Transcript Math 120 Chapter 1-1.1-1.3

Quantitative Literacy
Problem Solving Method
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Polya’s Procedure

George Polya (1887-1985) developed a
general procedure for solving problems.
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Slide 5-2
Guidelines for Problem Solving
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Understand the Problem.
Devise a Plan.
Carry Out the Plan.
Check the Results.
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Slide 5-3
1. Understand the Problem.
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Read the problem carefully, at least twice.
Try to make a sketch of the problem. Label the
given information given.
Make a list of the given facts that are pertinent
to the problem.
Decide if you have enough information to solve
the problem.
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Slide 5-4
2. Devise a Plan to Solve the Problem.
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Can you relate this problem to a previous problem that
you’ve worked before?
Can you express the problem in terms of an algebraic
equation?
Look for patterns or relationships.
Simplify the problem, if possible.
Use a table to list information to help solve.
Can you make an educated guess at the solution?
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Slide 5-5
3. Carrying Out the Plan.

Use the plan you devised in step 2 to solve the
problem.
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Slide 5-6
4. Check the Results.

Ask yourself, “Does the answer make sense?” and “Is it
reasonable?”
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If the answer is not reasonable, recheck your method and
calculations.
Check the solution using the original statement, if
possible.
Is there a different method to arrive at the same
conclusion?
Can the results of this problem be used to solve other
problems?
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Slide 5-7
Example: Selling a House
The Sharlow’s are planning to sell their home. They
want to be left with $139,500 after paying commission
to the realtor.
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If a realtor receives 7% of the selling price, how much
must they sell the house for?
If a realtor receives a flat $5000 and then 3% of the
selling price, how much must they sell their house for?
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Slide 5-8
Solution

Translate the problem in algebraic terms as follows:
selling price less commission is amount left.
x - 0.07x = $139,500
Thus, the Sharlow’s need to sell their home for
$150,000.
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Slide 5-9
Solution continued

If the realtor receives a flat fee of $5000, then
3% commission, first add 3% to the amount
they wish to be left with.
$139,500 + 3%  $143,814.
Then add $5000 to this total.
$143,814 + $5000 = $148,814.
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Slide 5-10
Example: Taxi Rates
In Mexico, a taxi ride costs $4.80 plus $1.68
for each mile traveled. Diego and Juanita
budgeted $25 for a taxi ride (excluding tip).
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How far can they travel on their $25 budget?
If they include a $2 tip, then how far can they
travel?
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Slide 5-11
Solution

We know that the initial charge plus the mileage
charge can equal $25. So, if we let x equal the
distance, in miles, driven by the taxi for $25, then
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If they wish to give a $2 tip, we solve the same way
only allowing only $23 to be the budget.
$4.80  x($1.68)  $25
x  12 miles.
$4.80  x($1.68)  $23
x  10.8 miles.
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Slide 5-12
Integers
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The set of integers consists of 0, the natural
numbers, and the negative natural numbers.
Integers = {…-4,-3,-2,-1,0,1,2,3,4,…}
On a number line, the positive numbers extend
to the right from zero; the negative numbers
extend to the left from zero.
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Slide 5-13
Addition of Integers
1.
2.
If the signs of the numbers are the same, add the
numbers and the sign remains the same.
If the signs of the numbers are different, subtract the
larger value minus the smaller value. The result
inherits the sign of the value farthest from zero on the
number line.
Evaluate:
a) 20 + (–140) = –120 c) 75 + 98 = 173
b) 270 + (–170) =100
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d) -183 + -160 = -343
Slide 5-14
Subtraction of Integers
1.
2.
3.
Write the first number with no changes.
Change the subtraction symbol to an addition symbol
and then replace the second number with its opposite.
Carry out the rules for the addition of signed numbers.
a – b = a + (b) = solution
Evaluate:
a) –7 – 3 = –7 + (–3) = –10
b) –7 – (–3) = –7 + 3 = –4
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Slide 5-15
Properties
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Multiplication Property of Zero
a0  0a  0
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Division
a
For any a, b, and c where b  0,  c means
b
that c • b = a.
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Slide 5-16
Rules for Multiplication
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The product of two
numbers with like signs
(positive  positive or
negative  negative) is a
positive number.
The product of two
numbers with unlike
signs (positive  negative
or negative  positive) is
a negative number.
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46  84  3,864
 70  3  210
 46  84  3,864
70  3  210
Slide 5-17
Examples
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Evaluate:
a) (3)(4)
c) 8 • 7
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b) (7)(5)
d) (5)(8)
Slide 5-18
Examples
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Evaluate:
a) (3)(4)
b) (7)(5)
c) 8 • 7
d) (5)(8)
Solution:
a) (3)(4) = 12
b) (7)(5) = 35
c) 8 • 7 = 56
d) (5)(8) = 40
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Slide 5-19
Rules for Division

The quotient of two numbers with like signs
(positive  positive or negative  negative) is a
positive number.

