Unit 5: Proportions and Equations

Lesson1: Reduce RatiosRatio: a relationship between 2 numbers 3 ways to write a ratio: Example: there are 8 white dogs for every 2 black dogs

1. As a Fraction 82

2. With a colon 8:2

3. With the word to 8 to 2

Reduce Ratios: the same way you reduce fractions. Find a number that will divide evenly into both

Example: 8:2 GCF (Greatest Common Factor) is 2 8/2 = 4 and 2/2 = 1 4:1 is the reduced ratio (be sure to leave the 1 so we can compare)

Tip for Ratio Word Problems: before you write the ratio, ask yourself this question: HOW MANY ARE THERE for each item.

Example: there are 30 kids in a class and 17 boys. What is the ration of girls to boys?How many girls are there: 30 – 17 = 13 girlsHow many boys are there: 17Ratio of boys to girls: 13:17

Examples and Notes from Class:

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Lesson 2: Unit RatesUnit Rate: a ration that compares 2 numbers and has a 1 as the second value. In other words, the rate for 1 thing

To convert a Ratio to a Unit Rate: divide both numbers by the second number

Example: Sarah drove 120 miles in 3 hours. What is the Unit Rate? (The unit rate is the same as saying, how far did Sarah drive in 1 hour?)

120miles3hours ÷ 33 = 40miles1hour Divide the top and bottom by 3

The Unit Rate is 40 miles in 1 hour or 40 miles per hour (mph)

Example with Decimals: David was paid $106 for 8 hours of work. What is the Unit Rate:

The Unit Rate is $13.25 per hour

Notes from Class:

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Lesson 3: Solve Applications with Unit RatesExample: Max’s family is driving across the country. They drove 400 miles in 8 hours.

1. Calculate the Unit Rate: 400miles8hours ÷ 88 = 50 miles in 1 hour

2. How far would they travel in 15 hours?Multiply the unit rate times the number of hours:

50 miles x 15 hours = 750 miles in 15 hours

Determine a Unit Rate from a Graph: this graph shows the distance Mickey rode on his bike in different amounts of time

Choose any point that is at the intersection of 2 lines

This point shows 12 miles in 40 minutes

Unit Rate: 12miles40minutes ÷ 4040 = .3 miles in 1 min

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Notes and Examples from Class

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Lesson 4: Equations and Proportional RelationshipsProportion: a comparison of 2 equal ratios

Determine if 2 Ratios are a Proportion:

Example:

Method 1: Find the Unit Rate: The Unit Rates for both are $.5 for 1 banana, so this is a proportion

Method 2: Cross Multiply

You get 36 when you cross multiply, so this is a proportion

Notes and Examples from Class

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Determine if an Equation represents a Proportional Relationship: in other words, will x and y increase or decrease at the same rate?

Example: y = 6x Choose 2 values for x and solve the equation for y

SHORTCUT: if the equation is in the form of y = kx (k is any rational number), then the equation represents a proportional relationship. Get y alone on one side. If the other side has something times x, it’s proportional

Example: y + 4 = 3xy + 4 – 4 = 3x – 4 subtract 4 from both sidesy = 3x – 4 this is NOT in the form of y = kx so it is NOT proportional

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Lesson 6: Identify if a Graph or Table shows a Proportional RelationshipProportional Relationship: means that rate is the same for each pair of values

Example: Does this table show a proportional relationship?Express each pair of values is a ratio and simplify

15 is already simplified

210 ÷ 22 = 15

315 ÷ 33 = 15

420 ÷ 44 = 15

Since the ratio for each is 15 , the table shows a proportional relationship

In other words, the y value will always be 5 times bigger than the x value

Notes and Examples from Class:

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Proportional Relationships as a GraphLinear: all points are on the same straight line

How to determine if a graph represents a proportional relationship: Determine if the rates from each pair in the table are the same.

Example 1:

21,42 ,63 all simplify to the same rate of 2. Therefore, this is a proportional

relationship and the graph will go through the origin (0,0)

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Example 2:

24 simplifies to 12;

46 simplifies to 23;

68 simplifies to 34

Since the rates are different, this is not a proportional relationship and the graph does not go through the origin (0,0)

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Lesson 7: Compare ProportionsCopy notes and examples from class

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Lessons 8: Solve One Step Linear EquationsVariable: the unknown value in an equation. Represented by a letterSolve an Equation: determine the value for the variable that makes the equation

trueOpposite Operations: Addition and Subtraction are opposites Multiplication and Division are opposites

To solve an Equation: Get the variable alone on one side of the equal sign by using

opposite operationsExample: x – 4 = 7 to get x alone on the left side, get rid of the 4 x – 4 + 4 = 7 + 4 Since 4 is being subtracted, add it to both sides x = 11 Simplify 11 – 4 = 7 Replace x with 11 in the original equation

7 = 7 11 is the correct answer

Example: 5x = 60 to get x alone on the left side, get rid of the 5

5x5 = 605 since 5 is being multiplied, divide both sides by 5

x = 12 5/5 = 1, so it cancels out. 60/5 = 12 5 • 12 = 60 replace x with 12 in the original equation

60 = 60 60 is the correct answer

Example: -3x = -18 to get x alone on the left side, get rid of the -3

−3 x−3 = −18−3 since -3 is being multiplied, divide both sides by -3

x = 6 -3/-3 = 1 so it cancels out. -18/-3 = 6 -3 • 6 = -18 replace x with 6 in the original equation -18 = -18 6 is the correct answer

EQUATION SHORTCUT: you can move any term to the other side of the equation, as long as you change the sign

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Notes and Examples from Class:

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Lessons 9: Two Step EquationsTo solve an Equation: Get the variable alone on one side of the equal sign by using

opposite operations

Example: 4x + 7 = 31 4x + 7 – 7 = 31 – 7 get rid of the 7 by subtracting 7 from both sides 4x = 24

4 x4 = 244 divide both sides by 4

x = 6 4 • 6 + 7 = 31 substitute 6 for x in the original equation 31 = 31 6 is the correct answer

EQUATION SHORTCUT: you can move any term to the other side of the equation, as long as you change the sign Notes and Examples from Class

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Lessons 11: Solve Equations with Rational (a Fraction) CoefficientsThis lesson will not be graded. We will review in class.

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Lessons 12: Solve Equations with Variables on Both SidesExample:

2m + 14 = 4m - 16 2m – 4m + 14 = 4m – 4m – 16 Subtract 4m from both sides-2m + 14 = -16-2m + 14 – 14 = -16 – 14 Subtract 14 from both sides-2m = -30-2m = -30 Divide both sides by -2 -2 -2

m = 15

REMEMBER: you can move any term to the other side of the equation, as long as you change the sign

Practice and Examples from Class:

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Lessons 13: Write and Solve Equations from Words

Example:

Example:

Example:

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Examples from Class

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