Lesson 10
A STORY OF RATIOS
8•7
Lesson 10: Converting Repeating Decimals to Fractions
Student Outcomes

Students know intuitively that every repeating decimal is equal to a fraction. Students convert a decimal that
eventually repeats into a fraction.

Students know that the decimal expansions of rational numbers repeat eventually.

Students understand that irrational numbers are numbers that are not rational. Irrational numbers cannot be
represented as a fraction and have infinite decimals that never repeat.
Classwork
Discussion (4 minutes)

We have just seen that every fraction (therefore every rational number) is equal to a repeating decimal, and
we have learned strategies for determining the decimal expansion of fractions. Now we must learn how to
write a repeating decimal as a fraction.

We begin by noting a simple fact about finite decimals: Given a finite decimal, such as 1.2345678, if we
multiply the decimal by 105 , we get 123,456.78. That is, when we multiply by a power of 10, in this case 105 ,
the decimal point is moved 5 places to the right, i.e.,
1.2345678 × 105 = 123 456.78
This is true because of what we know about the laws of exponents:
105 × 1.2 345 678 = 105 × (12 345,678 × 10−7 )
= 12 345 678 × 10−2

= 123 456.78
We have discussed in previous lessons that we treat infinite decimals as finite decimals in order to compute
with them. For that reason, we will now apply the same basic fact we observed about finite decimals to
infinite decimals. That is,
1.2345678 … × 105 = 123 456.78 ….
We will use this fact to help us write infinite decimals as fractions.
Example 1 (10 minutes)
Example 1
����.
Find the fraction that is equal to the infinite decimal 𝟎𝟎. 𝟖𝟖𝟖𝟖

����. Why might we want to write an infinite
We want to find the fraction that is equal to the infinite decimal 0. 81
decimal as a fraction?
Lesson 10:
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Converting Repeating Decimals to Fractions
126
Lesson 10
A STORY OF RATIOS
8•7
Provide time for students to discuss a reason for wanting to do this. In previous lessons, we said that infinite decimals
must be approximated in order to compute with them. When we compute with approximations, our answers are not
precise. If we can rewrite an infinite decimal as a fraction, we can more easily compute, thus leading to a more precise
answer. It may be necessary to simply tell them, but let them think first.


����.
For the stated reason, we must figure out how to find the fraction that is equal to the infinite decimal 0. 81
���.
We let 𝑥𝑥 = 0. �81
���. Students should recognize that
Allow students time to work in pairs or small groups to write the fraction equal to 0. �81
the preceding discussion has something to do with this process and should be an entry point for finding the solution.
MP.1
����, an equation of some form will lead them to the fraction. Give
They should also recognize that since we let 𝑥𝑥 = 0. 81
them time to make sense of the problem. Make a plan for finding the fraction, and then attempt to figure it out.

����, we will multiply both sides of the equation by 102 and then solve for 𝑥𝑥. We will multiply by
Since 𝑥𝑥 = 0. 81
102 because there are two decimal digits that repeat immediately following the decimal point.
����
𝑥𝑥 = 0. 81
𝑥𝑥 = 0.818 181 81 …
102 𝑥𝑥 = (102 )0.818 181 81 …
100𝑥𝑥 = 81.818 181 …
Ordinarily, we would finish solving for 𝑥𝑥 by dividing both sides of the equation by 100. Do you see why that is
not a good plan for this problem?

If we divide both sides by 100, we would get 𝑥𝑥 =
81.818181…
, which does not really show us that the
100
repeating decimal is equal to a fraction (rational number) because the repeating decimal is still in the
numerator.

We know that 81.818181 … is the same as 81 + 0.818181 …. Then by substitution, we have
100𝑥𝑥 = 81 + 0.818181 ….
MP.7
����?
How can we rewrite 100𝑥𝑥 = 81 + 0.818181 … in a useful way using the fact that 𝑥𝑥 = 0. 81
We can rewrite 100𝑥𝑥 = 81 + 0.818181 … as 100𝑥𝑥 = 81 + 𝑥𝑥 because 𝑥𝑥 represents the repeating
decimal block 0.818181 ….
����:
Now we can solve for 𝑥𝑥 to find the fraction that represents the repeating decimal 0. 81

