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Pre Algebra
Section 3.3
Unit 3 - Decimals
Section 3 – Multiplying and Dividing Decimals
Multiplying Decimals
When multiplying decimals we multiply the numbers normally, then we add together the number of
decimal places in our original problem and add that many decimal places to our answer.
૜. ૝૞ ∙ ૛. ૜
3.45
× 2.3
1035
6900
7.935
2 ݈݀݁ܿ݅݉ܽ‫ݏ݈݁ܿܽ݌ ݏ‬
1 ݈݀݁ܿ݅݉ܽ ‫ݏ݈݁ܿܽ݌‬
3 ݈݀݁ܿ݅݉ܽ ‫ݏ݈݁ܿܽ݌‬
Since there are 3 decimal places in the original numbers, we add in three decimal points to the
answer
Example1)
ሺ−૛. ૟ૠሻሺ૞. ૜ሻ
First we write the problem vertically
−2.67
× 5.3
801
13350
−14.151
The original problem is a negative times a
positive, therefore our answer is negative. The
original problem had 2 decimal places in one
number and 1 decimal place in the second , so
the answer has 3 decimal places
Example 2)
૙. ૙૙૙૜૝ ሺ૙. ૙૙૙૛૜ሻ
0.00034
× 0.00023
102
680
0.0000000782
5 decimal places
5 decimal places
10 decimal places
In this problem we had to add zeros in front of the 782 so we could ensure we have 10 decimal places in
your answer.
American River College
119
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Pre Algebra
Section 3.3
Multiplying Decimals by a power of 10
Consider the following problems and their solutions
2.34567 × 10 = 23.4567
2.34567 × 100 = 234.567
2.34567 × 1000 = 2345.67
2.34567 × 10000 = 23456.7
2.34567 × 100000 = 234567.
2.34567 × 1000000 = 2345670.
⋮
Notice in each case the numbers stayed the same, but the number of places the decimal place
moved to the right matched the number of 0’s in the power of 10 you are multiplying by.
The pattern continues in the opposite direction as well
2.34567 × .1 = 0.234567
2.34567 × .01 = 0.0234567
2.34567 × .001 = 0.00234567
2.34567 × .0001 = 0.000234567
⋮
In this case the number the decimal places you move the decimal to the left matches the number of
decimal places the 1 is over.
Example 3)
ሺ−૞૟. ૠૡૢሻሺ૚૙૙ሻ
ሺ−56.789ሻሺ100ሻ
Notice that in this problem we are multiplying by 100
−5678.9
Two zeros, moves the decimal place 2 places
Example 4)
−૛૝૞. ૟ ∙ ૙. ૙૙૙૚
−245.6 ∙ 0.0001
0.0001 will move the decimal 4 places to the left.
−0.02456
The leading zero is not necessary
American River College
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Pre Algebra
Section 3.3
Circumference of a circle
A circle is a set of points a set distance(radius) from a given point (center). The diameter of a circle is
the distance across the entire circle.
r
Where r is the radius.
ௗ
Where d is the diameter. Notice ݀ = 2‫ = ݎ ݎ݋ ݎ‬.
d
ଶ
The Circumference of a circle is like perimeter of a rectangle.
To calculate Circumference we use the formula
‫ = ܥ‬2ߨ‫ߨ݀ = ܥ ݎ݋ ݎ‬
ܹℎ݁‫ = ߨ ݁ݎ‬3.14159265 … , ‫ݏݑ݅݀ܽݎ = ݎ‬, ܽ݊݀ ݀ = ݀݅ܽ݉݁‫ݎ݁ݐ‬.
For this class we will use the approximation ߨ ≈ 3.14.
Example 5)
Find the circumference of the circle whose radius is 7cm.
‫ = ܥ‬2ߨ‫ݎ‬
6.28
×7
43.96
‫ ≈ ܥ‬2 ∙ 3.14 ∙ 7ܿ݉
‫ ≈ ܥ‬6.28 ∙ 7ܿ݉
‫ ≈ ܥ‬43.96ܿ݉
Example 6)
Find the Circumference of the following circle.
7ft
In this example we are given the diameter so
we will use the formula ‫ߨ݀ = ܥ‬.
