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Solving Equations

Solving One-Step Equations

Objective: To solve one-step equations in one variable.

Objectives

• I can solve equations using addition and subtraction.

• I can solve equations using multiplication and division.

• I can solve equations using reciprocals.

• I can use a one – step equation as a model for real world problems.

Vocabulary

• Equivalent equations are equations that have the same solution(s). You can find the solution of a one – step equation using the properties of equality and inverse operations to write a simpler equivalent equation.

• Addition Property of Equality: Adding the same number to each side of an equation produces an equivalent equation.– Algebra: For any real numbers a, b, and c, if a = b, then a + c = b + c

– Example: x – 3 = 2

x – 3 + 3 = 2 + 3

Vocabulary

• Subtraction Property of Equality: Subtracting the same number from each side of an equation produces an equivalent equation.– Algebra: For any real numbers a, b, and c, if a = b, then a – c = b – c

– Example: x + 3 = 2

x + 3 – 3 = 2 – 3

• To solve an equation, you must isolate the variable. You do this by getting the variable with a coefficient of 1 alone on one side of the equation.

Vocabulary• You can isolate a variable using the properties of equality and

inverse operations.

• An inverse operation undoes another operation.

• Subtraction is the inverse of addition where as addition is the inverse of subtraction.

• Multiplication is the inverse of division where as division is the inverse of multiplication.

• Each inverse operation you perform should produce a simpler equivalent equation.

Solving an Equation Using Subtraction

What is the solution?

1. 27 + n = 46

2. 6 = x + 2

3. 23 = v + 5

4.2

3+ 𝑗 =

5

6

5. 15 + s = 42

6. y + 2 = –6

Solving an Equation Using AdditionWhat is the solution?

1. –7 = b – 3

2. m – 8 = –14

3.1

2= 𝑦 −

3

2

4. 8 = v – 5

5. –13 = –4 + v

6. –12 + x = 17

Practice

Solve each equation using addition or subtraction.1. 6 = x + 2

2. 23 = v + 5

3. –17 = 3 + k

4. j – 3 = –7

5. 5.5 = – 2 + d

6. 67 = w – 65

Vocabulary• Multiplication Property of Equality: Multiplying each side of

an equation by the same nonzero number produces an equivalent equation.– Algebra: For any real numbers a, b, and c, if a = b, then a ∙ c = b ∙ c

– Example: 𝑥

3= 2

𝑥

3∙ 3 = 2 ∙ 3

• Division Property of Equality: Dividing each side of an equation by the same nonzero number produces an equivalent equation.– Algebra: For any real numbers a, b, and c, such that c ≠ 0, if a = b,

then 𝑎

𝑐=

𝑏

𝑐.

– Example: 5x = 205𝑥

5=

20

5

Solving an Equation Using Division


1. 4x = 64

2. 10 = 15x

3. –3.2z = 14

4. –8n = –64

5. 5b = 145

6. –96 = 4c

Solving an Equation Using Multiplication


1.x

4= −9

2. 19 =r

3

3.x

−9= 8

4.x

5= −12

5. y ÷ 5 = 20

6. z÷ 15 = 3

Practice

Solve each equation using multiplication or division.1. –8n = –64

2. 17.5 = 5s

3. 11 = 2.2t

4.𝑘

7= 13

5. 14 =𝑧

2

6. −13 =𝑚

−5

Vocabulary

When the coefficient of the variable in an equation is a fraction, you can use the reciprocals of the fraction to solve the equation.

4

5𝑚 = 28

×5

4×

5

4

m = 35

Solving Equations Using Reciprocals


1.2

3q = 18

2. 12 =3

4k

3.1

5x =

2

7

4.5

8v = −1

5. 36 =4

9n

6.3

2d =

1

2

PracticeSolve each equation.

1.5

6𝑤 = 25

2.2

3𝑟 = 6

3.3

8ℎ = 9

4.1

5𝑥 =

2

7

5. 36 =4

9𝑑

Using a One-Step Equation as a Model

Toucans and blue-and-yellow macaws are both tropical birds.

The length of the average toucan is about 2

3of the length of an

average blue-and-yellow macaw. Toucans are about 24 inches long. What is the length of an average blue-and-yellow macaw?

Using a One-Step Equation as a Model

An online DVD company offers gift certificates that you can use to purchase rental plans. You have a gift certificate for $30. The plan you select costs $5 per month. How many months can you purchase with the gift certificate?

Practice

• You have a rack that can hold 30 CDs. You can fit 7 more CDs on the rack before the rack is full. How many CDs are in the rack?

• Between 1788 and 2008, the U.S. Constitution was amended 27 times. How many years have passed on average between one amendment and the next, to the nearest tenth of a year?

Exit Ticket

1. 6 = x + 2

2. y – 19 = 37

3. 5b = 145

4. m ÷ 7 = 12

5. −6 =3

7n

Solving Two-Step Equations

Objective: To solve two-step equations in one variable.

Objectives

• I can solve two – step equations.

• I can use an equation as a model for solving real world problems.

• I can solve equations with two terms in the numerator.

• I can use deductive reasoning to solve equations.

Vocabulary• To solve two-step equations, you can use the properties

of equality and inverse operations to form a series of simpler equivalent equations. You can use the properties of equality repeatedly to isolate the variable.