100𝑥𝑥 = 81 + 𝑥𝑥
100𝑥𝑥 − 𝑥𝑥 = 81 + 𝑥𝑥 − 𝑥𝑥
(100 − 1)𝑥𝑥 = 81

���� =
Therefore, the repeating decimal 0. 81
9
.
11
99𝑥𝑥 = 81
99𝑥𝑥 81
=
99
99
81
𝑥𝑥 =
99
9
𝑥𝑥 =
11
Have students use calculators to verify that we are correct.
Lesson 10:
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Converting Repeating Decimals to Fractions
127
Lesson 10
A STORY OF RATIOS
8•7
Exercises 1–2 (5 minutes)
Students complete Exercises 1–2 in pairs. Allow them to use calculators to check their work.
Exercises 1–2
1.
a.
Let 𝒙𝒙 = 𝟎𝟎. ������
𝟏𝟏𝟏𝟏𝟏𝟏. Explain why multiplying both sides of this equation by 𝟏𝟏𝟎𝟎𝟑𝟑 will help us determine the
fractional representation of 𝒙𝒙.
When we multiply both sides of the equation by 𝟏𝟏𝟎𝟎𝟑𝟑 , on the right side we will have 𝟏𝟏𝟏𝟏𝟏𝟏. 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 …. This is
helpful because we will be able to subtract the repeating decimal from both sides by subtracting 𝒙𝒙.
b.
After multiplying both sides of the equation by 𝟏𝟏𝟎𝟎𝟑𝟑 , rewrite the resulting equation by making a substitution
that will help determine the fractional value of 𝒙𝒙. Explain how you were able to make the substitution.
𝒙𝒙 = 𝟎𝟎. ������
𝟏𝟏𝟏𝟏𝟏𝟏
������
𝟏𝟏𝟎𝟎𝟑𝟑 𝒙𝒙 = (𝟏𝟏𝟎𝟎𝟑𝟑 )𝟎𝟎. 𝟏𝟏𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏𝟏𝟏. ������
𝟏𝟏𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝟎𝟎. 𝟏𝟏𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏𝟏𝟏 …
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝒙𝒙
c.
d.
������, we can substitute the repeating decimal 𝟎𝟎. 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 … with 𝒙𝒙.
Since we let 𝒙𝒙 = 𝟎𝟎. 𝟏𝟏𝟏𝟏𝟏𝟏
Solve the equation to determine the value of 𝒙𝒙.
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 − 𝒙𝒙 = 𝟏𝟏𝟏𝟏𝟏𝟏 + 𝒙𝒙 − 𝒙𝒙
𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗 = 𝟏𝟏𝟏𝟏𝟏𝟏
𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗 𝟏𝟏𝟏𝟏𝟏𝟏
=
𝟗𝟗𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗𝟗𝟗
𝟏𝟏𝟏𝟏𝟏𝟏
𝒙𝒙 =
𝟗𝟗𝟗𝟗𝟗𝟗
𝟒𝟒𝟒𝟒
𝒙𝒙 =
𝟑𝟑𝟑𝟑𝟑𝟑
Is your answer reasonable? Check your answer using a calculator.
Yes, my answer is reasonable and correct. It is reasonable because the denominator cannot be expressed as a
product of 𝟐𝟐’s and 𝟓𝟓’s; therefore, I know that the fraction must represent an infinite decimal. It is also
reasonable because the decimal value is closer to 𝟎𝟎 than to 𝟎𝟎. 𝟓𝟓, and the fraction
to
2.
𝟏𝟏
. It is correct because the division of
𝟐𝟐
𝟒𝟒𝟒𝟒
𝟑𝟑𝟑𝟑𝟑𝟑
using a calculator is 𝟎𝟎. 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 ….
𝟒𝟒𝟒𝟒
𝟑𝟑𝟑𝟑𝟑𝟑
is also closer to 𝟎𝟎 than
�. Check your answer using a calculator.
Find the fraction equal to 𝟎𝟎. 𝟒𝟒
�
Let 𝒙𝒙 = 𝟎𝟎. 𝟒𝟒
�
𝒙𝒙 = 𝟎𝟎. 𝟒𝟒
�
𝟏𝟏𝟏𝟏𝟏𝟏 = (𝟏𝟏𝟏𝟏)𝟎𝟎. 𝟒𝟒
�
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟒𝟒. 𝟒𝟒
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟒𝟒 + 𝒙𝒙
𝟏𝟏𝟏𝟏𝟏𝟏 − 𝒙𝒙 = 𝟒𝟒 + 𝒙𝒙 − 𝒙𝒙
𝟗𝟗𝟗𝟗 = 𝟒𝟒
𝟗𝟗𝟗𝟗 𝟒𝟒
=
𝟗𝟗
𝟗𝟗
𝟒𝟒
𝒙𝒙 =
𝟗𝟗
Lesson 10:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
Converting Repeating Decimals to Fractions
128
Lesson 10
A STORY OF RATIOS
8•7
Example 2 (6 minutes)
Example 2
�.
Find the fraction that is equal to the infinite decimal 𝟐𝟐. 𝟏𝟏𝟏𝟏𝟖𝟖

We want to find the fraction that is equal to the infinite decimal 2.138� . Notice that this time there is just one
digit that repeats, but it is three places to the right of the decimal point. If we let 𝑥𝑥 = 2.138� , by what power of
10 should we multiply? Explain.
The goal is to multiply by a power of 10 so that the only remaining decimal digits are those that repeat.
For that reason, we should multiply by 102 .
We let 𝑥𝑥 = 2.138� , and multiply both sides of the equation by 102 .


𝑥𝑥 = 2.138�
102 𝑥𝑥 = (102 )2.138�
100𝑥𝑥 = 213. 8�
100𝑥𝑥 = 213 + 0. 8�
This time, we cannot simply subtract 𝑥𝑥 from each side. Explain why.
Subtracting 𝑥𝑥 in previous problems allowed us to completely remove the repeating decimal. This time,
𝑥𝑥 = 2.138� , not just 0. 8� .
What we will do now is treat 0. 8� as a separate, mini-problem. Determine the fraction that is equal to 0. 8� .



Let 𝑦𝑦 = 0. 8� .
𝑦𝑦 = 0. 8�
10𝑦𝑦 = 8. 8�
10𝑦𝑦 = 8 + 0. 8�
10𝑦𝑦 = 8 + 𝑦𝑦
10𝑦𝑦 − 𝑦𝑦 = 8 + 𝑦𝑦 − 𝑦𝑦
9𝑦𝑦 = 8
9𝑦𝑦 8
=
9
9
8
𝑦𝑦 =
9
Lesson 10:
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Converting Repeating Decimals to Fractions
129
Lesson 10
A STORY OF RATIOS

