Chapter 4
Miss Klicka
Academic Algebra 1a
Chapter 4, Section 1: Ratios and Proportions
Pre-read
Read pages 195 - 200. Copy definitions, properties and examples 1 and 4.
Ratio: ____________________________________________________________________________
Proportion: _______________________________________________________________________
Means: ____________________________________________________________________________
Extremes: __________________________________________________________________________
Means-Extremes Property of Proportions:
Example 1)
Chapter 4
Miss Klicka
Academic Algebra 1a
Example 4)
a.)
Class Notes
b.)
Chapter 4, Section 1: Ratios and Proportions
To determine if a Proportion is a True Proportion we simply cross multiply.
2 = 5
6
15
Ask yourself does…
2 * 15 = 6 * 5
30 = 30 Yes Therefore, it is a true proportion
3 = 9
7
21
Does 3 * 21 = 7 * 9?
63 = 63 True proportion
1 = 3
4
8
1*8=4*3
8 ≠ 12 Not a proportion
We can also use cross multiplication to solve for unknowns in as proportion.
4 = x
6
18
Cross Multiply
4 * 18 = 6 * x
72 = 6x
6
Check
4 * 18 = 6 * 12
72 = 72 True
6
12 = x
3 = 7.5
4
x
3 * x = 4 * 7.5
3x = 30
3
2 = 2
3
3 True
3 * 10 = 4 * 7.5
30 = 30 True
3
x = 10
2 = x
3
10.5
To Check w/ Ab/c key
4 = 12
6
18
2 * 10.5 = 3 * x
21 = 3x
3
3
7 = x
2 * 10.5 = 3 * 7
21 = 21 True
Chapter 4
Miss Klicka
Academic Algebra 1a
Examples with an addition/subtraction in the numerator or denominator:
YOU MUST PUT IT IN PARENTHESIS WHEN YOU CROSS MULTIPLY
4 = 2
3 x+2
4(x + 2) = 3 * 2
4x + 8 = 6
-8
-8
4(-1/2 + 2) = 3 * 2
4 (1.5) = 6
6 = 6 True
4x = -2
4
4
x=-½
3 = x–3
5
15
3 * 15 = 5(x – 3)
45 = 5x – 15
+15
+15
3 * 15 = 5(12 – 3)
45 = 5(9)
45 = 45 True
60 = 5x
5
5
12 = x
4 =
7
7
2x + 1
4(2x + 1) = 7 * 7
8x + 4 = 49
-4
-4
8x = 45
8
4[2(5 5/8) + 1] = 7 * 7
4(11 ¼ + 1) = 49
4 (12 ¼ ) = 49
49 = 49 True
8
x = 5 5/8
Practice Problems: Page 198 - 200 (1, 5 – 34) 4.1SG, P. 764 (4.1) 1-18
Chapter 4
Miss Klicka
Academic Algebra 1a
Chapter 4, Section 4: Percents
Pre-read
Read pages 215 - 217. Copy definitions, properties and examples 1 - 3.
Percent: _______________________________________________________________________________
Percentage: ____________________________________________________________________________
Base: __________________________________________________________________________________
Rate: __________________________________________________________________________________
Percent Proportion:
Simple Interest: _________________________________________________________________________
Example 1)
Example 2)
Chapter 4
Miss Klicka
Academic Algebra 1a
Example 3)
Class Notes
Chapter 4, Section 4: Percents
Percent: a ratio that compares a number to 100
Percent = Part
= Is
Base
Whole
Of
In this section we will use the Is/Of Proportion
**It is very important that you MEMORIZE THIS PROPORTION and how to use it**
Is/Of Proportion:
Is = %
Of
100
Remember the denominator of the ratio
on the right IS ALWAYS 100
Examples:
Set up your proportion (Place an “x” where the unknown value is)
30 is what % of 80?
30 = x
80 100
Solutions:
30 * 100 = 80 * x
3000 = 80x
80
80
37.5 = x
46 is what % of 80?
46 = x
80 100
46 * 100 = 80 * x
4600 = 80x
80
80
57.5 = x
70% of what number is 56?