• To solve a two – step equation, identify the operations and undo them using inverse operations. You can undo the operations in the reverse order of the order of operations.– Example, you use subtraction first to undo addition, and then

division to undo multiplication. Or you can use addition to undo subtraction, and then multiplication to undo division.

Solving a Two–Step Equation


1. 2x + 3 = 15

2. 5 =t

2− 3

3. 3n – 4 = 11

4.y

5+ 2 = −8

5. –5x – 2 = 13

6. 14 = –2k + 3

Practice

Solve each equation. Check your answer.

1. 2 +𝑎

4= −1

2. −1 = 7 + 8𝑥

3. 4𝑏 + 6 = −2

4. 10 =𝑥

4− 8

5.𝑎

5− 18 = 2

6. −14 = −5 + 3𝑐

Using an Equation as a Model

You are making a bulletin board to advertise community service opportunities in your town. You plan to use half a sheet of construction paper to each ad. You need 5 sheets of construction paper for a title banner. You have 18 sheets of construction paper. How many ads can you make?

Suppose you need one quarter of a sheet of paper for each ad and four full sheets for the title banner. How many ads could you make?

Practice

• You have $16 and a coupon for a $5 discount at a local supermarket. A bottle of olive oil costs $7. How many bottles of olive oil can you buy?

• Two college friends rent an apartment. They have to pay the landlord two months’ rent and a $500 security deposit when they sign the lease. The total amount they pay the landlord is $2800. What is the rent for one month?

Vocabulary

• When one side of an equation is a fraction with more than one term in the numerator, you can still undo division by multiplying each side by the denominator.

Solving With Two Terms in the Numerator


1.x−7

3= −12

2. 6 =x−2

4

3.y−4

2= 10

4. −2 =d−7

7

5.x+5

6= 15

6. 71

2=

x+3

2

Practice


1.𝑧+10

9= 2

2.𝑏+3

5= 1

3. 7 =𝑥−8

3

4. −2 =𝑑−7

7

5.𝑔−3

3=

5

3

6. 4 =𝑎+10

2

Vocabulary

• When you use deductive reasoning, you must state your steps and your reason for each step using properties, definitions, or rules.

Using Deductive Reasoning

What is the solution? Justify each step.

1. –t + 8 = 3

2.x

3− 5 = 4

3. 26 =m

6+ 5

4. 20 – 3h = 2

Practice

Solve each equation. Justify each step.

1. 14 − 𝑏 = 19

2. 3 −𝑥

2= 6

3. −1 = 4 +𝑥

3

4. −2 = 3𝑥 + 4

Exit Ticket• 2x + 3 = 15

• 3n – 4 = 11

• 26 =m

6+ 5

•a

5− 18 = 2

•y−4

2= 10

• 7 =x−8

3

• −1 = 4 +x

3

•q

−3+ 12 = 2

Solving Multi-Step Equations

Objective: To solve multi-step equations in one variable.

Objectives

I can combine like terms to solve equations.

I can solve multi-step equations using real world problems.

I can solve an equation using the distributive property.

I can solve an equation that contains fractions.

I can solve an equation that contains decimals.

Vocabulary

To solve multi-step equations, you form a series of simpler equivalent equations.

To do this, use the properties of equality, inverse operations, and properties of real numbers.

You use the properties until you isolate the variable.

Combining Like Terms


1. 5 = 5m – 23 +2m

2. 11m – 8 – 6m = 22

3. -2y + 5 + 5y = 14

4. 7p + 8p – 12 = 59

5. 12 + 3p − 5p = −13

6. -25 = 4r – 7 – 8r

Practice


1. 7 − 𝑦 − 𝑦 = −1

2. 72 + 4 − 14𝑑 = 36

3. 6𝑝 − 2 − 3𝑝 = 16

4. 17 = 𝑝 − 3 − 3𝑝

5. 𝑥 + 2 + 𝑥 = 22

6. 9𝑡 − 6 − 6𝑡 = 6

Solving a Multi-Step Equation

Martha takes her niece and nephew to a concert. She buys t-shirts and bumper stickers for them. The bumper stickers costs $1 each. Martha’s niece wants 1 shirt and 4 bumper stickers, and her nephew wants 2 shirts but no bumper stickers. If Martha’s total is $67, what is the cost of one shirt?

Solving a Multi-Step Equation

Noah and Kate are shopping for new guitar strings in a music store. Noah buys 2 packs of strings. Kate buys 2 packs of strings and a music book. The book costs $16. Their total cost is $72. How much is on pack of strings?

Practice

• You have a part-time job. You work for 3 hours on Friday and 6 hours on Saturday. You also receive an allowance of $20 per week. You earn $92 per week. How much do you earn per hour at your part-time job?

• A family buys airline tickets online. Each ticket costs $167. The family buys travel insurance with each ticket that costs $19 per ticket. The Web Site charges a fee of $16 for the entire purchase. The family is charged a total of $1132. How many tickets did the family buy?

Solving an Equation Using the Distributive Property


1. -8(2x – 1) = 36

2. 18 = 3(2x – 6)

3. -2(3x + 9) = 24

4. 12(3 – x) = 60

5. n + 5(n – 1) = 7

6. 64 = 8(r + 2)

Practice


1. 26 = 6 5 − 4𝑓

2. −4 𝑟 + 6 = −63

3. 5 2 + 4𝑧 = 85

4. 15 = 5 2𝑥 − 3

5. 7 𝑓 − 1 = 45

Solving an Equation that Contains FractionsWhat is the solution?