8•7
8
Now that we know that 0. 8� = , we will go back to our original problem.
9
100𝑥𝑥 = 213 + 0. 8�
8
100𝑥𝑥 = 213 +
9
213 × 9 8
+
100𝑥𝑥 =
9
9
213 × 9 + 8
100𝑥𝑥 =
9
1925
100𝑥𝑥 =
9
1925 1
1
(100𝑥𝑥) =
�
�
9
100
100
1925
𝑥𝑥 =
900
77
𝑥𝑥 =
36
Exercises 3–4 (6 minutes)
Students complete Exercises 3–4 independently or in pairs. Allow students to use calculators to check their work.
Exercises 3–4
3.
����. Check your answer using a calculator.
Find the fraction equal to 𝟏𝟏. 𝟔𝟔𝟐𝟐𝟐𝟐
����
Let 𝒙𝒙 = 𝟏𝟏. 𝟔𝟔𝟐𝟐𝟐𝟐
����
𝒙𝒙 = 𝟏𝟏. 𝟔𝟔𝟐𝟐𝟐𝟐
����
Let 𝒚𝒚 = 𝟎𝟎. 𝟐𝟐𝟐𝟐
����
𝟏𝟏𝟏𝟏𝟏𝟏 = (𝟏𝟏𝟏𝟏)𝟏𝟏. 𝟔𝟔𝟐𝟐𝟐𝟐
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏. ����
𝟐𝟐𝟐𝟐
𝒚𝒚 = 𝟎𝟎. ����
𝟐𝟐𝟐𝟐
����
𝟏𝟏𝟎𝟎𝟐𝟐 𝒚𝒚 = (𝟏𝟏𝟎𝟎𝟐𝟐 )𝟎𝟎. 𝟐𝟐𝟐𝟐
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟐𝟐𝟐𝟐. ����
𝟐𝟐𝟐𝟐
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟐𝟐𝟐𝟐 + 𝒚𝒚
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 − 𝒚𝒚 = 𝟐𝟐𝟐𝟐 + 𝒚𝒚 − 𝒚𝒚
𝟗𝟗𝟗𝟗𝟗𝟗 = 𝟐𝟐𝟐𝟐
𝟗𝟗𝟗𝟗𝟗𝟗 𝟐𝟐𝟐𝟐
=
𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗
𝟐𝟐𝟐𝟐
𝒚𝒚 =
𝟗𝟗𝟗𝟗
���� =
𝟏𝟏. 𝟔𝟔𝟐𝟐𝟐𝟐
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏
𝟗𝟗𝟗𝟗𝟗𝟗
Lesson 10:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
Converting Repeating Decimals to Fractions
����
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏. 𝟐𝟐𝟐𝟐
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏 + 𝒚𝒚
𝟐𝟐𝟐𝟐
𝟗𝟗𝟗𝟗
𝟏𝟏𝟏𝟏 × 𝟗𝟗𝟗𝟗 𝟐𝟐𝟐𝟐
𝟏𝟏𝟏𝟏𝟏𝟏 =
+
𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗
𝟏𝟏𝟏𝟏 × 𝟗𝟗𝟗𝟗 + 𝟐𝟐𝟐𝟐
𝟏𝟏𝟏𝟏𝟏𝟏 =
𝟗𝟗𝟗𝟗
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏 =
𝟗𝟗𝟗𝟗
𝟏𝟏
𝟏𝟏 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏
(𝟏𝟏𝟏𝟏𝟏𝟏) =
�
�
𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏 𝟗𝟗𝟗𝟗
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏
𝒙𝒙 =
𝟗𝟗𝟗𝟗𝟗𝟗
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏 +
130
Lesson 10
A STORY OF RATIOS
����. Check your answer using a calculator.
Find the fraction equal to 𝟐𝟐. 𝟗𝟗𝟔𝟔𝟔𝟔
4.
����
Let 𝒙𝒙 = 𝟐𝟐. 𝟗𝟗𝟔𝟔𝟔𝟔
����
𝒙𝒙 = 𝟐𝟐. 𝟗𝟗𝟔𝟔𝟔𝟔
����
𝟏𝟏𝟏𝟏𝟏𝟏 = (𝟏𝟏𝟏𝟏)𝟐𝟐. 𝟗𝟗𝟔𝟔𝟔𝟔
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟐𝟐𝟐𝟐. ����
𝟔𝟔𝟔𝟔
����
Let 𝒚𝒚 = 𝟎𝟎. 𝟔𝟔𝟔𝟔
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟔𝟔𝟔𝟔. ����
𝟔𝟔𝟔𝟔
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟔𝟔𝟔𝟔 + 𝒚𝒚
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 − 𝒚𝒚 = 𝟔𝟔𝟔𝟔 + 𝒚𝒚 − 𝒚𝒚
𝟗𝟗𝟗𝟗𝟗𝟗 = 𝟔𝟔𝟔𝟔
���� =
𝟐𝟐. 𝟗𝟗𝟔𝟔𝟔𝟔
𝟗𝟗𝟗𝟗𝟗𝟗
𝟑𝟑𝟑𝟑𝟑𝟑
����
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟐𝟐𝟐𝟐. 𝟔𝟔𝟔𝟔
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟐𝟐𝟐𝟐 + 𝒚𝒚
����
𝒚𝒚 = 𝟎𝟎. 𝟔𝟔𝟔𝟔
����
𝟏𝟏𝟎𝟎𝟐𝟐 𝒚𝒚 = (𝟏𝟏𝟎𝟎𝟐𝟐 )𝟎𝟎. 𝟔𝟔𝟔𝟔
𝟗𝟗𝟗𝟗𝟗𝟗 𝟔𝟔𝟔𝟔
=
𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗
𝟔𝟔𝟔𝟔
𝒚𝒚 =
𝟗𝟗𝟗𝟗
𝟐𝟐𝟐𝟐
𝒚𝒚 =
𝟑𝟑𝟑𝟑
8•7
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟐𝟐𝟐𝟐 +
𝟐𝟐𝟐𝟐
𝟑𝟑𝟑𝟑
𝟐𝟐𝟐𝟐 × 𝟑𝟑𝟑𝟑 + 𝟐𝟐𝟐𝟐
𝟏𝟏𝟏𝟏𝟏𝟏 =
𝟑𝟑𝟑𝟑
𝟗𝟗𝟗𝟗𝟗𝟗
𝟏𝟏𝟏𝟏𝟏𝟏 =
𝟑𝟑𝟑𝟑
𝟏𝟏
𝟏𝟏 𝟗𝟗𝟗𝟗𝟗𝟗
� � 𝟏𝟏𝟏𝟏𝟏𝟏 = � �
𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏 𝟑𝟑𝟑𝟑
𝟗𝟗𝟗𝟗𝟗𝟗
𝒙𝒙 =
𝟑𝟑𝟑𝟑𝟑𝟑
𝟏𝟏𝟏𝟏𝟏𝟏 =
𝟐𝟐𝟐𝟐 × 𝟑𝟑𝟑𝟑
𝟐𝟐𝟐𝟐
𝟑𝟑𝟑𝟑
+
Discussion (4 minutes)

