3.03 Converting between fractions and decimals
Introduction
In chapter 2 we saw how we can write parts of numbers smaller than 1 using fractions, and in 3.01 we saw how we can extend the place value table to the left of the ones (units), giving us tenths, hundredths, thousandths and so on.
In this lesson we will look at how to convert between fractions and decimals.
Ideas
Convert fractions to decimals
Think about the names of the columns in our place value table: tenths, hundredths, thousandths and so on. To change a fraction to a decimal, it's easiest if we have a denominator that is a power of 10 so that it matches the fractions in the place value table.
Examples
Example 1
Write the fraction \dfrac{47}{100} as a decimal.
If the denominator of a fraction is a power of 10, write the digits in a place value table to convert it into a decimal.
Fractions without powers of ten in the denominator
If we have a fraction that does not have a power of 10 in the denominator, then we want to find an equivalent fraction that does. We can follow these simple steps:
Find a suitable number to multiply or divide the denominator by to make it a power of 10.
Multiply or divide both the numerator and denominator by this number.
Using a place value table or otherwise, write out the number in decimal form.
Examples
Example 2
Write the fraction \dfrac{17}{25} as a decimal.
To convert a fraction where the denominator is not a power of 10 to a decimal:
Find a suitable number to multiply or divide the denominator by to make it a power of 10.
Multiply or divide both the numerator and denominator by this number.
Using a place value table or otherwise, write out the number in decimal form.
Convert decimals to fractions
We know how to convert between decimals and fractions with powers of 10 in the denominator by using a place value table. This means we can very easily convert any decimal to a fraction with a denominator of 10,\,100,\,1000 and so on. We normally want to write our fractions in simplified form, so we may need to then further simplify the fraction by cancelling common factors from the numerator and denominator.
A benchmark is a reference point which we can use to make converting between various forms easier. There are a few common fractions and decimals that we use as benchmarks that are worth keeping in mind.
From the place value table, we know that 0.1=\dfrac{1}{10}, but what is the decimal equivalent of \dfrac12 and \dfrac14?
Well, \dfrac12we know is equivalent to \dfrac{5}{10}=0.5.
\dfrac14 or "one quarter" is exactly half of one half, so if one half is 50 hundredths, then half of that must be 25 hundredths, or \dfrac{25}{100}=0.25.
Using these benchmarks we can see that \dfrac{3}{4}=0.75 as it is three groups of one quarter, or 3\times 0.25=0.75.
Examples
Example 3
Write the decimal 0.29 as a simplified fraction.
Example 4
Write the decimal 0.535 as a simplified fraction.
To convert decimals to fractions easily, remember that the number of decimal places is equal to the number of zeros required in the denominator.