1.3x

4−

x

3= 10

2.2b

5+

3b

4= 3

3.1

9=

5

6−

m

3

4.2m

7+

3m

14= 1

5.3x

4−

x

8= 5

Practice

Solve each equation. Choose the method you prefer to use. Check your answer.

1.𝑏

13−

3𝑏

13=

8

13

2.𝑛

5−

3𝑛

10=

1

5

3.2

3+

3𝑚

5=

31

15

4. 5𝑦 −3

5=

4

5

5.𝑏

3+

1

8= 19

Solving an Equation That Contains DecimalsWhat is the solution?

1. 3.5 – 0.2x = 1.24

2. 0.5x – 2.325 = 3.95

3. 2.3 – 0.05x = 4.80

4. 1.2 = 2.4 – 0.6x

5. 1.3 + 0.5x= -3.41

6. 25.24 = 5g + 3.89

Practice


1. 25.24 = 5𝑔 + 3.89

2. 0.11𝑘 + 1.5 = 2.49

3. 0.25𝑛 + 0.1𝑛 = 9.8

4. 1.025𝑣 + 2.458 = 7.583

5. 1.06𝑔 − 3 = 0.71

Assignment

1. 9t – 6 – 6t = 6

2. -23 = -2a – 10 + a

3. 7(f – 1) = 45

4. -4(r + 6)= -63

5. 1.025v + 2.458 = 7.583

6. 0.11k + 1.5 = 2.49

Solving Equations with Variables on Both Sides

Objective: To solve equations with variables on both sides. To identify equations that are identities or have no solution.

Objectives

• I can solve an equation with variables on both sides.

• I can use a real life equation with variables on both sides.

• I can solve an equation with grouping symbols.

• I can use identities and determine if an equation does not have a solution.

Vocabulary

• To solve equation with variables on both sides, you can use the properties of equality and inverse operations to write a series of simpler equivalent equations.

Solving an Equation withVariables on Both Sides


1. 5x – 1 = x + 15

2. 2b + 4 = –18 – 9b

3. –n – 24 = 5 – n

4. 7k + 2 = 4k – 10

5. 5s + 13 = 2s + 22

6. 3x + 4 = 5x – 10

Practice

Solve each equation . Check your answer.

1. 3 +5q = 9 + 4q

2. 4p + 2 = 3p – 7

3. 8 – 2y = 3y – 2

4. –3c – 12 = –5 + c

5. 6m – 2 = 2m + 6

6. 3n – 5 = 7n + 11

Using an Equation with Variables on Both Sides

It takes a graphic designer 1.5 hours to make one page of a Web site. Using new software, the designer could complete each page in 1.25 hours, but it takes 8 hours to learn the software. How many web pages would the designer have to make in order to save time using the new software?

Using an Equation with Variables on Both Sides

An office manager spent $650 on a new energy-saving copier that will reduce the monthly electric bill for the office from $112 to $88. In how many months will the copier pay for itself?

Practice

• An architect is designing a rectangular greenhouse. Along one wall is a 7 foot storage area and 5 sections for different kinds of plants. On the opposite wall is a 4 foot storage area and 6 sections for plants. All the sections for plants are of equal length. What is the length of each wall?

• A hairdresser is deciding where to open her own studio. If the hairdresser chooses Location A, she will pay $1200 per month in rent and will charge $45 per haircut. If she chooses Location B, she will pay $1800 per month in rent and will charge $60 per haircut. How many haircuts would she have to give in one month to make the same profit at either location?

Solving an Equation with Grouping Symbols


1. 3(q – 5) = 2(q + 5)

2. (g + 4) – 3g = 1 + g

3. 2(k – 1) = 4(k – 2)

4. 2(5x – 1) = 3(x + 11)

5. 4(2y + 1) = 2(y – 13)

6. 7(4 – a) = 3(a – 4)

Practice


1. 7(6 – 2a) = 5(–3a + 1)

2. 2r – (5 – r) = 13 + 2r

3. 8 – (3 + b) = b – 9

4. 5g +4(–5 +3g) = 1 – g

5. 5(y – 4) = 7(2y + 1)

Vocabulary

• An equation that is true for every possible value of the variable is an identity. An example of this is x + 1 = x + 1.

• An equation has no solution if there is no value of the variable that makes the equation true, such as x + 1 = x + 2.

Identities and Equations With No Solution


1. 10x + 12 = 2(5x + 6)

2. 9m – 4 = –3m + 5 + 12m

3. 3(4b – 2) = –6 + 12b

4. 2x + 7 = –(3 – 2x)

5. 4x – 5 = 2(2x + 1)

6. 2(a – 4) = 4a – (2a + 4)

Practice

Determine whether each equation is an identity or whether it has no solution.

1. k – 3k = 6k + 5 – 8k

2. –6a + 3 = –3(2a – 1)

3. 5y + 2 = 1

2(10y + 4)

4. 2(2k – 1) = 4(k – 2)

5. 4 – d = –(d – 4)

Concept Summary – Solving Equations

1. Use the distributive property to remove any grouping symbols. Use properties of equality to clear decimals and fractions.