The quotient of two numbers with unlike signs
(positive  negative or negative  positive) is a
negative number.
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Slide 5-20
Example
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Evaluate:
a) 72  8
b)
72
 8
9
d)
72
 9
8
9

c)
72
9
8
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Slide 5-21
Order of Operations
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The order in which we
carry out operations is
important!
Different methods result
in different solutions.
We must agree upon
only one solution for
every problem.
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3 2  7
3 2  7
3 2  7
67
3  2  7
6  7
3  5
1
 15
Slide 5-22
Order of Operations
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Rank 1: Grouping Symbols
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(Parentheses, Brackets, etc. innermost first)
Rank 2: Exponents (Powers) & Roots (from left to right)
Rank 3: Multiplication & Division (from left to right)
Rank 4: Addition & Subtraction (from left to right)
Follow these in order of operation and you always arrive at the
standard solution for a problem.
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Slide 5-23
Ratio & Proportion
1.3
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The Rational Numbers

The set of rational numbers, denoted by Q,
is the set of all numbers of the form p/q,
where p and q are integers and q  0.
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Slide 5-25
Fractions

Fractions are numbers such as:
1 2
9
, , and
.
3 9
53
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The numerator is the number above the fraction
line.
The denominator is the number below the
fraction line.
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Slide 5-26
Reducing Fractions
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In order to reduce a fraction, we divide both the
numerator and denominator by the greatest
common divisor.
72
Example: Reduce
to its lowest terms.
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Solution: 72  72  9  8

81
81
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81  9
9
Slide 5-27
Mixed Numbers
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A mixed number consists of an integer and a
fraction. For example, 3 ½ is a mixed number.
3 ½ is read “three and one half” and means
“3 + ½”.
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Slide 5-28
Improper Fractions
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Rational numbers greater than 1 or less than -1
that are not integers may be written as mixed
numbers, or as improper fractions.
An improper fraction is a fraction whose
numerator is greater than its denominator.
An example of an improper fraction is 12/5.
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Slide 5-29
Fundamental Property of Proportions

The ratios
a
c
and
b
d
, form the proportion
a c

b d
If and only if ad = bc (cross products are equal) and
b ≠ 0, d ≠ 0
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Slide 5-30
Multiplication/Division Principle of Equality

If a = b, the ac = bc, or
a b

c c
For all real numbers a, b, c with c ≠ 0
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Slide 5-31
Proportion Models
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Percent Problems
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Geometric Problems
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Diluted Mixture Problems
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DS = “Desired Strength” of solution to be made (Percent)
AS = “Available Strength” of solution to be used (Percent)
AU = “Amount” of available solution “Used”
AM = “Amount” of desired solution “Made”
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Part
a
b


Whole 100 c
1
measuremen t

scale actual distance
DS AU

AS AM
Slide 5-32
Multiplication of Fractions
a c a  c ac
 

, b  0, d  0.
b d b  d bd

Division of Fractions
a c a d ad
   
, b  0, d  0, c  0.
b d b c bc
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Slide 5-33
Example: Multiplying Fractions

Evaluate the following.
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a) 2  7
3 16
2 7
2  7 14
7




3 16 3  16 48 24
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
b)
 3   1
1 4    2 2 
  

 3   1 7 5
1 4    2 2   4  2
  

35
3

4
8
8
Slide 5-34
Example: Dividing Fractions
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Evaluate the following.
a) 2 6

3 7
2 6 2 7
  
3 7 3 6
2  7 14 7



3  6 18 9
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
b) 5 4

8 5
5 4 5 5
 

8 5
8 4
5  5 25


84
32
Slide 5-35
Addition and Subtraction of Fractions
a b ab
 
, c  0.
c c
c
a b ab
 
, c  0.
c c
c
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Slide 5-36
Example: Add or Subtract Fractions

Add: 4  3
9 9
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4 3 43 7
 

9 9
9
9

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Subtract:
11 3

16 16
11 3 11  3 8



16 16
16
16
1

2
Slide 5-37