Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
8•7
Lesson 12: Decimal Expansions of Fractions, Part 2
Classwork
Example 1
Find the decimal expansion of
35
11
Lesson 12:
.
Decimal Expansions of Fractions, Part 2
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NYS COMMON CORE MATHEMATICS CURRICULUM
Lesson 12
8•7
Exercises 1–3
1.
5
Find the decimal expansion of without using long division.
3
Lesson 12:
Decimal Expansions of Fractions, Part 2
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NYS COMMON CORE MATHEMATICS CURRICULUM
2.
Find the decimal expansion of
Lesson 12:
5
11
Lesson 12
8•7
without using long division.
Decimal Expansions of Fractions, Part 2
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NYS COMMON CORE MATHEMATICS CURRICULUM
3.
Find the decimal expansion of the number
Lesson 12:
23
99
Lesson 12
8•7
first without using long division and then again using long division.
Decimal Expansions of Fractions, Part 2
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
8•7
Lesson Summary
For rational numbers, there is no need to guess and check in which interval of tenths, hundredths, or thousandths
the number will lie.
For example, to determine where the fraction
1
8
lies in the interval of tenths, compute using the following
inequality:
𝑚 1 𝑚+1
< <
10 8
10
10
𝑚<
< 𝑚+1
8
1
𝑚 < 1 < 𝑚+1
4
Use the denominator of 10 because we need to find the tenths
1
digit of .
8
Multiply through by 10.
Simplify the fraction
10
8
.
1
4
The last inequality implies that 𝑚 = 1 and 𝑚 + 1 = 2 because 1 < 1 < 2. Then, the tenths digit of the decimal
1
expansion of is 1.
8
To find in which interval of hundredths
1
8
lies, we seek consecutive integers 𝑚 and 𝑚 + 1 so that
1
𝑚
1
1
𝑚+1
+
< <
+
.
10 100 8 10
100
This is equivalent to
𝑚
1 1
𝑚+1
< −
<
,
100 8 10
100
so we compute
1
8
−
1
10
=
2
80
=
1
40
. We have
𝑚
1
𝑚+1
<
<
.
100 40
100
Multiplying through by 100 gives
𝑚<
10
< 𝑚 + 1.
4
1
2
The last inequality implies that 𝑚 = 2 and 𝑚 + 1 = 3 because 2 < 2 < 3. Then, the hundredths digit of the
1
decimal expansion of is 2.
8
We can continue the process until the decimal expansion is complete or until we suspect a repeating pattern that
we can verify.
Lesson 12:
Decimal Expansions of Fractions, Part 2
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Lesson 12
NYS COMMON CORE MATHEMATICS CURRICULUM
8•7
Problem Set
1.
Without using long division, explain why the tenths digit of
2.
Find the decimal expansion of
3.
Find the decimal expansion of
4.
Which number is larger, √10 or
5.
Sam says that
7
11
25
9
11
41
11
is a 2.
without using long division.
to at least 5 digits without using long division.
28
9
? Answer this question without using long division.
= 0.63, and Jaylen says that
Lesson 12:
3
7
11
= 0.636. Who is correct? Why?
Decimal Expansions of Fractions, Part 2
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