2. Combine like terms on each side of the equation.

3. Use the properties of equality to get the variable terms on one side of the equation and the constants on the other.

4. Use the properties of equality of equality to solve for the variable.

5. Check your solution in the original equation.

Assignment

1. 6m – 2 = 2m + 6

2. 2b + 4 = -18 – 9b

3. 2(3x + 1) = 4(x – 5)

4. 5g + 4(-5 + 3g) = 1 – g

5. t + 8 = -t + 6 + 2t

6. 3(m + 1.5) = 1.5(2m + 3)

Literal Equations and Formulas

Objective: To rewrite and use literal equations and formulas.

Objectives

• I can rewrite a literal equation.

• I can rewrite literal equations with only variables.

• I can rewrite a geometric formula.

• I can rewrite a formula for real world problems.

Vocabulary

• A literal equation is an equation that involves two or more variables.

• When you rewrite literal equations, you may have to divide by a variable or variable expression.

Rewriting a Literal Equation

The equation 10x + 5y = 80, where x is the number of pizza and y is the number of sandwiches. How many sandwiches can you buy if you buy 3 pizzas? 6 pizzas?

Solve the equation 4 = 2m – 5n for m. What are the values of m when n = –2, 0, and 2?

Practice

Solve each equation for y. Then find the value of y for each value of x.

1. y + 2x = 5; x = –1, 0, 3

2. 3x – 5y = 9; x = –1, 0, 1

3. 5x = –4y + 4; x = 1, 2, 3

4. x – 4y = –4; x = –2, 4, 6

5. 2y + 4x = 8; x = –2, 1, 3

6. 4x = 3y – 7; x = 4, 5, 6

7. 2y + 7x = 4; x = 5, 10, 15

8. 6x = 7 – 4y; x = –2, –1, 0

Vocabulary

• When you rewrite literal equations, you may have to divide by a variable or variable expression.

• During this lesson we will say that the variable or variable expression is not equal to zero because division by zero is not defined.

Rewriting a Literal Equation With Only Variables

Solve each equation for x.

1. ax – bx = c

2. –t = r + px

3. ac – dx = b

4. xy = z

5. we – x = u

Practice

Solve each equation for x.

1. 𝑚𝑥 + 𝑛𝑥 = 𝑝

2. 𝑆 = 𝐶 + 𝑥𝐶

3. 𝐴 = 𝐵𝑥𝑡 + 𝐶

4.𝑥

𝑎=

𝑦

𝑏

5. 𝑦 =𝑥−𝑣

𝑏

6.𝑥+𝑘

𝑦−𝑝= 𝑟

Vocabulary

• A formula is an equation that states a relationship among quantities.

• Formulas are special types of literal equations.

• Some common formulas are on the next slide.

• Notice that some of the formulas use the same variables, but the definitions of the variables are different.

Formula Name Formula Definitions of variables

Perimeter of a rectangle P = 2L + 2W P = perimeter, L = length, W = width

Circumference of a circle C = 2𝜋r C = circumference, r = radius

Area of a rectangle A = LW A = area, L = length, W = width

Area of a triangle A =1

2BH A = area, B = base, H =

height

Area of a circle A = 𝜋r2 A = area, r = radius

Distance traveled D = Rt D = distance, R = rate, t = time

Temperature C = 5

9(F – 32) C = degrees Celsius, F =

degree Fahrenheit

Rewriting a Geometric Formula

• What is the radius of a circle with circumference 64 feet? Round to the nearest tenth.

• What is the height of a triangle that has an area of 24 in² and a base with a length of 8in?

Practice

Solve each problem. Round to the nearest tenth, if necessary. Use 3.14 for 𝜋.

1. What is the radius of a circle with circumference 22m?

2. What is the length of a rectangle with width 10 in and area 45 in²?

3. A triangle has height 4 ft and area 32 ft². What is the length of its base?

4. A rectangle has perimeter 84 cm and length 35 cm. What is its width?

Rewriting a Formula

• The monarch butterfly is the only butterfly that migrates annually north and south. The distance that a particular group of monarch butterflies travel is 1700 miles. It takes a typical butterfly about 120 days to travel one way. What is the average rate at which a butterfly travels in miles per day? Round to the nearest mile per day.

Rewriting a Formula

• Pacific gray whales migrate annually from the waters near Alaska to the waters near Baja California, Mexico, and back. The whales travel a distance of about 5000 miles each way at an average rate of 91 miles per day. About how many days does it take the whales to migrate one way?

PracticeSolve each problem. Round to the nearest tenth, if necessary.

1. A vehicle travels on a highway at a rate of 65 miles per hour. How long does it take the vehicle to travel 25 miles?

2. You can use the formula 𝑎 =ℎ

𝑛to find the batting average a of

a batter who has h hits in n times at bat. Solve the formula for h. If the batter has a batting average of .290 and has been at bat 300 times, how many hits does the batter have?

3. Bricklayers use the formula 𝑛 = 7𝑙ℎ to estimate the number n of bricks needed to build a wall of length l and height h, where l and h are in feet. Solve the formula for h. Estimate the height of a wall 28 ft long that requires 1568 bricks.

