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Pre Algebra
Section 1.3
Unit 1 - Integers
Section 3 – Multiplying and Dividing Integers
MULTIPLICATION
Consider the product 3 ∙ −5
Remember that means that you have three (-5) added together, in other words
(−5 ) + (−5 ) + (−5)
= −10 + −5
= −15
We know that 3 ∙ 5 = 15 . From the above we can see that if one number is a negative the answer will
become negative. Since multiplication in commutative (we can change the order and get the same
result) −5 ∙ 3 = -15 as well.
When you multiply two numbers that have opposite signs, the result is negative.
Remember that a negative sign means opposite. Therefore if you have two negatives you can think of it
as taking the opposite of a number and then take the opposite of that which leaves you with the original
number. For example
−2 ∙ −5 = 10
When you multiply two numbers that have the same sign, the result is positive.
One fun way to remember that multiplying two negatives makes a positive is to think of one of the
negative signs turning vertical to combine with the second negative to make the plus sign.
RULE:
When you multiply two numbers that have opposite signs, the results is negative.
When you multiply two numbers with the same sign, the result is positive.
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Section 1.3
Example 1)
ࡹ࢛࢒࢚࢏࢖࢒࢟ ૡ ∙ ૠ
8∙ 7
= 56
Same sign so the answer is positive
Example 2)
ࡹ࢛࢒࢚࢏࢖࢒࢟ − ૝ ∙ ૡ
−4 ∙ 8
= −32
Opposite signs so the answer is negative
Example 3)
ࡹ࢛࢒࢚࢏࢖࢒࢟ ૢ ∙ −૝
9 ∙ −4
= −36
Opposite signs so the answer is negative
Example 4)
ࡹ࢛࢒࢚࢏࢖࢒࢟ − ૟ ∙ −ૠ
−6 ∙ −7
= 42
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Same sign so the answer is positive
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Pre Algebra
Section 1.3
Properties of Multiplication
Let ܽ be any real number, then the following properties are true.
The
Multiplication
Property of Zero
ܽ ∙ 0 = 0
The
Multiplication
Property of One
ܽ∙1 = ܽ
The Associative Property
of Multiplication
(ܽ ∙ ܾ) ∙ ܿ = ܽ ∙ (ܾ ∙ ܿ)
The Commutative
Property of
Multiplication
ܽ∙ܾ = ܾ∙ܽ
So what do you do if there are more than 2 numbers? The following is an example of that.
Example 5)
૜(−૛)(૟)(−૝)
We will start with the first two numbers 3(-2) which means 3 ∙ −2, which is -6, so the problems
changes from
3(−2)(6)(−4)
= −6 (6)(−4)
Now combine the next two
= −36 (−4)
and the last two
= 144
is the final answer
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Pre Algebra
Section 1.3
Evaluating expressions with Multiplication
We can also evaluate expressions that contain multiplication.
Example 6)
Evaluate ૝࢞ , for ࢞ = −ૢ
4‫ ݔ‬means 4 ∙ ‫ݔ‬
Therefore replacing ‫ ݔ‬with the −9 gives
4∙‫ݔ‬
= 4 ∙ −9
= − 36
which is the final answer.
DIVISION
Every Division problem can be rewritten as a multiplication problem. Therefore the rules of
multiplication can be extended to division as well.
Notice the similarities between the following
−2 ∙ 3 = −6
and
−6 ÷ 3 = −2
We can even goes as far as saying −6 ÷ 3 = −2 because −2 ∙ 3 = −6. Therefore it makes sense
the rules will be the same as well.
RULE:
When you Divide two numbers that have opposite signs, the results is negative.
When you Divide two numbers with the same sign, the result is positive.
Example7)
−૞૞ ÷ ૞
−55 ÷ 5 = −11
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Pre Algebra
Section 1.3
Example 8)
−ૠ૛ ÷ −ૡ
−72 ÷ −8 = 9
Example 9)
૚૝૛
ଵସଶ
ି૛
ିଶ
is the same as 142 ÷ −2
142 ÷ −2 = − 71
Example 10)
ିૡ૚
ିૢ
ି଼ଵ
ିଽ
=9
Properties of Division
When dividing it is important to remember you cannot divide by zero!
•
•
•
௔
଴
௔
ଵ
݅‫݂݀݁݊݅݁݀݊ݑ ݏ‬
=ܽ
݂݅ ܽ ≠ 0, ‫ݐ‬ℎ݁݊
௔
௔
= 1 ܽ݊݀
଴
௔
=0
Evaluating expressions with division
We can evaluate expressions that contain division in the same way we have evaluated other
expressions.
Example 11 )
࢞
ࡱ࢜ࢇ࢒࢛ࢇ࢚ࢋ , ࢌ࢕࢘ ࢞ = −૚૛૞ ࢇ࢔ࢊ ࢟ = ૞.
࢟
௫
Replacing x with -125 and y with 5 gives
௬
=
ିଵଶହ
ହ
= −25
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Pre Algebra
Section 1.3
Area of a rectangle
The area of a rectangle can be found by multiplying the base times the height.
b
h
‫ ∙ ܾ = ܣ‬ℎ or ‫ݓ ∙ ݈ = ܣ‬
Example 12)
Find the area of a rectangle whose bas is 10 cm and Height is 3 cm.
5cm
4cm
‫∙ܾ=ܣ‬ℎ
‫ = ܣ‬5ܿ݉ ∙ 4 ܿ݉
‫ = ܣ‬20ܿ݉ ଶ
The area in 20 square cm.
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Pre Algebra
Section 1.3
Exercise 1.3
NAME:_________________________________
Multiply
1. 3(−4)
2. −4 ∙ 6
3. -4 ∙ 7
4. −3( −5)
5. −8(−8)
6. −11(−7)
7. −8 ∙ 3(−2)
8. 6 ∙ (−2) ∙ 5
9. (−5)(−6)(−2)
10. Find the product of 8 and -13.
11. Find the product of -7 and – 4
12. Find the product of -6 and 9
13. ‫ݕݔ ݁ݐܽݑ݈ܽݒܧ‬, ݂‫ = ݔ ݎ݋‬−6 ܽ݊݀ ‫ = ݕ‬9
14. ‫ ݁ݐܽݑ݈ܽݒܧ‬6ܾܽ, ݂‫ = ܽ ݎ݋‬−3 ܽ݊݀ ܾ = −7
15. Evaluate −7‫ݕݔ‬, ݂‫ = ݔ ݎ݋‬−1 ܽ݊݀ ‫ = ݕ‬6
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Pre Algebra
Section 1.3
Dividing
16. −33 ÷ 3
17. −95 ÷ −5
ିଷ଺
19. ିଵଶ
22. Evaluate
20.
ି௔
௕
18. 240 ÷ −12
ିଶ଼
21.
଻
଺ସ
ି଼
௔
, ݂‫ = ܽ ݎ݋‬72, ܽ݊݀ ܾ = −4.
23. Evaluate ௕ , ݂‫ = ܽ ݎ݋‬22 ܽ݊݀ ܾ = −2.
24. Evaluate ܽ ÷ ܾ, ݂‫ = ܽ ݎ݋‬56, ܽ݊݀ ܾ = 8.
For problem 25-27, consider the following rectangle.
h
b
Find the area of the rectangle with dimensions below.
25. ܾ = 3݉, ℎ = 4݉
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26.ܾ = 7݂‫ݐ‬, ℎ = 5݂‫ݐ‬
32
27. ܾ = 11ܿ݉, ℎ = 4ܿ݉
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