1.1 Problem Solving with Fractions

• Addition words Plus, more, more than, added to, increased by,

sum, total, sum of, increase of, gain of

• Subtraction words Less, subtract, subtracted from, difference,

less than, fewer, decreased by, loss of, minus, take away

1.1 Problem Solving with Fractions• Multiplication words

Product, double, triple, times, of, twice, twice as much

• Division words Divided by, divided into, quotient, goes into,

divide, divided equally per

• Equals Is, the same as, equals, equal to, yields,

results in, are

1.1 Problem Solving

• Changing word phrases to expressions:

The sum of a number and 9 x + 9

7 minus a number 7 - x

Subtract 7 from a number x – 7

The product of 11 and a number 11x

5 divided by a number

The product of 2 and the sum of a number and 8

2(x + 8)x5

1.1 Problem Solving

• Equation: statement that two algebraic expressions are equal.

Expression Equation

x – 7 x – 7 = 3

No equal sign Contains equal sign

Can be evaluated or simplified

Can be solved

1.1 Problem Solving with Fractions

• Solving Application Problems1. Read and understand the problem

2. Know what is given and work out a plan to answer what is to be found.

3. Estimate a reasonable answer

4. Solve the problem by using the facts given and your plan

1.1 Problem Solving with Fractions

• Estimating a reasonable answer: which of the following would be a reasonable cost for a man’s shirt?

1. $.65

2. $1

3. $20

4. $1000

1.2 Adding and Subtracting Fractions

• Adding fractions with the same denominator:

• Subtracting fractions with the same denominator:

b

ca

b

c

b

a

b

ca

b

c

b

a

1.2 Adding and Subtracting Fractions – Factor Trees

18

2

63

3

1.2 Adding and Subtracting Fractions

• To add or subtract fractions with different denominators - get a common denominator.

• Using the least common denominator:1. Factor both denominators completely2. Multiply the largest number of repeats of each

prime factor together to get the LCD3. Multiply the top and bottom of each fraction

by the number that produces the LCD in the denominator

1.2 Adding and Subtracting Fractions – no common factors in denominator

• Adding fractions with different denominators:

• Subtracting fractions with different denominators:

db

cbda

d

c

b

a

db

cbda

d

c

b

a

1.2 Adding and Subtracting Fractions

• Try these:?

9

5

9

1

?21

2

7

5

?4

1

9

5

1.2 Adding and Subtracting Fractions

• Proper fraction – numerator is less than the denominator

• Improper fraction - numerator is greater than the denominator

• Mixed fraction – sum of a fraction and a whole number

1.2 Adding and Subtracting Fractions

• Converting a mixed fraction to an improper fraction:

• Converting an improper fraction to a mixed fraction:

Divide 9 into 35:

8

27

8

383

8

33

9

35

9

83

9

35 8

27

3359

1.3 Multiplying and Dividing Fractions

• Multiplying or dividing the numerator (top) and the denominator (bottom) of a fraction by the same number does not change the value of a fraction.

• Writing a fraction in lowest terms:1. Factor the top and bottom completely

2. Divide the top and bottom by the greatest common factor

1.3 Multiplying and Dividing Fractions

• Multiplying fractions:

• Dividing fractions (multiply by the reciprocal):

db

ca

d

c

b

a

cb

da

c

d

b

a

d

c

b

a

1.3 Multiplying and Dividing Fractions

• Try these:

(simplify) 16

12

?14

3

9

7

?5

3

10

9

1.3 Multiplying and Dividing Fractions

• Converting decimals fractions:

• Converting fractions to decimals:

8

1

40

5

200

25

1000

125125.0

040

040

56

60

42

37500038

8

3

.

.

.

.

.

.

.

2.1 Solving Equations

• A linear equation in one variable can be written in the form: Ax + B = 0

• Linear equations are solved by getting “x” by itself on one side of the equation

• Addition Property of Equality: if A=B then A+C=B+C

• Multiplication Property of Equality: if A=B and C is non-zero, then AC=BC

• General rule: Whatever you do to one side of the equation, you must also do it to the other side.

2.1 Solving Equations• Some equations have more than one term

with the same variable. These are called “like terms”

• Like terms can be combined by adding the coefficients:

zzz

yyy

xxxx

15123

369

5528

xxx 528

2.1 Solving Equations

• Example of solving an equation:

4

205

8125

82127

k

k

k

kk

2.2 Applications of Equations

• Translate the following:

1. The sum of a number and 16

2. Subtract a number from 5.4

3. The product of a number and 9

4. The quotient of a number and 11

5. Four-thirds of a number

2.2 Applications of Equations• When 5 times a number is added to twice the

number, the result is 10. Find the number.

1. x is the variable representing the number.