Ratios, Rates, and Conversions

Objective: To find ratios and rates. To convert units and rates.

Objectives

• I can compare unit rates.

• I can convert units.

• I can convert units between systems.

• I can convert rates.

Vocabulary

• A ratio compares two numbers by division.

• The ratio of two numbers a and b, where b ≠ 0, can be written in three ways: a/b, a : b, and a to b.

• A ratio that compares quantities measured in different units is called a rate.

• A rate with a denominator of 1 unit is a unit rate.

• You can write ratios to find unit rates to compare quantities. You can also convert units and rates to solve problems.

Comparing unit rates

You are shopping for t-shirts. Which store offers the best deal?

Store A: $25 for 2 shirts

Store B: $45 for 4 shirts

Store C: $30 for 3 Shirts

Comparing unit rates

If Store B lowers its price to $42 for 4 shirts, does the solution change? Explain.

Store A: $25 for 2 shirts

Store C: $30 for 3 shirts

Practice

• Trisha ran 10 km in 2.5 h. Jason ran 7.5 km in 2 h. Alex ran 9.5 km in 2.25 h. Who had the fastest time?

• Bellingham, Washington, had an area of 25.4 mi² and a population of 74,547 during one year. Bakersfield, California, had an area of 113.1 mi² and a population of 295,536 during the same year. Which city had a greater number of people per square mile?

Vocabulary

• To convert from one unit to another, such as feet to inches, you multiply the original unit by a conversion factor that produces the desired unit.

• A conversion factor is a ratio of two equal measures in different units.

• A conversion factor is always equal to 1, such as 1 𝑓𝑡

12 𝑖𝑛.

Converting units

What is the given amount converted to the given units?

1. 330 min = _____ hours

2. 15 kg = _____ grams

3. 5 ft 3 in = _____ inches

4. 1250 cm = _____ m

5. 2.5 lb = _____ ounces

6. 9 yd = _____ feet

7. 79 dollars = _____ nickels

8. 63 yd = _____ feet

9. 168 h = _____ days

Practice

Convert the given amount to the given unit.

1. 200 cm = _____ m

2. 4 min = _____ seconds

3. 1500 mL = _____ L

4. 2 mi = _____ ft

5. 8000 ft = _____ mi

Vocabulary

• Notice that the units for each quantity are included in the calculations to help determine the units for the answer. This process is called unit analysis, or dimensional analysis.

Converting Units Between Systems

• The CN Tower in Toronto, Canada, is about 1815 ft tall. About how many meters tall is the tower? Use the fact that 1 m ≈3.28 ft.

• A building is 1450 ft tall. How many meters tall is the building?Use the fact that 1 m ≈ 3.28 ft.

Practice

Convert the given amount to the given unit.

1. 5 kg = _____ pounds (lb)

2. 3 qt = _____ L

3. 89 cm = _____ in

4. 2 ft = _____ cm

5. 10 lb = _____ kg

6. 12 m = _____ yd

7. 13 yd = _____ m

Vocabulary

• You can also convert rates.

• You can convert a speed in miles per hour to feet per second.

• Because rates compare measures in two different units, you must multiply by two conversion factors to change both of the units.

Converting Rates

• A student ran the 50 – yd dash in 5.8 s. At what speed did the student run in miles per hour? Round to the nearest tenth.

• An athlete ran a sprint of 100 ft in 3.1 s. At what speed was the athlete running in miles per hour? Round to the nearest mile per hour.

Practice

• The janitor at a school discovered a slow leak in a pipe. The janitor found that it was leaking 4 fl oz per minute. How fast was the pipe leaking in gallons per hour?

• Mr. Swanson bought a package of 10 disposable razors for $6.30. He found that each razor lasted for 1 week. What is the cost per day?

Solving Proportions

Objective: To solve and apply proportions.

Objectives

• I can solve a proportion using the multiplication property and cross products property.

• I can solve a multi – step proportion.

• I can use a proportion to solve a problem.

Vocabulary

• A proportional relationship can produce an infinite number of equivalent ratios.

• A proportion is an equation that states that two ratios are

equal. For example, 𝑎

𝑏=

𝑐

𝑑, where b ≠ 0 and d ≠0, is a

proportion.

• If two ratios are equal and a quantity in one of the ratios is unknown, you can write and solve a proportion to find the unknown quantity.

Solving a Proportion Using the Multiplication Property


1.7

8=

𝑚

12

2.𝑥

7=

4

5

3.𝑏

6=

4

5

4.𝑞

8=

4

5

5.3

16=

𝑥

12

Practice

Solve each proportion.

1.−3

4=

𝑥

26

2.9

2=

𝑘

25

3.3

4=

𝑥

5

4.𝑥

120=

1

24

5.𝑚

7=

3

5

6.2

15=

ℎ

125

Vocabulary

• In the proportion 𝑎

𝑏=

𝑐

𝑑, the products ad and bc are called

cross products.

• Cross Products Property of a proportion

– Words: The cross products of a proportion are equal.

– Algebra: If a/b = c/d, where b ≠ 0 and d ≠0, then ad = bc.

Vocabulary

• Here is why it works: You can use the Multiplication Property of Equality to prove the Cross Products Property.

𝑎

𝑏=

𝑐

𝑑Assume this equation is true.

