One Amount, Two Names: Fractions and Decimals | Grade 5 Math | LevelKite

A fraction and a decimal can look very different. Yet they can name the very same amount. It is like one person who has two names.

In this lesson you will convert between the two. To convert a number means to write the same amount in a different form. The amount does not change. Only the way you write it changes.

Start with a picture, so you can see the amount before you name it.

Six of the ten equal parts are shaded:

\frac{6}{10} = 0.6

.

Look at the bar. It is one whole cut into 10 equal parts, so each part is one tenth of the whole. Six parts are shaded.

You can name the shaded amount in three ways:

In words: six tenths
As a fraction: $\frac{6}{10}$
As a decimal: $0.6$

The fraction says 6 parts out of 10 equal parts. The decimal has a 6 in the tenths place, so it also says 6 tenths. Both names point to the same shaded amount, so $\frac{6}{10} = 0.6$ .

Forty-five of the hundred equal parts are shaded:

\frac{45}{100} = 0.45

.

Now look at the grid. It is one whole cut into 10 rows and 10 columns. That makes 100 equal parts, so each small part is one hundredth of the whole. There are 45 shaded parts.

In words: forty-five hundredths
As a fraction: $\frac{45}{100}$
As a decimal: $0.45$

In $0.45$ , the last digit sits in the hundredths place. That is why you read it as 45 hundredths. Once again the fraction and the decimal name the same shaded amount, so $\frac{45}{100} = 0.45$ .

Key concept

One amount, two names

A fraction and a decimal can name the same amount. A decimal with one, two or three digits after the decimal point is a short way to write a fraction whose denominator is 10, 100 or 1,000. The denominator is the bottom number of a fraction. It tells how many equal parts make one whole.

The place of the decimal's last digit tells you the denominator:

tenths place: 10
hundredths place: 100
thousandths place: 1,000

Converting changes the name of the amount. It never changes the amount.

From a fraction to a decimal. The first three decimal places count tenths, hundredths and thousandths. So when a fraction has a denominator of 10, 100 or 1,000, you can write it as a decimal right away. You saw this with $\frac{6}{10} = 0.6$ .

What if the denominator is a different number, such as 5 or 8? Then you rename the fraction first. To rename a fraction means to write an equal fraction that has a different denominator. Here you rename by multiplying the numerator (the top number, which counts the parts you have) and the denominator by the same whole number greater than 1. This cuts every part into that many smaller equal pieces. You count more pieces, but they cover the same amount.

This method works when you can multiply the denominator by a whole number to get 10, 100 or 1,000. Try 10 first, then 100, then 1,000. Not every denominator can do this. No whole number times 3 gives 10, 100 or 1,000, so this method does not work for $\frac{1}{3}$ .

Worked example

Rename fifths as tenths

Convert $\frac{3}{5}$ to a decimal.

Look for 10, 100 or 1,000 as the new denominator. Try 10 first. Since 5 times 2 is 10, you can rename fifths as tenths.
$5 \times 2 = 10$
Multiply the numerator and the denominator by 2. This cuts each fifth into 2 equal pieces, so 3 fifths become 6 tenths. The amount stays the same.
$\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}$
Write 6 tenths as a decimal. The tenths place is the first place after the decimal point, so the 6 goes there. This is the same 6 tenths you saw shaded in the bar at the start of the lesson.
$\frac{6}{10} = 0.6$

Answer

The fraction $\frac{3}{5}$ is equal to the decimal $0.6$ .

Worked example

Rename eighths as thousandths

Convert $\frac{3}{8}$ to a decimal.

Look for a new denominator. Try 10: 8 times 1 is 8 and 8 times 2 is 16, so 10 is skipped. Try 100: 8 times 12 is 96 and 8 times 13 is 104, so 100 is skipped. Try 1,000: 8 times 125 is exactly 1,000. So you can rename eighths as thousandths.
$8 \times 125 = 1{,}000$
Multiply the numerator and the denominator by 125. The pieces get smaller, but the amount stays the same.
$\frac{3}{8} = \frac{3 \times 125}{8 \times 125} = \frac{375}{1{,}000}$
Write 375 thousandths as a decimal. The thousandths place is the third place after the decimal point, so the last digit, 5, must sit there.
$\frac{375}{1{,}000} = 0.375$

Answer

The fraction $\frac{3}{8}$ is equal to the decimal $0.375$ .

More fractions renamed. In the last row, a 0 holds the tenths place so the 8 sits in the hundredths place.

Fraction	Multiply numerator and denominator by	Renamed fraction	Decimal
$\frac{1}{2}$	$5$	$\frac{5}{10}$	$0.5$
$\frac{3}{4}$	$25$	$\frac{75}{100}$	$0.75$
$\frac{13}{20}$	$5$	$\frac{65}{100}$	$0.65$
$\frac{2}{25}$	$4$	$\frac{8}{100}$	$0.08$

From a decimal to a fraction. Now go the other way. A decimal tells you its fraction when you read it aloud. Read $0.75$ aloud and you say seventy-five hundredths, which is $\frac{75}{100}$ . Find that row in the table and read it from right to left. Dividing 75 and 100 by 25 takes you back to $\frac{3}{4}$ . You are undoing the renaming.

These steps work for a decimal between 0 and 1 that has one, two or three digits after the decimal point:

Find the place of the last digit. It gives the denominator: tenths means 10, hundredths means 100, thousandths means 1,000.
Use the number after the decimal point as the numerator. Leave off any zeros at the front, so $0.08$ has a numerator of 8.
Write the fraction in lowest terms. A fraction is in lowest terms when 1 is the only whole number that divides both the numerator and the denominator evenly (with nothing left over). To get there, divide both by a whole number greater than 1 that divides each of them evenly. Repeat until no such number is left.

Worked example

From a decimal to lowest terms

Convert $0.45$ to a fraction in lowest terms.

Find the place of the last digit. In 0.45, the last digit, 5, is in the hundredths place. So the decimal says 45 hundredths: the numerator is 45 and the denominator is 100. These are the 45 shaded parts out of 100 that you saw in the grid.
$0.45 = \frac{45}{100}$
Find a whole number greater than 1 that divides both 45 and 100 evenly. Both numbers end in 5 or 0, so 5 divides both. Divide the numerator and the denominator by 5. This joins every 5 hundredths into 1 twentieth, so the amount stays the same.
$\frac{45}{100} = \frac{45 \div 5}{100 \div 5} = \frac{9}{20}$
Check for lowest terms. The whole numbers that divide 9 evenly are 1, 3 and 9. Of these, only 1 also divides 20 evenly. No whole number greater than 1 divides both 9 and 20, so the fraction is in lowest terms.

Answer

The decimal $0.45$ is equal to the fraction $\frac{9}{20}$ , which is in lowest terms.

A fraction and a decimal can be two names for one amount. A whole cut into 10 or 100 equal parts lets you see both names at once.

To convert a fraction to a decimal, rename it with a denominator of 10, 100 or 1,000, then write the decimal. This works when a whole number times the denominator gives 10, 100 or 1,000.
To convert a decimal that is between 0 and 1 into a fraction, let the place of the last digit give the denominator. Then write the fraction in lowest terms.

Next, you will use these ideas with mixed numbers, which have a whole number part and a fraction part. You will also compare and order fractions and decimals by writing them in one form.