1. 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

2. 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

3. 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)

4. 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

5. 1.1 Problem Solving with Fractions
Solving Application Problems
Read and understand the problem
Know what is given and work out a plan to answer what is to be found.
Estimate a reasonable answer
Solve the problem by using the facts given and your plan

6. 1.1 Problem Solving with Fractions
Estimating a reasonable answer: which of the following would be a reasonable cost for a man’s shirt?
$.65 $1
$20
$1000

7. 1.2 Adding and Subtracting Fractions
Adding fractions with the same denominator:
Subtracting fractions with the same denominator:

8. 1.2 Adding and Subtracting Fractions – Factor Trees18 6 3 2 3

9. 1.2 Adding and Subtracting Fractions
To add or subtract fractions with different denominators - get a common denominator.
Using the least common denominator:
Factor both denominators completely
Multiply the largest number of repeats of each prime factor together to get the LCD
Multiply the top and bottom of each fraction by the number that produces the LCD in the denominator

10. 1.2 Adding and Subtracting Fractions – no common factors in denominator
Adding fractions with different denominators:
Subtracting fractions with different denominators:

11. 1.2 Adding and Subtracting Fractions
Try these:

12. 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

13. 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:

14. 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:
Factor the top and bottom completely
Divide the top and bottom by the greatest common factor

15. 1.3 Multiplying and Dividing Fractions
Multiplying fractions:
Dividing fractions (multiply by the reciprocal):

16. 1.3 Multiplying and Dividing Fractions
Try these:

17. 1.3 Multiplying and Dividing Fractions
Converting decimals fractions:
Converting fractions to decimals:

18. 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.

19. 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:

20. 2.1 Solving Equations
Example of solving an equation:

21. 2.2 Applications of Equations
Translate the following:
The sum of a number and 16
Subtract a number from 5.4
The product of a number and 9
The quotient of a number and 11
Four-thirds of a number

22. 2.2 Applications of Equations
When 5 times a number is added to twice the number, the result is 10. Find the number.
x is the variable representing the number.
Equation:
Solve:
Check:

23. 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

24. 2.3 Formulas Example: Solve for T in the formula: Distribute:
Subtract P from both sides
Divide by PR

25. 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:

26. 2.4 Ratios and Proportions
Proportion – statement that two ratios are equal Examples: Cross multiplication: if then

27. 2.4 Ratios and Proportions
Solve for x: Cross multiplication: so x = 63

28. 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:

29. 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:

30. 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:

31. 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:

32. 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:

33. 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

34. 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

35. 3.2 Identifying the Base and the Part
Usually 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

36. 3.3 Finding the Base
Using the Basic Percent Equation to solve for Base: 22.5 is 30% of _____

37. 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?

38. 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?

39. 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 = = 75%

40. 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

41. 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?

42. 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.

43. 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

44. 3.5 Increase and Decrease Problems
Increase Problem: Original + Increase = New value (base) (part)
Decrease Problem: Original - Decrease = New value (base) (part)

45. 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

46. 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:

47. 3.5 Increase and Decrease Problems
Widget problem (continued). To get the # of widgets produced 2 years ago:

48. 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?