𝑏𝑑 ×𝑎

𝑏= 𝑏𝑑 ×

𝑐

𝑑Multiplication Property of Equality

𝑏𝑑 ×𝑎

𝑏= 𝑏𝑑 ×

𝑐

𝑑Divide the common factors

𝑑𝑎 = 𝑏𝑐 Simplify

𝑎𝑑 = 𝑏𝑐 Communitive Property of Multiplication

• For this proportion, a and d are called the extremes of the proportion and b and c are called the means. Notice that in the Cross Products Property the product of the means equals the product of the extremes.

Solving a Proportion Using the Cross Products Property


1.4

3=

8

𝑥

2.𝑦

3=

3

5

3.5

9=

15

𝑥

4.3

𝑣=

8

13

5.−9

𝑏=

5

6

Practice


1.15

𝑎=

3

2

2.8

𝑝=

3

10

3.3

8=

30

𝑚

4.−3

4=

𝑚

22

5.2

7=

4

𝑑

6.2

−5=

6

𝑡

Solving a Multi-Step Proportion


1.𝑏−8

5=

𝑏+3

4

2.𝑛

5=

2𝑛+4

6

3.𝑤+3

4=

𝑤

2

4.3

𝑥+1=

1

2

5.𝑎−2

9=

2

3

6.7

𝑘−2=

5

8

Practice


1.𝑏+4

5=

7

4

2.3

3𝑏+4=

2

𝑏−4

3.3

7=

𝑐+4

35

4.𝑞+2

5=

2𝑞−11

7

5.2𝑐

11=

𝑐−3

4

6.𝑐+1

𝑐−2=

4

7

Vocabulary

• When you model a real – world situation with a proportion, you must write the proportion carefully.

• You can write the proportion so that the numerators have the same units and the denominators have the same units.

– Correct: 100 𝑚𝑖

2 ℎ=

𝑥 𝑚𝑖

5 ℎ

– Incorrect: 100 𝑚𝑖

2 ℎ=

5 ℎ

𝑥 𝑚𝑖

Using a Proportion to Solve a Problem

• A portable media player has 2 GB of storage and can hold about 500 songs. A similar but larger media player has 80 GB of storage. About how many songs can the larger media player hold?

• An 8 oz can or orange juice contains about 97 mg of vitamin C. About how many mg of vitamin C are there in a 12 oz can of orange juice?

Practice

• A gardener is transplanting flowers into a flowerbed. She has been working for an hour and has transplanted 14 flowers. She have 35 more flowers to transplant. If she works at the same rate, how many more hours will it take her?

• If 5 lb of pasta salad serves 14 people, how much pasta salad should you bring to a picnic with 49 people?

• A florist is making centerpieces. He uses 2 dozen roses for every 5 centerpieces. How many dozens of roses will he need to make 20 centerpiece?

Proportions and Similar Figures

Objective: To find missing lengths in similar figures. To use similar figures when measuring indirectly.

Objectives

• I can find the length of a side.

• I can apply similarity.

• I can interpret scale drawings.

• I can use scale models in real world problems.

Vocabulary

• Similar figures have the same shape but not necessarily the same size.

• You can use proportions to find missing side lengths in similar figures. Such figures can help you measure real – world distances indirectly.

• The symbol ~ means “is similar to.”

• In similar figures, the measures of corresponding angles are equal, and corresponding side lengths are in proportion.

Vocabulary

• The order of the letters when you name similar figures is important because it tells which parts of the figures are corresponding parts.

– Because ∆𝐴𝐵𝐶~∆𝐹𝐺𝐻, the following is true.

– 𝐴 ≅ 𝐹, 𝐵 ≅ 𝐺, 𝐶 ≅ 𝐻

–𝐴𝐵

𝐹𝐺=

𝐴𝐶

𝐹𝐻=

𝐵𝐶

𝐺𝐻

Finding the Length of a Side

• In the diagram, ∆𝐴𝐵𝐶~∆𝐷𝐸𝐹. What is DE?

• What is the length of AC?

A

B

C D

E

F

10 16 12

18

Practice

• The figures in each pair are similar. Find the missing length.

7.5

18x

5

28

3542

z

Practice

• The figures in each pair are similar. Find the missing length.

32

36

y

45

60

36

60

k

Vocabulary

• You can also use proportions to solve indirect measurement problems like finding a distance using a map.

• You can use similar figures and proportions to find lengths that you cannot measure directly.

Applying Similarity

• The sun’s rays strike the building and the girl at the same angle, forming two similar triangles. How tall is the building?

– The girl’s height is 5 feet.

– The girl’s shadow is 3 feet.

– The building’s shadow is 15 feet.

Applying Similarity

A man who is 6 feet is standing next to a flagpole. The shadow of the man is 3.5 feet and the shadow of the flagpole is 17.5 feet. What is the height of the flagpole?

Practice

In the diagram of the park, ∆𝐴𝐷𝐹~∆𝐵𝐶𝐹. The crosswalk at point A is about 20 yards long. A bridge across the pond will be built, from point B to point C. What will the length of the bridge be?

F

A

D

B

C50 yards

120 yards

Vocabulary

• A scale drawing is a drawing that is similar to an actual object or place.

• Floor plans, blueprints, and maps are all examples of scale drawings.