2. Equation:

3. Solve:

4. Check:

1025 xx

73

710 1

107

x

x

10)(2)(5 770

720

750

710

710

2.3 Formulas

• I = PRT• M = P(1 + RT)• G = NP

• S = C + M

• Interest = principal x rate x time• Maturity value• Gross sales = number of items

sold x price per item• Selling price = cost of the item +

markup

2.3 Formulas

• Example: Solve for T in the formula:

1. Distribute:

2. Subtract P from both sides

3. Divide by PR

)1( RTPM PRTPM

PRTPM

PR

PMT

2.4 Ratios and Proportions

• Ratio – quotient of two quantities with the same units

Examples: a to b, a:b, or

Note: percents are ratios where the second number is always 100:

ba

10035%35

2.4 Ratios and Proportions

• Proportion – statement that two ratios are equal

Examples:

Cross multiplication:

if then

dc

ba

bcad dc

ba

2.4 Ratios and Proportions

• Solve for x:

Cross multiplication:

so x = 63

7981 x

x9567x 9781

3.1 Writing Fractions and Decimals as Percents

• Write a decimal as a percent by moving the decimal point 2 places to the right and attaching a percent sign:

• Example:%2.38382.0

3.1 Writing Fractions as Percents• Write a fraction as a percent by converting

the fraction to a decimal and then converting the decimal to a percent:

• Example:

%5.37375.0 040

040

56

60

42

37500038

8

3

.

.

.

.

.

.

.

3.1 Writing Fractions and Decimals as Percents

• Write a percent as a decimal by moving the decimal point 2 places to the left and removing the percent sign:

• Example:%34141.3

3.1 Writing Fractions and Decimals as Percents

• Write a percent as a fraction by first changing the percent to a decimal then changing the decimal to the fraction and reduce:

• Example:

20

9

520

59

100

4545.0%45

3.1 Writing Fractions and Decimals as Percents

• Write a fractional percent as a decimal by first changing the fractional part to a decimal and leaving the percent sign. Then move the decimal point 2 places to the left and removing the percent sign:

• Example: 0325.%25.3%3 41

3.2 Finding the Part

• B = Base – the whole or the total

• R = Rate – a number followed by “%” or “percent”

• P = Part – the result of multiplying base times rate

)()()( rateRbaseBpartP

3.2 Finding the Part for a Business Problem

• B = Base – sales, R = Rate – sales tax rate, P = Part – sales tax

• Example: If the sales tax rate is 5%, what is the sales tax and total sale on $133 of merchandise

65.6$

05.133$%5133$

P

P

RBP

3.2 Identifying the Base and the PartUsually the Base Usually the Part

Sales Sales Tax

Investment Return

Savings Interest

Retail Price Amount of Discount

Value of Real Estate Rents

Total Sales Commission

Value of Stocks Dividends

Earnings Expenditures

Original Change

3.3 Finding the Base• Using the Basic Percent Equation to solve

for Base:22.5 is 30% of _____

753

2253.0

5.22

3.05.22

%305.22

B

B

B

B

3.3 Finding the Base• Finding sales when sales tax rate is given:

The 5% sales tax collected by a store was $380. What was the total amount of sales?

7600$05.0

380$

380$05.

380$B of %5

B

B

3.3 Finding the Base• Finding the amount of an investment:

The yearly maintenance cost of an apartment is 2½% of its value. If maintenance is $37,000 per year, what is the value of the apartment complex?

000,480,1$025.0

37000$

37000$025.

37000$B of %2 21

B

B

3.3 Finding the Base• Finding the base if rate and part are different

quantities:United Hospital finds that 25% of its employees are men and 720 are women are women. What is the total number of employees?First – if 25% are men, then the percent of women = 100-25 = 75%

96075.0

720

72075.

720B of %75

B

B

3.4 Finding the Rate

• Using the percent equation to solve for rate:45 is what percent of 180?

Note: Rate is always expressed as a percent

%2525.04

1

180

45

45180

45180 of _____%

R

R

3.4 Finding the Rate

• Finding rate of return when the amount of return and the investment are known:$3400 is invested in a new computer yielding additional income of $1700. What is the rate of return?

%5050.02

1

3400

1700

17003400

1700$$3400 of _____%

R

R

3.4 Finding the Rate• Solving for the percent remaining:

A car is expected to last 10 years before it needs replacement. If the car is 7 years old, what percent of the car’s life remains?To find the number of years remaining subtract 7 from 10 to get 3 years left.

%3030.010

3

310

310 of _____%

R

R

3.4 Finding the Rate• Find the percent of increase/decrease:

Sales of digital cameras went from $40,000 to $100,000. Find the percent increase.Increase = $100,000 - $40,000 = $60,000

%1505.1000,40

000,60

000,60000,40

000,60$$40,000 of _____%

R

R

3.5 Increase and Decrease Problems

• Increase Problem:

Original + Increase = New value (base) (part)

• Decrease Problem:

Original - Decrease = New value (base) (part)

3.5 Increase and Decrease Problems

• The value of a house is $143,000 this year. That is 10% more than last year’s value. What was the value of the home last year?Last year’s value + 10% of last year’s value = this year’s value

000,130$1.1

000,143

000,143$1.1

000,143$%110

000,143$%10100%

B

B

B

BB

3.5 Increase and Decrease Problems

• Finding the base after 2 increases:This year’s production of widgets was 144,000. It is 20% more than last year’s production which was also 20% more than the previous year’s production. Find the number of widgets produced 2 years ago. To find last year’s # of widgets:

000,1202.1

000,144

000,1442.1

000,144%120

000,144%20100%

B

B

B

BB

3.5 Increase and Decrease Problems

• Widget problem (continued). To get the # of widgets produced 2 years ago:

000,1002.1

000,120

000,1202.1

000,120%120

000,120%20100%

B

B

B

BB

3.5 Increase and Decrease Problems

• Decrease problem:Craig paid $450 for an LCD TV set. The price he paid was 10% less than the original price. What was the original price?

500$9.

450$

450$9.

450$%90

450$%10100%

B

B

B

BB