# Converting between decimals and fractions

### Converting between decimals and fractions

#### Lessons

In this lesson, we will learn:

• What are equivalent decimal fractions (denominators that are powers of 10)
• How to convert between decimals and decimal fractions
• How to convert between decimals and non-decimal fractions
• How to represent decimal fractions with base ten (block) models

Notes:

• Decimals are also the short form of a fraction! We talked about decimals as the place values smaller than the ones place, but they are also another way to write fractions.

• Decimal fractions are fractions with a denominator that is a power of 10 (Ex. $\frac{5}{10}, \frac{73}{100}, \frac{426}{1000}$)

• How do we convert between decimals and decimal fractions?
• To convert a decimal into a decimal fraction:
• Take all the digits of the decimal and put them in the fraction’s numerator as a whole number (and remove any leading zeroes)
• Ex. 0.073= $\frac{73}{?}$
• Look at the number of decimal place values in your decimal, that’s how many zeroes you will put in your fraction’s denominator
• Ex. 0.07 3 =$\frac{73}{1000}$
• To convert a decimal fraction into a decimal:
• Look at the number zeroes in the denominator, that’s how many decimal place values you will have in your decimal
• Ex. $\frac{73}{1000}$ = 0. _ _ _
• Take all the numbers in your numerator; start from the smallest place values on the right and fill in the number backwards (back-fill them); any empty place values will be filled in with leading zeroes.
• Ex. $\frac{73}{1000}$ = 0. _ _ _ $\,$$\,$ 0. _ _ 3 $\,$$\,$ 0. _ 7 3 $\,$$\,$ 0. 0 7 3

• Trailing zeroes are not important in the value of a decimal number.
• Ex. 0.5 and 0.50 are the same! This is because $\frac{5}{10}$ = $\frac{50}{100}$
• Tenths and hundredths are easily converted into equivalent fractions (factor of 10). This is also true for hundredths and thousandths
• Ex. 0.50 = 0.500 because $\frac{50}{100}$ = $\frac{500}{1000}$

• What are NOT decimal fractions?
• Other fractions with denominators that are NOT powers of ten are non-decimal fractions (Ex. $\frac{1}{2}$, $\frac{2}{5}$, $\frac{5}{18}$, $\frac{7}{25}$, $\frac{547}{900}$)
• Some non-decimal fractions have denominators that are factors or multiples of powers of 10
• They can be converted into equivalent decimal fractions
• The common fractions that you should know the decimal values for are:

• To represent decimal fractions with base ten (block) models:
• (1) figure out what pieces represents what place value
• (2) tally the number of each tenth, hundredth, thousandth
• (3) and finally, write in standard form and/or fraction form

• We can represent more than one whole for base ten models and decimals.
• Fractions that represent greater than one whole are mixed fractions
• Follow the same three steps as before
• The number of complete wholes is written as a big number on the left side
• The decimal fraction is written on the right side
• Introduction
Introduction to converting between decimals and fractions:
a)
What are the different ways to show decimals?

b)
What are decimal fractions?

c)
How do we convert between decimal fractions and decimals?

d)
What are leading and trailing zeroes?

e)
What are equivalent decimal fractions?

f)
What are non-decimal fractions and how do we convert them into decimals?

g)
How do we show decimal fractions using base ten (block) models?

• 1.
Converting between decimals and decimal fractions
Recall that decimal fractions are fractions with denominators that are powers of 10 (ex. 10, 100, 1000, etc.).
a)
Turn each decimal fraction into a decimal:
1. $\frac{85}{100}$
2. $\frac{6}{1000}$
3. $\frac{76}{10}$

b)
Turn each decimal into a decimal fraction:
1. .836
2. 0.4
3. .75

c)
Use decimal fractions to write this decimal in expanded form: 1.529

• 2.
Equivalent tenth and hundredth decimal fractions
Fill in the chart to understand equivalent tenths and hundredths:

a)
$\frac{1}{10}$

b)
$\frac{8}{10}$

c)
$\frac{10}{10}$

• 3.
Converting between decimals and non-decimal fractions
Recall that non-decimal fractions are fractions with denominators that are NOT powers of 10 (i.e. any other numbers besides 10, 100, 1000, etc.)
a)
Turn each fraction into a decimal:
1. $\frac{1}{2}$
2. $\frac{3}{4}$
3. $\frac{4}{5}$
4. $\frac{1}{25}$

b)
Turn each decimal into a fraction in lowest terms:
1. 0.50
2. 2.75
3. 1.08
4. 3.6

• 4.
Writing fractions from base ten (block) models
Write the decimal and fraction represented by the shaded parts of each base ten (block) model:
a)

b)

c)

d)

e)

• 5.
Decimals and fractions word problem:
Jimmy and his friends are drinking juice together. After 10 minutes, they measure how much juice they each have left over in their cups and turn those amounts into decimal fractions. Who changed their decimals into fractions correctly?
a)
Jimmy's glass has 0.2 L of orange juice and he says, "that's a fraction of 0.20 L".

b)
Noah has 0.68 L of apple juice in his glass and he says, "that's a fraction of $\frac{68}{10}$".

c)
Ben has 0.075 L of mango juice and he says, "that's a fraction of $\frac{75}{1000}$".