copyright © 2005 pearson education, inc. problem solving method quantitative literacy

Copyright © 2005 Pearson Education, Inc.

Problem Solving Method

Quantitative Literacy

2 Copyright © 2005 Pearson Education, Inc.

Polya’s Procedure

George Polya (1887-1985) developed a general procedure for solving problems.


Guidelines for Problem Solving

Understand the Problem. Devise a Plan. Carry Out the Plan. Check the Results.


1. Understand the Problem.

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.


2. Devise a Plan to Solve the Problem.

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?


3. Carrying Out the Plan.

Use the plan you devised in step 2 to solve the problem.


4. Check the Results.

Ask yourself, “Does the answer make sense?” and “Is it reasonable?” 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?


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.

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?


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.


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.


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).

How far can they travel on their $25 budget? If they include a $2 tip, then how far can they

travel?


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

If they wish to give a $2 tip, we solve the same way only allowing only $23 to be the budget.

$4.80 ($1.68) $25

12 miles.

x

x

$4.80 ($1.68) $23

10.8 miles.

x

x


Integers

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.


Addition of Integers

1. If the signs of the numbers are the same, add the numbers and the sign remains the same.

2. 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 d) -183 + -160 = -343


Subtraction of Integers

1. Write the first number with no changes.2. Change the subtraction symbol to an addition symbol

and then replace the second number with its opposite.3. 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


Properties

Multiplication Property of Zero

Division

For any a, b, and c where b 0, means that c • b = a.

0 0 0a a

a

cb


Rules for Multiplication

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.

864,38446

210370

864,38446

210370


Examples

Evaluate: a) (3)(4) b) (7)(5) c) 8 • 7 d) (5)(8)


Examples

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


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.


Example

Evaluate: a) b)

c) d)

72

98

72

98

72

89

72

89


Order of Operations

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.

723

1

76

76

723

15

53

723

723


Order of Operations

Rank 1: Grouping Symbols (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.

Copyright © 2005 Pearson Education, Inc.

Ratio & Proportion

1.3


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.


Fractions

Fractions are numbers such as:

The numerator is the number above the fraction line.

The denominator is the number below the fraction line.

1 2 9, , and .

3 9 53


Reducing Fractions

In order to reduce a fraction, we divide both the numerator and denominator by the greatest common divisor.

Example: Reduce to its lowest terms.

Solution:

72

81

72 72 9 8

81 81 9 9


Mixed Numbers

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”.


Improper Fractions

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.


Fundamental Property of Proportions

The ratios , form the proportion

If and only if ad = bc (cross products are equal) and

b ≠ 0, d ≠ 0

d

cand

b

a

d

c

b

a


Multiplication/Division Principle of Equality

If a = b, the ac = bc, or

For all real numbers a, b, c with c ≠ 0

c

b

c

a


Proportion Models

Percent Problems

Geometric Problems

Diluted Mixture Problems

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”

c

ba

100Whole

Part

distanceactual

tmeasuremen

scale

1

AM

AU

AS

DS


Division of Fractions

Multiplication of Fractions

, 0, 0.

a c a c acb d

b d b d bd

, 0, 0, c 0.a c a d ad

b db d b c bc


Example: Multiplying Fractions

Evaluate the following.

a)

b)

2 7

3 16

3 11 2

4 2

2 7 2 7 14 7

3 16 3 16 48 24

3 1 7 51 2

4 2 4 2

35 34

8 8


Example: Dividing Fractions

Evaluate the following. a)

b)

2 6

3 7

2 6 2 7

3 7 3 62 7 14 7

3 6 18 9

5 4

8 5

5 4 5 5

8 5 8 45 5 25

8 4 32


Addition and Subtraction of Fractions

, c 0.

, c 0.

a b a b

c c c

a b a b

c c c


Example: Add or Subtract Fractions

Add:

Subtract:

4 3

9 9 11 3

16 16

4 3 4 3 7

9 9 9 9

11 3 11 3 8

16 16 16 161

2

copyright © 2005 pearson education, inc. problem solving method quantitative literacy

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