , share one point.

Different forms can hide which point is farther right. Which is greater, $\frac{2}{5}$ or $0.25$ ? It is hard to tell just by looking. So start by writing both numbers in the same form. The same form is only the first move, though. Before you compare by counting parts, the parts must be the same size. Watch for that in the next example.

Worked example

Compare a fraction with a decimal

Which is greater, $\frac{2}{5}$ or $0.25$ ?

Write both numbers in the same form. Here, change the fraction into a decimal. Each fifth is 2 tenths, so 2 fifths is 4 tenths.
$\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10} = 0.4$
Both numbers are decimals now, but they do not count parts of the same size. The decimal 0.4 counts tenths and the decimal 0.25 counts hundredths. Make the parts the same size. One tenth is 10 hundredths, so 4 tenths is 40 hundredths. Write 0.4 with two decimal places, which means two digits after the decimal point.
$0.4 = \frac{4}{10} = \frac{40}{100} = 0.40$
Now both numbers count hundredths, so you can compare the counts: 40 hundredths is more than 25 hundredths. You can also check place by place from the left. Both numbers have 0 wholes. In the tenths place, 4 is more than 2.
$0.40 > 0.25$
Show both numbers on the number line below. It goes from 0 to 1 in steps of 0.05, which is 5 hundredths. The decimal 0.25 is 25 hundredths, so it is 5 steps from 0. The decimal 0.40 is 40 hundredths, so it is 8 steps from 0. The point 0.4 is farther to the right, so it is the greater number.

Answer

$\frac{2}{5}$ is greater than $0.25$ . In symbols, $\frac{2}{5} > 0.25$ .

From 0 to 1, each fifth is 4 steps, so

\frac{2}{5}

is the point

0.4

, 8 steps from 0. The point

0.25

is 5 steps from 0, to the left of

0.4

.

Key concept

One form first, then same-size parts

A number keeps its value when you change its form.

To convert a mixed number or a decimal, keep the whole number part and convert only the part less than $1$ .

To compare or order numbers, write them all in one form first. That alone does not make the parts the same size: $0.25$ counts hundredths and $0.4$ counts tenths. To compare by counting parts, first make the parts the same size. Use equal decimal places ( $0.40$ and $0.25$ ) or a common denominator, the same bottom number ( $\frac{40}{100}$ and $\frac{25}{100}$ ).

With decimals you can also go place by place from the left: whole number parts, then tenths, then hundredths. The first place where the digits differ shows which number is greater. Or see which point is farther right on one number line.

Worked example

Put four numbers in order

Put these numbers in order from least to greatest: $1.4$ , $\frac{4}{5}$ , $1\frac{1}{4}$ , $0.65$ .

Write every number in one form. Decimals work well here, because fifths change into tenths and fourths change into hundredths. Start with the fraction. Each fifth is 2 tenths, so 4 fifths is 8 tenths.
$\frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10} = 0.8$
Now the mixed number. Keep the whole number part, 1, and convert only the fraction part. One fourth is 25 hundredths, because 4 groups of 25 hundredths make 100 hundredths, which is 1 whole.
$1\frac{1}{4} = 1 + \frac{1}{4} = 1 + \frac{25}{100} = 1 + 0.25 = 1.25$
The four decimals are 1.4, 0.8, 1.25 and 0.65. Two of them count tenths and two count hundredths, so the parts are not all the same size yet. Give every decimal two decimal places. Four tenths is 40 hundredths, and 8 tenths is 80 hundredths.
$1.4 = 1.40 \qquad 0.8 = 0.80$
Compare the whole number parts first. The decimals 0.65 and 0.80 have 0 wholes, so they are less than 1. The decimals 1.25 and 1.40 have 1 whole and some more, so they are greater than 1 and come after the other two. Then compare the hundredths inside each pair: 65 is less than 80, and 25 is less than 40.
$0.65 < 0.80 < 1.25 < 1.40$
Write the order again in the forms you were given. The decimal 0.80 came from four fifths, 1.25 came from 1 and one fourth, and 1.40 came from 1.4.
$0.65 < \frac{4}{5} < 1\frac{1}{4} < 1.4$

Answer

From least to greatest, the order is $0.65$ , $\frac{4}{5}$ , $1\frac{1}{4}$ , $1.4$ .

From left to right, the points are

0.65

,

0.8

,

1.25

and

1.4

. In the forms from the example, that is

0.65

,

\frac{4}{5}

,

1\frac{1}{4}

,

1.4

.

The number line above shows the part from $0.5$ to $1.5$ . Each step is $0.05$ , which is $5$ hundredths. Find each number from the example.

$0.65$ is $3$ steps to the right of $0.5$ .
$\frac{4}{5}$ is the point $0.8$ , because $4$ fifths is $8$ tenths.
$1\frac{1}{4}$ is the point $1.25$ . Start at $1$ and move $5$ steps to the right, because $5$ steps of $0.05$ make $0.25$ , which is one fourth.
$1.4$ is $8$ steps to the right of $1$ .

Read the points from left to right: $0.65$ , $\frac{4}{5}$ , $1\frac{1}{4}$ , $1.4$ . That is the same order as in the example. The two points to the left of $1$ are the numbers with $0$ wholes. The two points to the right of $1$ are the numbers with $1$ whole.

Remember: a number keeps its value when you change its form.

To convert between a mixed number and a decimal, keep the whole number part and convert only the part less than $1$ .
To compare or order numbers in different forms, write them all in one form first. Then make the parts the same size, with equal decimal places or a common denominator, before you compare by counting parts.
You can also compare whole number parts first and then go place by place from the left, or see which point is farther right on one number line.