56 = 70
56 * 100 = x * 70
x
100
5600 = 70x
70
70
x = 80
Chapter 4
Miss Klicka
Find 42% of 80.
x = 42
80
100
Academic Algebra 1a
x * 100 = 80 * 42
100x = 3360
100
100
x = 33.6 or 33 3/5
Converting Fraction to a Decimal:
Example:
3 = 0.75
4
Part
= Decimal Equivalent
Whole
Simply divide 3 (Part) by 4 (Whole)
3 = 0.375 = 37.5%
8
3 =
4
Converting to a percent from a fraction
First convert to decimal then multiply by 100
3 ÷ 4 = 0.75
0.75 • 100 = 75%
Simple Interest:
I = PRT
I = Interest in $
P = Principal amount (Amount of loan or deposit balance)
R = Rate (Interest Rate as a DECIMAL)
T = Time (In YEARS)
Examples:
How much interest would I earn if I had $2,300 or deposit for 6 months with 4.25% interest?
I = PRT
I = (2,300)(0.0425)( ½ )
I = $48.88
How much money (principal) would I need to have on deposit for 3 years at 4% to earn $450 in
interest?
I = PRT
$450 = P (0.04)(3)
Isolate Variable
450
Undo multiplication by dividing
=
[(0.04)(3)]
$3750 = p
P (0.04)(3)
(0.04)(3)
Must put use double parenthesis
Chapter 4
Miss Klicka
I = $225, p = $9,000, t = 2.5, r = ?
225 = (9000)r(2.5)
Isolate Variable
225
Undo Multiplication with division
= (9000)r(2.5)
[(9000)(2.5)]
0.01 = r
1% = r
(9000)(2.5)
Must use double parenthesis
Interest needs to be in percent
Move decimal to right 2 times or multiply by 100
Practice Problems: Pages 218 - 221 6-45 4.4SG, P. 765 (4.4) 1-20
Academic Algebra 1a
Chapter 4
Miss Klicka
Academic Algebra 1a
Chapter 4, Section 5: Percent of Change
Pre-read
Read pages 222 - 224. Copy definitions, properties and example 1.
Percent of Decrease: _____________________________________________________________________
Percent of Increase: _____________________________________________________________________
Example 1)
Class Notes
New – Old
Old
Chapter 4, Section 5: Percent of Change
or
Amount of Change
Original Amount
**If New – Old is negative it is a DECREASE
**If New – Old is positive it is an INCREASE
Multiply by 100 to find
percent at the end
Percent Change
Amount of Change • 100
Original Amount
Example:
First test was a 72. Second test was an 81. What is the percent of change?
Use: (New – Old) (100) = (81 – 72) • 100
Old
72
= (.125) • 100
= 12.5% Increase
Chapter 4
Miss Klicka
Academic Algebra 1a
What was the percent change?
Old Price = $40
Sale Price = $30
30 – 40 • 100
40
-10 • 100
40
-10 • 100
40
= (-0.25) • 100 = -25
= 25% Decrease
What was the percent change?
Old Price = $30
New Price = $40
40 – 30 • 100
30
10 • 100
30
= (.333) • 100 = 33.3333
= 33.3% Increase
(not a negative means it will be an increase)
Finding Final Price
Price – Discount = Sale Price
Discount = Price • Discount Rate
Sale Price + Tax = Final Price
Tax Amount = Sale Price • Tax Rate
* Remember Discount Rate and Tax Rate MUST BE DECIMALS.
To convert from % to decimal you simply divide by 100
or
shift the decimal two places to the left
Chapter 4
Miss Klicka
Academic Algebra 1a
Example:
Jimmy finds a sale on sneakers.
Originally $95, they are 20% off.
PA sales tax is 6%.
What is his final price?
Discount = Original Price • Discount Rate
? = $95 • 20%
$19 = 95 • 0.20
Sale Price = 0riginal Price – Discount
? = $95 – $19
$76 = 95 – 19
Tax Amount = Sale Price • Tax Rate
? = $76 • 6%
$4.56 = 76 • 0.06
Final Price = Sale Price + Tax Amount
? = $76 + $4.56
$80.56 = 76 + 4.56
Brittany & Michelle go shopping for math supplies. They find a big sale on a “Deluxe Math Supply
Box” that contains everything they need for math. She has $76.50. She wants to know if she has
enough money to buy the box. She really really wants it! The box costs $80. However, it is currently
10% off. PA sales tax is 6%. Does she have enough money to buy the “box” for her favorite class?