‫ߨ݀ = ܥ‬
‫ ≈ ܥ‬7݂‫ ∙ ݐ‬3.14
‫ ≈ ܥ‬21.98 ݂‫ݐ‬
American River College
121
Milano
Pre Algebra
Section 3.3
Dividing Decimals
Consider the following
12.345
34 419.730
34
79
68
117
102
153
136
170
170
0
Notice the decimal point in the
answer is directly above the
decimal point in the dividend.
The number your are dividing by(the divisor) cannot have a decimal. If there is a decimal you must
move the decimal over in both the Divisor and the Dividend so that the divisor does not contain one.
Example 7)
ૡ. ૚૜૞ ÷ ሺ−. ૛૞ሻ
−.25. 8.13.50
−32.54
−25 813.50
75
63
50
135
125
100
100
0
American River College
Notice that the answer is
− 32.54
Since a positive divided by a
negative is a negative.
122
Milano
Pre Algebra
Section 3.3
Example 8)
−૚ ÷ ૜
Notice that the process is
repeating. We could continue
forever.
−0.333 …
−3 1.000
9
10
9
10
9
1
ത
−0.33333 … = −. 3
ത
So the answer is −. 3
Dividing Decimals by a Power of 10
Notice the following pattern
2.34567 ÷ 10 = 0.234567
2.34567 ÷ 100 = 0.0234567
2.34567 ÷ 1000 = 0.00234567
2.34567 ÷ 10000 = 0.000234567
2.34567 ÷ 100000 = 0.0000234567
2.34567 ÷ 1000000 = 0.000002345670
⋮
The pattern continues the other way as well
2.34567 ÷ 0.1 = 23.4567
2.34567 ÷ 0.01 = 234.567
2.34567 ÷ 0.001 = 2345.67
2.34567 ÷ 0.0001 = 23456.7
American River College
123
Milano
Pre Algebra
Section 3.3
⋮
Writing fractions as Decimals
We can rewrite fractions as decimals by treating the fraction as division.
Example9)
૞
Rewrite ૚૚ as a Decimal.
0.454 …
11 5.0000
44
60
55
50
44
60
തതത
Since the 4 and the 5 both repeat the answer is written 0. ത45
Comparing Number
If you are asked to order two numbers such as
ଶ
ଽ
and 0.23. You may find it easier to turn them into
decimals.
ଶ
ത , or 0.22222222….,
Since ଽ = 0. 2
ଶ
It is easy to see 0.23 >.222…, Therefore 0.23 > ଽ
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OR
124
ଶ
ଽ
< 0.23.
Milano
Pre Algebra
Section 3.3
Exercise 3.3
NAME:___________________________________
Multiply
1. 0.24ሺ5.23ሻ
2. 23 ∙ 13.26
3. 12.45ሺ3.1ሻ
4. −12.1 ∙ 5.2
5. 0.45ሺ−22.1ሻ
6. −7.1 ∙ .3
7.−5.2ሺ−3.34ሻ
8. ሺ−2.1ሻሺ−1.38ሻ
9.−0.3ሺ−0.7ሻ
10. ሺ−0.00023ሻሺ−0.0012ሻ
11.ሺ−0.000235ሻሺ0.0012ሻ
12.0.003ሺ−2.1ሻ
13.1.23 ∙ 0.00001
14. −2.135 ∙ 1000
15. −13.3 ∙ 10
American River College
125
Milano
Pre Algebra
Section 3.3
For the following calculate the Circumference.
16. r = 3m
17. d=7in
18. d=6ft
19. 45.36 ÷ 36
20. 27.048 ÷ ሺ−12ሻ
21.−2.55 ÷ 15
22.−0.27 ÷ 0.12
23. −25.83 ÷ ሺ−2.1ሻ
24. −3.9 ÷ 1.2
Divide
American River College
126
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Pre Algebra
Section 3.3
In the following round your answers to the nearest thousandths place.
25.7.2 ÷ 3.1
27. −1.23 ÷ ሺ−0.23ሻ
26.−5.23 ÷ 6
Write the following fractions as a decimal, If answer repeats show that in your answer using the bar
above the repeating portion.
ସ
ଶ
28.ଽ
ଶ
30. − ଷ
29.− ହ
Order the following by adding < or >
ଶ
31.ହ
0.46
American River College
ଵ
32.0.21
଼
127
ଵ
33. − ଻
− .14
Milano
Pre Algebra
Section 3.3
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American River College
128
Milano