• In a scale drawing, the ratio of any length on the drawing to the actual length is always the same. This ratio is called the scale of the drawing.

Interpreting Scale Drawings

• What is the actual distance from Jacksonville to Orlando? The ruler measures 1.25 inches. The scale is 1 inch : 110 miles.

• The distance from Jacksonville to Gainesville on the map is about 0.6 inches. What is the actual distance from Jacksonville to Gainesville.

Practice

The scale of a map is 1 cm : 15 km. Find the actual distance corresponding to each map distance.

1. 2.5 cm

2. 0.2 cm

3. 15 cm

4. 4.6 cm

Vocabulary

• A scale model is a three-dimensional model that is similar to a three-dimensional object.

• The ratio of a linear measurement of a model to the corresponding linear measurement of the actual object is always the same.

• This ratio is called the scale of the model.

Using Scale Models

• A giant model heart measures 14 feet tall. The heart is the ideal size for a person who is 170 feet tall. About what size would you expect the heart of a man who is 6 feet tall to be?

• A scale model of a building is 6 inches tall. The scale of the model is 1 inch : 50 feet. How tall is the actual building?

Practice

• A professional model – maker is building a giant scale model of a house fly to be used in a science fiction film. An actual fly is about 0.2 inches long with a wingspan of about 0.5 inches. The model fly for the movie will be 27 feet long. What will its wingspan be?

• Abbottsville and Broken Branch are 175 miles apart. On a map, the distance between the two towns in 2.5 inches. What is the scale of the map?

Percents

Objective: To solve percent problems using proportions. To solve percent problems using

the percent equation.

Objectives

• I can find a percent using the percent proportion.

• I can find a percent using the percent equation.

• I can find a part or a base.

• I can use the simple interest formula.

Vocabulary

• Percents are useful because they standardize comparisons to a common base of 100.

• You can solve problems involving percents using either proportions or the percent equation, which are closely related.

• If you write a percent as a fraction, you can use a proportion to solve a percent problem.

Vocabulary

• The Percent Proportion:

– You can represent “a is p percent of b” using the percent proportion shown below. In the proportion, b is the base, and a is a part of base b.

– Algebra: 𝑎

𝑏=

𝑝𝑒𝑟𝑐𝑒𝑛𝑡 %

100, where b ≠ 0

– Example: What percent of 75 is 30?30

75=

𝑝

100

–𝑖𝑠

𝑜𝑓=

%

100

Finding a Percent Using the Percent Proportion

1. What percent of 56 is 42?





Practice

Find each percent.




4. What percent of 15 is75?



Vocabulary

• You have used the percent proportion 𝑎

𝑏=

𝑝

100to find

a percent.

• When you write 𝑝

100as p% and solve for a, you get

the equation 𝑎 = 𝑝% × 𝑏.

• This equation is called the percent equation.

• You can use either the percent equation or the percent proportion to solve any percent problem.

Vocabulary

• The Percent Equation

– You can represent “a is p percent of b” using the percent equation shown below. In the equation, a is a part of the base b.

– Algebra: a = p% × b, where b ≠ 0

– Example: What percent of 60 is 25? 25 = p% x 60

Finding a Percent Using the Percent Equation

1. What percent of 40 is 2.5?





Practice

Find each percent.







Finding a Part

• A dress shirt that normally costs $38.50 is on sale for 30% off. What is the sale price of the shirt?

• A family sells a car to a dealership for 60% less than they paid for it. They paid $9000 for the car. For what price did they sell the car?

• A tennis racket normally costs $65. The tennis racket is on sale for 20% off. What is the sale price of the tennis racket?

Practice

Find each part.

1. What is 25% of 144?

2. What is 12.5% of 104?

3. What is 125% of 12.8?

4. What is 63% of 150?

5. What is 150% of 63?

6. What is 1% of 1?

7. A beauty salon buys bottles of styling gel for $4.50 per bottle and marks up the price by 40%. For what price does the salon sell each bottle?

Finding a Base

• 125% of what number is 17.5?

• 30% of what number is 12.5?

• 75% of what number is 36?

• 35% of what number is 80?

Practice

Find each base.

1. 20% of what number is 80?

2. 60% of what number is 13.5?





Vocabulary

• A common application of percents is simple interest, which is interest you earn on only the principal in an account.

• Simple Interest Formula:

– The simple interest formula is given below, where I is the interest, P is the prinicpal, r is the annual interest rate written as a decimal, and t is the time in years.

– Algebra: I = Prt

– Example: If you invest $50 at a simple interest rate of 3.5% per year for 3 years, the interest you earn is I = 50(0.035)(3) = $5.25.

Vocabulary

• When you solve problems involving percents, it is helpful to know fraction equivalents for common percents.

• You can use the fractions to check your answers for reasonableness.

• Here are some common percents represented as fractions.

1% =1

1005% =

1

2010% =

1

1020% =

1

525% =

1

4

33. 3% =1

350% =

1

266. 6% =

2

375% =

3

4100% = 1

Using the Simple Interest Formula

• You deposited $840 in a savings account that earns a simple interest rate of 4.5% per year. You want to keep the money in the account for 4 years. How much interest will you earn? Check your answer for reasonableness.