Discount = Original Price • Discount Rate
? = $80 • 10%
$8 = 80 • 0.10
Sale Price = Original Price – Discount
? = $80 – $8
$72 = 80 – 8
Tax Amount = Sale Price • Tax Rate
? = $72 • 6%
$4.32 = 72 • 0.06
Final Price = Sale Price + Tax Amount
? = $72 + $4.32
$76.32 = 72 + 4.32
 Yes, they will have enough money to buy the “Deluxe Math Supply Box” 
Chapter 4
Miss Klicka
Is
= %
Of
What number increased by 30% equals 260?
260 = 100 + 30
x
100
260 = 130
x
100
260 * 100 = x * 130
26000 = 130x
26000 = 130x
130
130
200 = x
What number decreased by 25% is 160?
160 = 100 – 25
x
100
160 = 75
x
100
160 * 100 = x * 75
16000 = 75x
16000
75
= 75x
75
213 1/3 = x
Academic Algebra 1a
& Cross Multiply
100
**equals = is**
Chapter 4
Miss Klicka
Academic Algebra 1a
An item sells for $70 after a 33 1/3 % discount
I.e. price is decreased by 331/3%
What is the original price?
70 = 100 – 33 1/3
x
100
70 = 66 2/3
x
100
70 * 100 = x * 66 2/3
7000 = 66 2/3 x
7000 = 66 2/3 x
66 2/3
66 2/3
$105 = x
Practice Problems: Pages 224 - 227 (1-3, 5-30) 4.5SG, P. 765 (4.5) 1-12
Chapter 4
Miss Klicka
Academic Algebra 1a
Chapter 4, Section 6: Probability and Odds
Pre-read
Read pages 228 - 230. Copy definitions, properties and example 3.
Probability: ____________________________________________________________________________
Probability of an Event: __________________________________________________________________
Definition of Probability
Equally Likely: _________________________________________________________________________
Random: ______________________________________________________________________________
Odds: _________________________________________________________________________________
Definition of Odds
Chapter 4
Miss Klicka
Academic Algebra 1a
Example 3)
Class Notes
Chapter 4, Section 6: Probability and Odds
Probability =
# of Favorable Outcomes
# of Total Outcomes
Example: P(event)
If I roll a die what is the probability of rolling…?
P(2) = 1/6 or 1:6
P(odd) = 3/6 = ½ or 1:2
P(2 or 4) = 2/6 = 1/3 or 1:3
P(7) = 0/6 or 0:6 will never happen
P(1-6) = 6/6 = 1 P(event) will occur
P(factors of 6) = 4/6 = 2/3 or 2:3
(1,2,3,6)
Ratio
Chapter 4
Odds =
Miss Klicka
# of Successful Outcomes
# of Unsuccessful Outcomes
Academic Algebra 1a
Numerator & Denominator
should add up to
Total number of outcomes
Examples:
If I roll a die what are the odds of rolling…?
Odds(2) = 1/5 NUMERATOR & DENOMINATOR SHOULD SUM TO TOTAL POSSIBLE OUTCOMES
Odds(2 or 4) = 2/4 = ½ or 1:2
Odds(odd) = 3/3 = 1/1 or 1:1 Do Not Change to just 1; must have the denominator
Practice Problems: Pages 230 - 232 (6 -36) 4.6SG, P. 766 (4.6) 1-18
Chapter 4
Miss Klicka
Academic Algebra 1a
Practice Problems are in GREEN
KNOW VOCAB& FORMULAS IN BOLD
Chapter 4 Study Guide
**Review Class Notes**
RATIO & PROPORTION
Ratio:
a comparison of 2 numbers by division
Example:
boys to girls can be expressed as...
b
g
b:g
b to g
Proportion: an equation stating that two ratios are equal
**You can check to see if two ratios are equal by cross multiplication**
(in other words they are a true proportion)
Example:
Are the following ratios equal?
