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3.9
YOU WILL NEED
• a calculator
GOAL
Use division to express fractions as decimals.
LEARN ABOUT the Math
Julie and her friends are going on a hike. Julie buys 2 kg of trail
mix to share.
How much trail mix will each hiker receive?
A. Suppose that there are three hikers. How can you use these
linking cubes to show that each hiker will get 2 of a kilogram
3
of trail mix?
B. Write a division sentence to describe how each share can be
repeating decimal
a decimal in which a
block of one or more
digits eventually repeats
in a pattern; for example,
25
0.252 525 ...,
99
31
0.861 111 ..., and
36
1
0.142 857 142 857
7
... . The dots mean that
the numbers continue
in the same pattern
without stopping.
132
Chapter 3
calculated.
C. Calculate 2.000 3 using pencil and paper. What do you
notice about the remainder after each step?
D. When you calculate 2 3 on your calculator, why does the
display show 0.66666667 rather than the repeating decimal
0.6666 …, which is the actual answer?
E. Suppose that a different number of hikers go on the hike. Each
hiker will receive an equal share of the 2 kg of trail mix.
Complete the table on the next page. Write enough digits so that
you can either see a pattern or see all the digits if the decimal is
terminating.
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terminating decimal
a decimal that is
complete after a certain
number of digits with
no repetition; for
example, 0.777
Mass of trail mix each
hiker receives (kg)
Number of hikers
As a fraction
As a decimal
2
1
2
2
2
3
1
2
3
4
5
6
7
8
9
10
Communication Tip
Some repeating decimals have large groups of repeating digits. This makes
them awkward to write out. Instead of writing the repeating digits several
times, we use a horizontal bar to mark them. This is called bar notation. For
example, write 0.143 514 351 435 … as 0.1
4
3
5, and write 0.999 … as 0.9
.
Reflecting
F. Why can a terminating decimal always be written over a
multiple of 10, such as ■, ■, or ■?
10 100
1
3
G. Why can’t be written in the
a terminating decimal?
1000
form ■ or ■?
10
100
Why can’t it be
H. Why do you divide the numerator by the denominator to write
a fraction as a decimal?
1
8
1
25
I. Why can a fraction such as or be written as an equivalent
fraction with a denominator of 1000?
NEL
Decimal Numbers
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WORK WITH the Math
Example 1
Comparing using equivalent decimals
Nolan has three bags of popcorn to share with seven friends.
Fiona has four bags of popcorn to share with eight friends. The
bags of popcorn are all the same size. Which group will receive
larger portions?
Nolan’s Solution
3
3 bags shared among 8 people is .
8
4
4 bags shared among 9 people is .
9
I included myself and Fiona in our groups. I
calculated the size of each share in bags for
both groups.
I knew that I could compare these numbers
more easily by writing them as decimals.
3 ÷ 8 = 0.375 bags
4 ÷ 9 = 0.4 bags
I used my calculator and divided the
numerator in each fraction by its denominator.
0.4 > 0.375
Fiona’s friends will receive the larger portions.
0.4
is greater than 0.4, and 0.375 is less
than 0.4.
Example 2
Determining whether a decimal repeats
Determine whether the decimal equivalent of each fraction
terminates or repeats. Order the fractions from least to greatest.
2
7
8
53
a) b) c) d) 10
9
42
80
Max’s Solution
If a decimal terminates, I will be able to express it as an equivalent fraction with a denominator
that is a multiple of 10, such as 10, 100, or 1000.
2
a) = 0.2; terminates
10
134
Chapter 3
2
10
is already in this form, so I know that it terminates.
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7
b) = 0.777 ...; repeats
9
I know that 1 0.111 …, so I know that 7
8
4
c) = ; repeats
42 21
Since 7 is a factor of 21, but 7 is not a
factor of a multiple of 10, such as 1000
or 10 000, the decimal repeats.
I can rewrite 8 as 4. I tried to write
9
9
must be 7 times as much.
53 6625
d) = or 0.6625; terminates
80 10 000
4
21
42
21
■, but there is no whole number I
100
can multiply 21 by to get 100. I couldn’t write
the fraction as ■ either.
1000
I tried writing 53 ■, but this didn’t work
80
100
since 100 80 is not a whole number. Then
I tried 53 ■. This didn’t work either.
80
1000
I tried one more time. This worked because
80 8 10 and 10 000 8 125 10.
I checked my predictions with a calculator.
2
8
c) = 0.190476
a) = 0.2
10
42
7
53
b) = 0.7
d) = 0.6625
9
80
I was right!
0.190476, 0.2, 0.6625, 0.7
I ordered the decimals from least to greatest.
8 2 53 7
, , , 42 10 80 9
Then I ordered the fractions from least to
greatest.
A Checking
1. Write each repeating decimal in bar notation.
a) 0.555 555 555 …
b) 0.134 561 345 613 456 …
2. Compare each pair of fractions using equivalent decimals.
Replace each ■ with , , or .
5
2
7
5
a) ■ b) ■ 9
8
16
11
NEL
17
20
11
14
c) ■ Decimal Numbers
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B Practising
3. Decide whether the decimal equivalent of each fraction
terminates or repeats.
3
5
a) b) 4
9
9
14
19
20
c) d) 4. Write each decimal as a fraction.
a) 0.1625
b) 0.8550
5. If possible, write each fraction as a terminating decimal.
14
25
5
b) 8
a) 1
16
4
d) 5
c) 19
20
22
f) 32
e) 6. If possible, write each fraction as a repeating decimal.
1
6
8
b) 9
a) 7
11
7
d) 15
c) 48
49
57
f) 111
e) 7. Sort the fractions based on whether they are equivalent to a
terminating decimal or a repeating decimal.
4
5
a) c) e)
9
6
3
15
b) d) f)
5
16
5
18
19
32
8. Order the fractions in question 7 from least to greatest.
8
8 8
9 99 999
9. a) Describe the following fraction pattern: , , , ….
Write the next three fractions in the pattern.
b) Rewrite the pattern using decimal equivalents for each of
the six fractions.
c) Describe the decimal pattern. Is the decimal pattern easier
or harder to describe than the fraction pattern?
10. Express each fraction as a repeating decimal.
1
7
a) 136
Chapter 3
2
7
b) 3
7
c) NEL
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11. a) Describe a pattern in your answers for the previous
question.
4
7
5
7
b) Predict the decimal equivalents of and .
12. Replace each ■ with , , or .
6
11
a) 0.2 ■ 0.2
7
13
d) ■ 45
99
b) ■ 0.4
5
e) 0.357 357 357 … ■ 0.3
7
5
4
5
c) 0.8
2
■ f)
2
■ 0.633
3
13. Order the numbers from least to greatest.
1 5
8 7
9
10
a) , , 0.35, 0.3
9
, 5 27
9 50
b) 0.56, 0.56
, 0.5
6
, , 14. Predict the decimal equivalent of each fraction, using the fact
that 1 0.333....
3
2
1
a) b) 3
9
4
3
1
30
c) d) 15. Calculate the decimal equivalent of each fraction.
1
12
a) 1
28
b) 1
44
1
52
c) d) 16. Look at your answers for the previous question.
a) How are the decimal equivalents alike?
b) What do the denominators have in common?
17. The cost of a new toy is $1 after taxes. You and two friends
want to split the cost evenly.
a) Express each person’s share as a fraction.
b) Express the fraction as a decimal.
c) How much should each of you pay? Explain your decision.
d) Create a similar problem with a different fraction and
solve it.
18. How can you tell, without calculating, that the decimal
equivalent of 1 repeats?
33
NEL
Decimal Numbers
137