Download Section 1.7 Multiplication and Division of Real Numbers

Math 123 – Section 1.7 - Multiplication & Division of Real Numbers - Page 1
Section 1.7
Multiplication and Division of Real Numbers
I. Rules
A.
B.
Signs alike - Answer is positive.
Signs different – Answer is negative.
II. Examples – Simplify each of the following
A. Multiplication
1. (10)(3)
Signs are alike, answer is positive.
Answer: +30
2. (8)(5)
Answer: 40
Signs are different, answer is negative.
3. (7)(15)
Signs are alike, answer is positive.
Answer: +105
4. Now you try one: 8(4)
Answer: 32
5. (8)(3)(5)
24(5)
Answer: 120
On the first two numbers, signs are alike, answer is positive.
Signs are alike, answer is positive.
6. You try one: (2)(3)(4)
Answer: 24
 4
7.    (30) Signs are alike, answer is positive. When a number does not have a
 5
denominator, it is understood to be 1.
 4  30 
Cross-cancel.
   

 5  1 
6
 4  30 
   

 5  1 
1
24
1
Answer: 24
Multiply numerators, multiply denominators.
Reduce.
8. Now you try one: 
Answer: 
72
 
93
14
27
9. (4)(4)(4)
Answer: 64
There is an odd number of negatives, the answer is negative.
© Copyright 2012 by John Fetcho. All rights reserved
Math 123 – Section 1.7 - Multiplication & Division of Real Numbers - Page 2
2(5)(2)(3)(1)
10.
There is an even number of negatives, the answer is
positive.
Answer: 60
 1
Now you try one: 12  
3
Answer: 4
11.
12.
9(3x)
Answer: 27x
Signs are different, answer is negative.
(3y 1)
13.
1(3y  1)
Answer: 3y + 1
14. Now you try one:
Answer: 6x  12
Remember that a negative sign in front of a parenthesis means
the same as 1 times the parenthesis. So we really have:
Distribute the 1.
3(2x + 4)
III. Division Rules
A. Signs alike – answer is positive.
B. Signs different – answer is negative.
C. Zero divided by anything (except 0) is 0.
D. Anything (but 0) divided by 0 is undefined.
IV. Examples – Simplify the following.
A. Division
35
Signs are different, answer is negative.
1.
7
Answer: 5
100
10
Answer: 10
2.
84
21
Answer: 4
3.
Signs are different, answer is negative.
Signs are the same, answer is positive.
4. Now you try one:
18
6
Answer: 3
5. 
7  1
  
9  6
2
7  6
   
9  1
3
Remember, to divide fractions we have to invert (find the
multiplicative inverse) of the fraction following the division
sign and multiply.
Cross-cancel. Signs are the same, answer is positive.
© Copyright 2012 by John Fetcho. All rights reserved
Math 123 – Section 1.7 - Multiplication & Division of Real Numbers - Page 3
Answer :
14
3
6. Now you try one:
Answer : 
V.
5  10 
÷ 
12  3 
1
8
Determining solutions.
A.
Remember that a solution to an equation is any number that, when substituted in for
the variable, gives us a true statement.
B.
Examples - Determine whether the given number is a solution of the equation.
1.
5x = 3x  6; 3
5(3) = 3(3)  6
15 = 9  6
15 = 15
Answer: It is a solution.
2.
Substitute 3 in for x on both sides of the equation.
Multiply. When the signs are different, the answer is
negative.
Add the right-hand side. The signs are alike, so take the
sign and add the numbers.
True.
6(w + 2) = 4w  10; 9
6(9 + 2) = 4(9)  10
6(7) = 36  10
42 = 46
Answer: It is not a solution.
3.
Now you try one:
Substitute 9 in for w on both sides.
On the left-hand side, add in the parenthesis first.
The signs are different, take the side of the biggest
and subtract. On the right-hand side, multiply first.
The signs are different, the answer is negative.
On the left-hand side, multiply. The signs are
different, the answer is negative. On the right-hand
side, add. The signs are alike, take the sign and add.
False.
5 m  1 3m  2

; 4
6
4
Answer: It is a solution.
V.
Applications
A.
Process
1.
Read the problem.
2.
Determine what is unknown.
3.
Define a variable for the unknown.
4.
Write an equation expressing the problem algebraically (i.e. - translate from English
into Algebra).
5.
Solve the equation.
6.
Write a complete sentence answering the question.
© Copyright 2012 by John Fetcho. All rights reserved
Math 123 – Section 1.7 - Multiplication & Division of Real Numbers - Page 4
B.
Examples - Follow the 6-step process from above to answer each question.
1.
In the years after warning labels were put on cigarette packs, the numbers of smokers
dropped from approximately two in five adults to one in five. The bar graph (left-hand
column, page 85) shows the percentage of American adults who smoked cigarettes
for selected years from 1965 through 2009.
Here is a mathematical model that approximates the data displayed by the bar graph:
C = 0.5x + 41.
Percentage of
Americans who
smoked cigarettes
a.
Number of years
after 1965
Use the mathematical model to determine the percentage of American adults who
smoked cigarettes in 2005. How does this compare with the actual percentage
displayed on the graph? (Page 85, #120a)
x = 2005 – 1965 = 40
So we will substitute 40 in for x in the formula:
C = 0.5(40) + 41
C = 20 + 41
C = 21
Multiply.
Add
Answer the question.
Answer: In 2005, 21% of American adults smoked cigarettes. This is exactly the
same as the percentage displayed on the graph.
b.
Now you do part (b): Use the mathematical model to project the percentage of
American adults who will smoke cigarettes in 2019. (page 85, #120b)
Answer: In 2019, 14% of American adults will be smoking cigarettes.
© Copyright 2012 by John Fetcho. All rights reserved