What we have observed so far is that when an infinite decimal repeats, it can be written as a fraction, which
means that it is a rational number. Do you think infinite decimals that do not repeat are rational as well?
Explain.
Provide students time to discuss with a partner before sharing their thoughts with the class.


Considering the work from this lesson, it does not seem reasonable that an infinite decimal that does
not repeat can be expressed as a fraction. We would not have a value that we could set for 𝑥𝑥 and use
to compute in order to find the fraction. For those reasons, we do not believe that an infinite decimal
that does not repeat is a rational number.
Infinite decimals that do not repeat are irrational numbers; in other words, when a number is not equal to a
rational number, it is irrational. What we will learn next is how to use rational approximation to determine the
approximate decimal expansion of an irrational number.
Closing (5 minutes)
Summarize, or ask students to summarize, the main points from the lesson:

We know that to work with infinite decimals we must treat them as finite decimals.

We know how to use our knowledge of powers of 10 and linear equations to write an infinite decimal that
repeats as a fraction.

We know that every decimal that eventually repeats is a rational number.
Lesson 10:
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Converting Repeating Decimals to Fractions
131
Lesson 10
A STORY OF RATIOS
8•7
Lesson Summary
Numbers with decimal expansions that repeat are rational numbers and can be converted to fractions using a linear
equation.
Example: Find the fraction that is equal to the number 𝟎𝟎. ������
𝟓𝟓𝟓𝟓𝟓𝟓.
������.
Let 𝒙𝒙 represent the infinite decimal 𝟎𝟎. 𝟓𝟓𝟓𝟓𝟓𝟓
������
𝒙𝒙 = 𝟎𝟎. 𝟓𝟓𝟓𝟓𝟓𝟓
𝟏𝟏𝟏𝟏 𝒙𝒙 = 𝟏𝟏𝟏𝟏𝟑𝟑 �𝟎𝟎. ������
𝟓𝟓𝟓𝟓𝟓𝟓�
𝟑𝟑
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟓𝟓𝟓𝟓𝟓𝟓. ������
𝟓𝟓𝟓𝟓𝟓𝟓
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟓𝟓𝟓𝟓𝟓𝟓 + 𝟎𝟎. ������
𝟓𝟓𝟓𝟓𝟓𝟓
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟓𝟓𝟓𝟓𝟓𝟓 + 𝒙𝒙
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 − 𝒙𝒙 = 𝟓𝟓𝟓𝟓𝟓𝟓 + 𝒙𝒙 − 𝒙𝒙
𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗 = 𝟓𝟓𝟓𝟓𝟓𝟓
𝟗𝟗𝟗𝟗𝟗𝟗
𝟓𝟓𝟓𝟓𝟓𝟓
𝒙𝒙 =
𝟗𝟗𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗𝟗𝟗
𝟓𝟓𝟓𝟓𝟓𝟓
𝟔𝟔𝟔𝟔
𝒙𝒙 =
=
𝟗𝟗𝟗𝟗𝟗𝟗 𝟏𝟏𝟏𝟏𝟏𝟏
Multiply by 𝟏𝟏𝟎𝟎𝟑𝟑 because there are 𝟑𝟑 digits that repeat.
Simplify.
By addition
By substitution; 𝒙𝒙 = 𝟎𝟎. ������
𝟓𝟓𝟓𝟓𝟓𝟓
Subtraction property of equality
Simplify.
Division property of equality
Simplify.
�, do not
This process may need to be used more than once when the repeating digits, as in numbers such as 𝟏𝟏. 𝟐𝟐𝟔𝟔