2? 5
6 = 15
2 * 15 =? 6 * 5
30 = 30 True / These ratios are equal and they are a true proportion
Using cross multiplication to find unknowns…
Example:
2= x
3 21
cross multiply and set up an equation
2 * 21 = 3 * x

2 ? 14
3 = 21
Simplify 2 = 2
3 3
or
2 * 21 = 3 * 14
42 = 42
42 = 3x
3
3
14 = x
USE ( ) WHEN YOU HAVE AN OPERATION IN THE NUMERATOR OR DENOMINATOR…
Example: Find x
Steps for solving an equation with
variables on both sides:
1. DISTRIBUTE (remove
parenthesis)
2. COMBINE LIKE TERMS (on
each side of the =)
3. GET THE VARIABLES ON ONE
SIDE (move smaller variable
by adding or subtracting)
4. UNDO
ADDITION/SUBTRACTION
5. UNDO
MULTIPLICATION/DIVISION
6. CHECK YOUR SOLUTION
Check: use ( ) when you
substitute your answer
x +7 = x + 1
6
3
3(x + 7) = 6(x + 1)
3x + 21 = 6x + 6
-3x
-3x
21 = 3x + 6
-6
-6
15 = 3x
3
3
5=x
Solving Proportion Practice Problems:
3 = 15
5
25
(Check solutions)
3–x = 8
4+x
48

(5) + 7 = (5) + 1
6
3
12 = 6
6
3
2 = 2 True
Chapter 4
Miss Klicka
Academic Algebra 1a
PROBABILITY & ODDS
Probability: the chance that a certain event will occur
**Written as P(event)
If P(event) = 0
If P(event) = 1
The event will never happen
The event will definitely happen
To calculate: # of favorable outcomes
# of total possible outcomes
Odds:
# of favorable outcomes
# of unfavorable outcomes
**Written as ½ or 1:2 /always keep as a ratio for odds
Probability & Odds of rolling a 3 on a die
Probability: # of favorable outcomes
# of total possible outcomes
1
6
Odds: # of favorable outcomes
1
# of unfavorable outcomes 5
Practice Problems for Probability & Odds:
There is a bowl of money. The bowl contains 50 quarters, 75 dimes, 100 nickels, 125 pennies.
What are the odds of choosing a penny?
What is the probability that a quarter will be chosen?
IS/OF PROPORTION
IS =
OF
%(percent)
100
Example:
What is 12% of 50?
 6 = 12
50 100
x = 12
50
100
x * 100 = 50 * 12
100x = 600
100
100
x=6
ISOLATE VARIABLE
6 * 100 = 50 * 12
600 = 600 True
Chapter 4
Miss Klicka
Academic Algebra 1a
Practice Problems:
What is 75% of 24?
Twelve is 20% of what number?
PERCENT OF CHANGE
amount of change = %
original price
100
or
new-old = %
old
100
ITEM SELLS FOR $45 AFTER A 20% DISCOUNT. FIND ORIGINAL PRICE.
45 = 80
x
100
80 = 100% - 20% discount
 $56.25 * .20 (20%) = $11.25
$56.25 - $11.25 = $45
Correct
4500 = 80x
80
80
$56.25 = original price
PRICE WAS DECREASED FROM $25 TO $10. FIND THE PERCENT OF DECREASE.
15 = x
25 100
15 = amount of change / new - old
1,500 = 25x
25
25
x =60
60% percent of decrease.
 $25 * .60 (60%) = $15
$25 - $15 = $10
Correct
Percent of Change Practice Problems:
Price was decreased from $120 to $114. Find the percent of change and tell if it is a decrease or
increase.
CALCULATING COST
SALE PRICE
= PRICE – DISCOUNT
DISCOUNT AMOUNT
= PRICE * DISCOUNT (AS A DECIMAL)
TAX AMOUNT
= SALE PRICE * TAX RATE (AS A DECIMAL)
Chapter 4
FINAL PRICE
Miss Klicka
Academic Algebra 1a
= SALE PRICE + TAX
Example:
CD:
Discount:
Tax:
19.99
19.99
14.99
14.99
$19.99
25%
6%
* .25 = $5.00
- 5.00 = 14.99
* .06 = .90
+ .90 = 15.89
Final Price of CD = $15.89
Practice Problem:
Class Ring:
Group Discount:
Sales tax:
$89.00
17%
5%
CALCULATING SIMPLE INTEREST
Simple Interest:
I=prt
(Interest = principle * rate (as a decimal) * time (always in years))
Example:
Find r if I = $780, p = $6500, t = 1 year
780 = 6500 * r * 1
780 = 6500r
6500 6500
.12 = r
12% is the rate
Practice Problem:
What interest rate does Dave need to get to earn $200 24 months after he deposits $5,000?
Chapter 4
***Reminder***
Percent
Base
Part
Whole
Is
Of
Miss Klicka
Academic Algebra 1a