• You deposited $125 in a savings account that earns a simple interest rate of 1.75% per year. You earned a total of $8.75 in interest. For how long was your money in the account?

Practice

• You deposit $1200 in a savings account that earns simple interest at a rate of 3% per year. How much interest will you have earned after 3 years?

• You deposit $150 in savings account that earns simple interest at a rate of 5.5% per year. How much interest will you have earned after 4 years?

Solving Percent Problems

Problem Type Example Proportion Equation

Find a percent. What percent of 6.3 is 3.5? 3.5

6.3=

𝑝

100

3.5 = p% x 6.3

Find a part. What is 32% of 125? 𝑎

125=

32

100

a = 32% x 125

Find a base. 25% of what number is 11? 11

𝑏=

25

100

11 = 25% x b

Change Expressed as a Percent

Objective: To find percent change.

Objectives

• I can find a percent decrease or percent increase.

• I can find percent error.

• I can find minimum and maximum dimensions.

• I can find the greatest possible percent error.

Vocabulary

• A percent change expresses an amount of change as a percent of an original amount.

• You can find a percent change when you know the original amount and how much it has changed.

• If a new amount is greater than the original amount, the percent change is called a percent increase.

• If the new amount is less than the original amount, the percent change is called a percent decrease.

Vocabulary

Percent Change:

– Percent change is the ratio of the amount of change to the original amount.

𝑝𝑒𝑟𝑐𝑒𝑛𝑡, 𝑝% =𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑒 𝑜𝑟 𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑒

𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝑎𝑚𝑜𝑢𝑛𝑡

– Amount of increase = new amount – original amount

– Amount of decrease = original amount – new amount

A common example of finding a percent decrease is finding a percent discount.

Finding a Percent Decrease

• A coat is on sale. The original price of the coat is $82. The sale price is $74.50. What is the discount expressed as a percent change?

• The average monthly precipitation for Chicago, Illinois, peaks in June at 4.1 inches. The average monthly precipitation in December is 2.8 inches. What is the percent decrease from June to December?

Finding a Percent Increase

• A store buys an electric guitar for $295. The store then marks up the price of the guitar to $340. What is the markup expressed as a percent change?

• In one year, the toll for passenger cars to use a tunnel rose from $3 to $3.50. What was the percent increase?

Practice

Tell whether each percent change is an increase or decrease. Then find the percent change. Round to the nearest percent.

1. Original: 12 New: 18

2. Original: 40.2 New 38.6


4. Original: 7.5 New: 9.5



7. Original: 14,500 New: 22,320

8. Original: 195.20 New: 215.25

9. Original: 1325.60 New: 1685.60

Vocabulary

• You can use percents to compare estimated or measured values to actual or exact values.

• Relative Error:

– Relative error is the ratio of the absolute value of the difference of a measured (or estimated) value and an actual value compared to the actual value.

– 𝑟𝑒𝑙𝑎𝑡𝑖𝑣𝑒 𝑒𝑟𝑟𝑜𝑟 =𝑚𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝑜𝑟 𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 −𝑎𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒

𝑎𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒

– When relative error is expressed as a percent, it is called percent error.

Finding Percent Error

• A decorator estimates that a rectangular rug is 5 ft by 8 ft. The rug is actually 4 ft by 8 ft. What is the percent error in the estimated area?

• You think that the distance between your house and a friend’s house is 5.5 mi. The actual distance is 4.75 mi. What is the percent error in your estimation?

Practice

Find the percent error in each estimation. Round to the nearest percent.

1. You estimate that your friend’s little brother is about 8 years old. He is actually 6.5 years old.

2. You estimate that your school is about 45 ft tall. Your school is actually 52 ft tall.

Vocabulary

• Sometimes we know the actual measurements of something. Sometimes we don’t know the actual measurements, but we know how precise they can be.

• Think of it this way, when we use a ruler, how precise are we?

• We could be off a little, meaning a fraction of a unit is off.

• The most any measurement can be off by is one half of the unit used in measuring.

Finding Minimum and Maximum Dimensions

• You are framing a poster and measure the length of the poster as 18.5 inch, to the nearest half inch. What are the minimum and maximum possible length of the poster?

• A student’s height is measured as 66 inches to the nearest inch. What are the student’s minimum and maximum possible heights?

Practice

A measurement is given. Find the minimum and maximum possible measurements.

1. A doctor measures a patient’s weight as 162 pounds to the nearest pound.

2. An ostrich egg has a mass of 1.1 kg to the nearest of a kg.

3. The length of an onion cell is 0.4 mm to the nearest tenth of a mm.

Finding the Greatest Possible Percent Error

The diagram shows the dimensions of a gift box to the nearest inch. What is the greatest possible percent error in calculating the volume of the gift box?

If the gift box’s dimensions were measured to the nearest half inch, how would the greatest possible error be affected?

5 in

6 in12 in

Practice

The table below shows the measured dimensions of a prism and the minimum and maximum possible dimensions based on the greatest possible error. What is the greatest possible percent error in finding the volume of the prism?

Dimensions Length Width Height

Measured 10 6 4

Minimum 9.5 5.5 3.5

Maximum 10.5 6.5 4.5

Practice

The side lengths of the rectangle have been measured to the nearest half of a meter. What is the greatest possible percent error in finding the area of the rectangle?

7.5 m

18.5 m

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