begin immediately after the decimal.
Irrational numbers are numbers that are not rational. They have infinite decimals that do not repeat and cannot be
represented as a fraction.
Exit Ticket (5 minutes)
Lesson 10:
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Converting Repeating Decimals to Fractions
132
Lesson 10
A STORY OF RATIOS
Name
8•7
Date
Lesson 10: Converting Repeating Decimals to Fractions
Exit Ticket
1.
�����.
Find the fraction equal to 0. 534
2.
����.
Find the fraction equal to 3.015
Lesson 10:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
Converting Repeating Decimals to Fractions
133
Lesson 10
A STORY OF RATIOS
8•7
Exit Ticket Sample Solutions
1.
Find the fraction equal to 𝟎𝟎. ������
𝟓𝟓𝟓𝟓𝟓𝟓.
Let 𝒙𝒙 = 𝟎𝟎. ������
𝟓𝟓𝟓𝟓𝟓𝟓.
𝒙𝒙 = 𝟎𝟎. ������
𝟓𝟓𝟓𝟓𝟓𝟓
������
𝟏𝟏𝟎𝟎𝟑𝟑 𝒙𝒙 = (𝟏𝟏𝟎𝟎𝟑𝟑 )𝟎𝟎. 𝟓𝟓𝟓𝟓𝟓𝟓
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟓𝟓𝟓𝟓𝟓𝟓. ������
𝟓𝟓𝟓𝟓𝟓𝟓
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟓𝟓𝟓𝟓𝟓𝟓 + 𝒙𝒙
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 − 𝒙𝒙 = 𝟓𝟓𝟓𝟓𝟓𝟓 + 𝒙𝒙 − 𝒙𝒙
𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗 = 𝟓𝟓𝟓𝟓𝟓𝟓
������ =
𝟎𝟎. 𝟓𝟓𝟓𝟓𝟓𝟓
2.
𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗 𝟓𝟓𝟓𝟓𝟓𝟓
=
𝟗𝟗𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗𝟗𝟗
𝟓𝟓𝟓𝟓𝟓𝟓
𝒙𝒙 =
𝟗𝟗𝟗𝟗𝟗𝟗
𝟏𝟏𝟏𝟏𝟏𝟏
𝒙𝒙 =
𝟑𝟑𝟑𝟑𝟑𝟑
𝟏𝟏𝟕𝟕𝟕𝟕
𝟑𝟑𝟑𝟑𝟑𝟑
����.
Find the fraction equal to 𝟑𝟑. 𝟎𝟎𝟏𝟏𝟏𝟏
����
Let 𝒙𝒙 = 𝟑𝟑. 𝟎𝟎𝟏𝟏𝟏𝟏
����
𝒙𝒙 = 𝟑𝟑. 𝟎𝟎𝟏𝟏𝟏𝟏
����
𝟏𝟏𝟏𝟏𝟏𝟏 = (𝟏𝟏𝟏𝟏)𝟑𝟑. 𝟎𝟎𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟑𝟑𝟑𝟑. ����
𝟏𝟏𝟏𝟏
����
Let 𝒚𝒚 = 𝟎𝟎. 𝟏𝟏𝟏𝟏
𝒚𝒚 = 𝟎𝟎. ����
𝟏𝟏𝟏𝟏
����
𝟏𝟏𝟎𝟎 𝒚𝒚 = (𝟏𝟏𝟎𝟎𝟐𝟐 )𝟎𝟎. 𝟏𝟏𝟏𝟏
𝟐𝟐
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏. ����
𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏 + 𝒚𝒚
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 − 𝒚𝒚 = 𝟏𝟏𝟏𝟏 + 𝒚𝒚 − 𝒚𝒚
𝟗𝟗𝟗𝟗𝟗𝟗 = 𝟏𝟏𝟏𝟏
𝟗𝟗𝟗𝟗𝟗𝟗 𝟏𝟏𝟏𝟏
=
𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗
𝟓𝟓
𝒚𝒚 =
𝟑𝟑𝟑𝟑
���� =
𝟑𝟑. 𝟎𝟎𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏
𝟔𝟔𝟔𝟔
Lesson 10:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
Converting Repeating Decimals to Fractions
����
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟑𝟑𝟑𝟑. 𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟑𝟑𝟑𝟑 + 𝒚𝒚
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟑𝟑𝟑𝟑 +
𝟓𝟓
𝟑𝟑𝟑𝟑
𝟑𝟑𝟑𝟑 × 𝟑𝟑𝟑𝟑 + 𝟓𝟓
𝟏𝟏𝟏𝟏𝟏𝟏 =
𝟑𝟑𝟑𝟑
𝟗𝟗𝟗𝟗𝟗𝟗
𝟏𝟏𝟏𝟏𝟏𝟏 =
𝟑𝟑𝟑𝟑
𝟏𝟏
𝟏𝟏 𝟗𝟗𝟗𝟗𝟗𝟗
(𝟏𝟏𝟏𝟏𝟏𝟏) =
�
�
𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏 𝟑𝟑𝟑𝟑
𝟗𝟗𝟗𝟗𝟗𝟗
𝒙𝒙 =
𝟑𝟑𝟑𝟑𝟑𝟑
𝟏𝟏𝟏𝟏𝟏𝟏
𝒙𝒙 =
𝟔𝟔𝟔𝟔
𝟏𝟏𝟏𝟏𝟏𝟏 =
𝟑𝟑𝟑𝟑 × 𝟑𝟑𝟑𝟑
𝟓𝟓
𝟑𝟑𝟑𝟑
+
134
Lesson 10
A STORY OF RATIOS
8•7
Problem Set Sample Solutions
1.
a.
Let 𝒙𝒙 = 𝟎𝟎. ������
𝟔𝟔𝟔𝟔𝟔𝟔. Explain why multiplying both sides of this equation by 𝟏𝟏𝟎𝟎𝟑𝟑 will help us determine the
fractional representation of 𝒙𝒙.
When we multiply both sides of the equation by 𝟏𝟏𝟎𝟎𝟑𝟑 , on the right side we will have 𝟔𝟔𝟔𝟔𝟔𝟔. 𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔 …. This is
helpful because we will be able to subtract the repeating decimal from both sides by subtracting 𝒙𝒙.
b.
After multiplying both sides of the equation by 𝟏𝟏𝟎𝟎𝟑𝟑 , rewrite the resulting equation by making a substitution
that will help determine the fractional value of 𝒙𝒙. Explain how you were able to make the substitution.
𝒙𝒙 = 𝟎𝟎. ������
𝟔𝟔𝟔𝟔𝟔𝟔
������
𝟏𝟏𝟎𝟎𝟑𝟑 𝒙𝒙 = (𝟏𝟏𝟎𝟎𝟑𝟑 )𝟎𝟎. 𝟔𝟔𝟔𝟔𝟔𝟔
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟔𝟔𝟔𝟔𝟔𝟔. ������
𝟔𝟔𝟔𝟔𝟔𝟔
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟔𝟔𝟔𝟔𝟔𝟔 + 𝟎𝟎. 𝟔𝟔𝟔𝟔𝟔𝟔 𝟔𝟔𝟑𝟑𝟑𝟑 …
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟔𝟔𝟔𝟔𝟔𝟔 + 𝒙𝒙
������, we can substitute the repeating decimal 𝟎𝟎. 𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔 … with 𝒙𝒙.
Since we let 𝒙𝒙 = 𝟎𝟎. 𝟔𝟔𝟔𝟔𝟔𝟔
c.
Solve the equation to determine the value of 𝒙𝒙.
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 − 𝒙𝒙 = 𝟔𝟔𝟔𝟔𝟔𝟔 + 𝒙𝒙 − 𝒙𝒙
𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗 = 𝟔𝟔𝟔𝟔𝟔𝟔
𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗 𝟔𝟔𝟔𝟔𝟔𝟔
=
𝟗𝟗𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗𝟗𝟗
𝟔𝟔𝟔𝟔𝟔𝟔
𝒙𝒙 =
𝟗𝟗𝟗𝟗𝟗𝟗
d.
Is your answer reasonable? Check your answer using a calculator.
Yes, my answer is reasonable and correct. It is reasonable because the denominator cannot be expressed as a
product of 𝟐𝟐’s and 𝟓𝟓’s; therefore, I know that the fraction must represent an infinite decimal. Also, the
𝟏𝟏
number 𝟎𝟎. 𝟔𝟔𝟔𝟔𝟔𝟔 is closer to 𝟎𝟎. 𝟓𝟓 than 𝟏𝟏, and the fraction is also closer to than 𝟏𝟏. It is correct because the
division of
𝟔𝟔𝟔𝟔𝟔𝟔
𝟗𝟗𝟗𝟗𝟗𝟗
using the calculator is 𝟎𝟎. 𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔𝟔 ….
Lesson 10:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
Converting Repeating Decimals to Fractions
𝟐𝟐
135
Lesson 10
A STORY OF RATIOS
2.
�. Check your answer using a calculator.
Find the fraction equal to 𝟑𝟑. 𝟒𝟒𝟒𝟒𝟖𝟖
�
Let 𝒙𝒙 = 𝟑𝟑. 𝟒𝟒𝟒𝟒𝟖𝟖
𝟐𝟐
�
𝒙𝒙 = 𝟑𝟑. 𝟒𝟒𝟒𝟒𝟖𝟖
𝟏𝟏𝟎𝟎 𝒙𝒙 =
(𝟏𝟏𝟎𝟎𝟐𝟐 )𝟑𝟑.
�
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟑𝟑𝟑𝟑𝟑𝟑. 𝟖𝟖
�
Let 𝒚𝒚 = 𝟎𝟎. 𝟖𝟖
�
𝒚𝒚 = 𝟎𝟎. 𝟖𝟖
�)
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟏𝟏𝟏𝟏(𝟎𝟎. 𝟖𝟖
�
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟖𝟖. 𝟖𝟖
�
𝟒𝟒𝟒𝟒𝟖𝟖
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟖𝟖 + 𝒚𝒚
𝟏𝟏𝟏𝟏𝟏𝟏 − 𝒚𝒚 = 𝟖𝟖 + 𝒚𝒚 − 𝒚𝒚
𝟗𝟗𝟗𝟗 = 𝟖𝟖
𝟗𝟗𝟗𝟗 𝟖𝟖
=
𝟗𝟗
𝟗𝟗
𝟖𝟖
𝒚𝒚 =
𝟗𝟗
�=
𝟑𝟑. 𝟒𝟒𝟒𝟒𝟖𝟖
3.
𝟕𝟕𝟕𝟕𝟕𝟕
𝟐𝟐𝟐𝟐𝟐𝟐
8•7
�
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟑𝟑𝟑𝟑𝟑𝟑. 𝟖𝟖
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟑𝟑𝟑𝟑𝟑𝟑 + 𝒚𝒚
𝟖𝟖
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟑𝟑𝟑𝟑𝟑𝟑 +
𝟗𝟗
𝟑𝟑𝟑𝟑𝟑𝟑 × 𝟗𝟗 𝟖𝟖
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 =
+
𝟗𝟗
𝟗𝟗
𝟑𝟑𝟑𝟑𝟑𝟑 × 𝟗𝟗 + 𝟖𝟖
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 =
𝟗𝟗
𝟑𝟑𝟑𝟑𝟑𝟑𝟑𝟑
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 =
𝟗𝟗
𝟏𝟏
𝟏𝟏 𝟑𝟑𝟑𝟑𝟑𝟑𝟑𝟑
�
� 𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = �
�
𝟏𝟏𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏𝟏𝟏
𝟗𝟗
𝟑𝟑𝟑𝟑𝟑𝟑𝟑𝟑
𝒙𝒙 =
𝟗𝟗𝟗𝟗𝟗𝟗
𝟕𝟕𝟕𝟕𝟕𝟕
𝒙𝒙 =
𝟐𝟐𝟐𝟐𝟐𝟐
Find the fraction equal to 𝟎𝟎. �������
𝟓𝟓𝟓𝟓𝟓𝟓𝟓𝟓. Check your answer using a calculator.
Let 𝒙𝒙 = 𝟎𝟎. �������
𝟓𝟓𝟓𝟓𝟓𝟓𝟓𝟓
�������
𝒙𝒙 = 𝟎𝟎. 𝟓𝟓𝟓𝟓𝟓𝟓𝟓𝟓
�������
𝟏𝟏𝟎𝟎 𝒙𝒙 = (𝟏𝟏𝟎𝟎𝟒𝟒 )𝟎𝟎. 𝟓𝟓𝟓𝟓𝟓𝟓𝟓𝟓
𝟒𝟒
�������
𝟏𝟏𝟏𝟏 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 = 𝟓𝟓𝟓𝟓𝟓𝟓𝟓𝟓. 𝟓𝟓𝟓𝟓𝟓𝟓𝟓𝟓
𝟏𝟏𝟏𝟏 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 = 𝟓𝟓𝟓𝟓𝟓𝟓𝟓𝟓 + 𝒙𝒙
𝟏𝟏𝟏𝟏 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 − 𝒙𝒙 = 𝟓𝟓𝟓𝟓𝟓𝟓𝟓𝟓 + 𝒙𝒙 − 𝒙𝒙
4.
𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗 = 𝟓𝟓𝟓𝟓𝟓𝟓𝟓𝟓
𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗 𝟓𝟓𝟓𝟓𝟓𝟓𝟓𝟓
=
𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗
𝟓𝟓𝟓𝟓𝟐𝟐𝟐𝟐
𝒙𝒙 =
𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗
����. Check your answer using a calculator.
Find the fraction equal to 𝟐𝟐. 𝟑𝟑𝟖𝟖𝟖𝟖
����
Let 𝒙𝒙 = 𝟐𝟐. 𝟑𝟑𝟖𝟖𝟖𝟖
����
𝒙𝒙 = 𝟐𝟐. 𝟑𝟑𝟖𝟖𝟖𝟖
����
𝟏𝟏𝟏𝟏𝟏𝟏 = (𝟏𝟏𝟏𝟏)𝟐𝟐. 𝟑𝟑𝟖𝟖𝟖𝟖
����
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟐𝟐𝟐𝟐. 𝟖𝟖𝟖𝟖
���� =
𝟐𝟐. 𝟑𝟑𝟖𝟖𝟖𝟖
𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐
𝟗𝟗𝟗𝟗𝟗𝟗
Lesson 10:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
Let 𝒚𝒚 = 𝟎𝟎. ����
𝟖𝟖𝟖𝟖
����
𝒚𝒚 = 𝟎𝟎. 𝟖𝟖𝟖𝟖
����
𝟏𝟏𝟎𝟎𝟐𝟐 𝒚𝒚 = (𝟏𝟏𝟎𝟎𝟐𝟐 )𝟎𝟎. 𝟖𝟖𝟖𝟖
����
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟖𝟖𝟖𝟖. 𝟖𝟖𝟖𝟖
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟖𝟖𝟖𝟖 + 𝒚𝒚
𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 − 𝒚𝒚 = 𝟖𝟖𝟖𝟖 + 𝒚𝒚 − 𝒚𝒚
𝟗𝟗𝟗𝟗𝟗𝟗 = 𝟖𝟖𝟖𝟖
𝟗𝟗𝟗𝟗𝟗𝟗 𝟖𝟖𝟖𝟖
=
𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗
𝟖𝟖𝟖𝟖
𝒚𝒚 =
𝟗𝟗𝟗𝟗
Converting Repeating Decimals to Fractions
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟐𝟐𝟐𝟐. ����
𝟖𝟖𝟖𝟖
𝟖𝟖𝟖𝟖
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟐𝟐𝟐𝟐 +
𝟗𝟗𝟗𝟗
𝟐𝟐𝟐𝟐 × 𝟗𝟗𝟗𝟗 𝟖𝟖𝟖𝟖
𝟏𝟏𝟏𝟏𝟏𝟏 =
+
𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗
𝟐𝟐𝟐𝟐 × 𝟗𝟗𝟗𝟗 + 𝟖𝟖𝟖𝟖
𝟏𝟏𝟏𝟏𝟏𝟏 =
𝟗𝟗𝟗𝟗
𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐
𝟏𝟏𝟏𝟏𝟏𝟏 =
𝟗𝟗𝟗𝟗
𝟏𝟏
𝟏𝟏 𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐
� � 𝟏𝟏𝟏𝟏𝟏𝟏 = � �
𝟏𝟏𝟏𝟏
𝟏𝟏𝟏𝟏 𝟗𝟗𝟗𝟗
𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐
𝒙𝒙 =
𝟗𝟗𝟗𝟗𝟗𝟗
136
Lesson 10
