Writing Decimals as Fractions
Objective Reinforce your understanding of the
meaning of decimal notation as an equivalent form for fractions
with a power of 10 in their denominator.
This lesson consists of reinforcement of students’
understanding of the decimal expansion. This is a good
opportunity to correct any misunderstandings they may have about
decimal expansion.
Decimal Notation
In a decimal, each digit to the right of the decimal point
represents a fractional value with a denominator that is a power
of 10 (10, 100, 1,000, …). For example, the value
represented by the digit 2 in 0.256 is "two tenths".
The values represented by the digits 5 and 6 are “five
hundredths” and “six thousandths”, respectively. This
means the decimal 0.256 can be represented by the sum .
Example 1
Represent 0.78 as the sum of fractions whose denominators are
powers of 10.
Solution
0.78 is equivalent to the sum .
It is only the digits to the right of the decimal point that
are represented by fractions in a decimal expansion. The digits
to the left of the decimal point represent the wholenumber part
of the decimal.
Example 2
Write 3.56 as a sum involving fractions whose denominators are
powers of 10.
Solution
3.56 can be represented by the sum .
Writing Decimals as Fractions
We can write the fractional parts of a decimal expansion as a
single fraction by writing them using a common denominator.
Example 3
Write 0.256 as a single fraction.
Solution
As shown previously, we know that 0.256 =
The common denominator for these fractions is 1,000.
So, 0.256 is equivalent to the fraction .
Example 4
Write 3.56 as a single fraction.
Solution
In Example 2, it was shown that . Since , the common
denominator is 100.
So, 3.56 is equivalent to .
Alternately, we could add just the fractions in the decimal
expansion , and get a mixed
number as our result.
So the decimal 3.56 is also equivalent to the mixed number .
Although it is easy to give a simple stepbystep procedure
for writing a decimal as a fraction, it is important to do
examples of the kind we have just done so that you can see why
the procedure works.
Writing Decimals as Fractions
1. Count the number of digits to the right of the decimal
point in the given decimal.
2. Drop the decimal point from the decimal, and place the
resulting whole number in the numerator of the fraction.
3. In the denominator, write the power of 10 that has as many
zeros as there were digits to the right of the decimal point in
the given decimal.
Example 5
Write 0.638 as a fraction.
Solution
Step 1 The number of digits to the right of
the decimal point is 3.
Step 2 Dropping the decimal point gives the
whole number 638. Use this number as the numerator of the
fraction.
Step 3 From Step 1, we know that the power of
10 used in the denominator should have 3 zeros. So 1,000 is the
denominator of the fraction.
Example 6
Write 21.67 as a fraction and a mixed number.
Solution
Step 1 The number of digits to the right of
the decimal point is 2.
Step 2 Dropping the decimal point gives the
whole number 2,167. Use this number as the numerator of the
fraction.
Step 3 The denominator of the fraction should
be 100.
As a mixed number, .