A STORY OF RATIOS
5.
8•7
Find the fraction equal to 𝟎𝟎. �����������
𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕. Check your answer using a calculator.
Let 𝒙𝒙 = 𝟎𝟎. �����������
𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕𝟕.
𝒙𝒙 = 𝟎𝟎. ������������
𝟕𝟕𝟕𝟕𝟕𝟕 𝟐𝟐𝟐𝟐𝟐𝟐
������������
𝟏𝟏𝟎𝟎 𝒙𝒙 = (𝟏𝟏𝟎𝟎𝟔𝟔 )𝟎𝟎. 𝟕𝟕𝟕𝟕𝟕𝟕
𝟐𝟐𝟐𝟐𝟐𝟐
𝟔𝟔
𝟏𝟏 𝟎𝟎𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 = 𝟕𝟕𝟕𝟕𝟕𝟕 𝟖𝟖𝟖𝟖𝟖𝟖. 𝟕𝟕𝟕𝟕𝟕𝟕 𝟐𝟐𝟐𝟐𝟐𝟐
𝟏𝟏 𝟎𝟎𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 = 𝟕𝟕𝟕𝟕𝟕𝟕 𝟐𝟐𝟐𝟐𝟐𝟐 + 𝒙𝒙
𝟏𝟏 𝟎𝟎𝟎𝟎𝟎𝟎 𝟎𝟎𝟎𝟎𝟎𝟎𝟎𝟎 − 𝒙𝒙 = 𝟕𝟕𝟕𝟕𝟕𝟕 𝟐𝟐𝟐𝟐𝟐𝟐 + 𝒙𝒙 − 𝒙𝒙
6.
𝟗𝟗𝟗𝟗𝟗𝟗 𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗 = 𝟕𝟕𝟕𝟕𝟕𝟕 𝟐𝟐𝟐𝟐𝟐𝟐
𝟗𝟗𝟗𝟗𝟗𝟗 𝟗𝟗𝟗𝟗𝟗𝟗𝟗𝟗 𝟕𝟕𝟕𝟕𝟕𝟕 𝟐𝟐𝟐𝟐𝟐𝟐
=
𝟗𝟗𝟗𝟗𝟗𝟗 𝟗𝟗𝟗𝟗𝟗𝟗
𝟗𝟗𝟗𝟗𝟗𝟗 𝟗𝟗𝟗𝟗𝟗𝟗
𝟕𝟕𝟕𝟕𝟕𝟕 𝟐𝟐𝟐𝟐𝟐𝟐
𝒙𝒙 =
𝟗𝟗𝟗𝟗𝟗𝟗 𝟗𝟗𝟗𝟗𝟗𝟗
𝟓𝟓
𝒙𝒙 =
𝟕𝟕
Explain why an infinite decimal that is not a repeating decimal cannot be rational.
Infinite decimals that do repeat can be expressed as a fraction and are therefore rational. The method we learned
today for writing a repeating decimal as a rational number cannot be applied to infinite decimals that do not repeat.
The method requires that we let 𝒙𝒙 represent the repeating part of the decimal. If the number has a decimal
expansion that does not repeat, we cannot express the number as a fraction ( i.e., a rational number).
7.
� = 𝟏𝟏. Use what you learned today to show
In a previous lesson, we were convinced that it is acceptable to write 𝟎𝟎. 𝟗𝟗
that it is true.
�
Let 𝒙𝒙 = 𝟎𝟎. 𝟗𝟗
�
𝒙𝒙 = 𝟎𝟎. 𝟗𝟗
�
𝟏𝟏𝟏𝟏𝟏𝟏 = (𝟏𝟏𝟏𝟏)𝟎𝟎. 𝟗𝟗
�
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟗𝟗. 𝟗𝟗
𝟏𝟏𝟏𝟏𝟏𝟏 = 𝟗𝟗 + 𝒙𝒙
𝟏𝟏𝟏𝟏𝟏𝟏 − 𝒙𝒙 = 𝟗𝟗 + 𝒙𝒙 − 𝒙𝒙
𝟗𝟗𝟗𝟗 = 𝟗𝟗
𝟗𝟗𝟗𝟗 𝟗𝟗
=
𝟗𝟗
𝟗𝟗
𝟗𝟗
𝒙𝒙 =
𝟗𝟗
𝒙𝒙 = 𝟏𝟏
8.
Examine the following repeating infinite decimals and their fraction equivalents. What do you notice? Why do you
think what you observed is true?
���� =
𝟎𝟎. 𝟖𝟖𝟖𝟖
𝟖𝟖𝟖𝟖
𝟗𝟗𝟗𝟗
�=
𝟎𝟎. 𝟒𝟒
𝟒𝟒
𝟗𝟗
𝟎𝟎. ������
𝟏𝟏𝟏𝟏𝟏𝟏 =
𝟏𝟏𝟏𝟏𝟏𝟏
𝟗𝟗𝟗𝟗𝟗𝟗
𝟎𝟎. ����
𝟔𝟔𝟔𝟔 =
𝟔𝟔𝟔𝟔
𝟗𝟗𝟗𝟗
� = 𝟏𝟏. 𝟎𝟎
𝟎𝟎. 𝟗𝟗
In each case, the fraction that represents the infinite decimal has a numerator that is exactly the repeating part of
the decimal and a denominator comprised of 𝟗𝟗’s. Specifically, the denominator has the same number of digits of 𝟗𝟗’s
as the number of digits that repeat. For example, 𝟎𝟎. ����
𝟖𝟖𝟖𝟖 has two repeating decimal digits, so the denominator has
� = 𝟏𝟏, we can make the assumption that repeating 𝟗𝟗’s, like 𝟗𝟗𝟗𝟗, could be expressed
two 𝟗𝟗’s. Since we know that 𝟎𝟎. 𝟗𝟗
as 𝟏𝟏𝟏𝟏𝟏𝟏, meaning that the fraction
Lesson 10:
© 2014 Common Core, Inc. All rights reserved. commoncore.org
𝟖𝟖𝟖𝟖
𝟗𝟗𝟗𝟗
is almost
𝟖𝟖𝟖𝟖
, which would then be expressed as 𝟎𝟎. 𝟖𝟖𝟖𝟖.
𝟏𝟏𝟏𝟏𝟏𝟏
Converting Repeating Decimals to Fractions
137