grade 8 pre-algebra rates, ratios, and proportions

CONFIDENTIAL 1

Grade 8 Pre-AlgebraGrade 8 Pre-Algebra

Rates, Ratios, and Rates, Ratios, and ProportionsProportions

CONFIDENTIAL 2

Warm UpWarm Up

1) 2c - 5 = c + 4

2) 8r + 4 = 10 + 2r

3) 2x -1 = x + 11

4) 28 - 0.3y = 0.7y - 12

Factor each trinomial by guess and check:

1) c = 9

2) r = 1

3) x = 12

4) y = 40

CONFIDENTIAL 3

Rates, Ratios, and Proportions

A ratio is a comparison of two quantities by division. The ratio of a to b can be written a:b or a ,

b

where b ≠ 0. Ratios that name the same comparison are said to be equivalent.

A statement that two ratios are equivalent,such as 1 = 2 , is called a proportion.

12 24

Read the proportion

1 = x 15 675

“1 is to 15 as x is to 675.”

CONFIDENTIAL 4

Using Ratios

The ratio of faculty members to students at a college is 1:15. There are 675 students. How many

faculty members are there?

Faculty = 1Students 15

1 = x15 675

1 = x 15 675675 675

x = 45

There are 45 faculty members.

Write a ratio comparing faculty to students.

Write a proportion. Let x be the numberof faculty members.

Since x is divided by 675, multiply both sides of the equation by 675.

CONFIDENTIAL 5

Now you try!

1) The ratio of games won to games lost for a baseball team is 3 : 2.

The team won 18 games. How many games did the team lose?

1) 12

CONFIDENTIAL 6

A rate is a ratio of two quantities with different units, such as or 34mi ,

2gal

Rates are usually written as unit rates. A unit rate is a rate with a second quantity of 1 unit, such as

34mi , 2gal

or 17 mi/gal. You can convert any rate to a unit rate.

CONFIDENTIAL 7

Garry ate 53.5 hot dogs in 12 minutes to win a contest.

Find the unit rate. Round your answer to the nearest hundredth.

Finding Unit Rates

4.46 ≈ x

Write a proportion to find an equivalent ratio with a second quantity of 1.

Divide on the left side to find x.

53.5 = x 12 1

The unit rate is approximately 4.46 hot dogs per minute.

CONFIDENTIAL 8

Now you try!

1) Cory earns $52.50 in 7 hours. Find the unit rate.

1) 7.5

CONFIDENTIAL 9

A rate such as 12in. , 1 ft

in which the two quantities are equal but use different units, is called a conversion factor.

To convert a rate from one set of unitsto another, multiply by a conversion factor.

Conversion factor

CONFIDENTIAL 10

Conversion factor

A) The earth’s temperature increases, as you go deeper underground. In some places, it may increase by 25°C per

kilometer. What is this rate in degrees per meter?

To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity.

25°C × 1km1km 1000m

The rate is 0.025°C per meter.

0.025°C 1m

CONFIDENTIAL 11

B) The dwarf sea horse Hippocampus zosterae swims at a rate of 52.68 feet per hour. What is this speed in

inches per minute?

To convert the first quantity in a rate, multiply by a conversion factor with that unit in the second quantity.

52.68ft × 12 in 1h 1ft

The speed is 632.16 inches per hour.

632.16 in. 1h

Step1: Convert the speed to inches per hour.

Next page

CONFIDENTIAL 12


632.16 in × 1 h 1h 60 min

The speed is 10.536 inches per minute.

10.536 in 1 min

Step2: Convert this speed to inches per minute.

Check that the answer is reasonable. The answer is about 10 in./min.

•There are 60 min in 1 h, so 10 in./min is 60 (10) = 600 in./h.

• There are 12 in. in 1 ft, so 600 in./h is 600 = 50 ft/h. This is close to the rate

12 given in the problem, 52.68 ft/h.

CONFIDENTIAL 13

Now you try!

1) A cyclist travels 56 miles in 4 hours. What is the cyclist’s speed in feet per second? Round your

answer to the nearest tenth, and show that your answer is reasonable.1 mile = 5280 feet

1) 20.53 ft/sec

CONFIDENTIAL 14

In the proportion a = c, b d

the products a · d and b · c are called cross products.You can solve a proportion for a missing value by using

the Cross Products Property.

Cross Products Property

WORDS NUMBERS ALGEBRA

In a proportion, cross products are equal. 53.5 = x

12 1

2 · 6 = 3 · 4

If 53.5 = x 12 1

and b ≠ 0 and d ≠ 0,

then ad = bc.

CONFIDENTIAL 15

Solving Proportions

A) 5 = 3 9 w

Use cross products.

w = 27 5

Solve each proportion.

5w = 27

Divide both sides by 5.

5 = 39 w

5 (w) = 9 (3)

5w = 27 5 5

CONFIDENTIAL 16

B) 8 = 1 x + 10 12

Use cross products.

86 = x

96 = x + 10Subtract 10 from both sides.

8 = 1 x + 10 12

8 (12) = 1 (x + 10)

-10 -10

CONFIDENTIAL 17

Now you try!

1) -5 = y 2 8

2) g + 3 = 7 5 4


1) y = -20

2) g = 5.75

CONFIDENTIAL 18

A scale is a ratio between two sets of measurements, such as 1 in : 5 mi.

A scale drawing or scale model uses a scale to represent an object as smaller or larger than the

actual object.

A map is an example of a scale drawing.

A scale written without units, such as 32 : 1, means that 32 units of any measure

correspond to 1 unit of that same measure.

CONFIDENTIAL 19

Solution:

x · 1 = 100 (0.8)

Let x be the actual distance.

The actual distance is 80 mi.

A) On the map, the distance from Houston to Beaumont is 0.8 in. What is the actual distance?

1 in : 100 mi

x = 80

Scale Drawings and Scale Models

Use cross products to solve.

Write the scale as a fraction. map = 1 in actual 100 mi

0.8 in = 1 in x 100 mi

CONFIDENTIAL 20

Solution:

127 = 100x

Let x be the distance on the map.


B) The actual distance between Bryan- College Station and Galveston is 127 mi. What is this distance on the map?

1 in : 100 mi

1.27 = x



x = 1 in 127 100 mi

127 = 100x100 100

Since x is multiplied by 100, divide both sides by 100 to undo the multiplication.

CONFIDENTIAL 21

1) A scale model of a human heart is 16 ft long. The scale is 32:1. How many inches long is the

actual heart it represents?

Now you try!

1) 0.5 inch

CONFIDENTIAL 22

Assessment

1) The ratio of the sale price of a jacket to the original price is 3 : 4. The original price is

$64. What is the sale price?

2) Find the unit rate.A computer’s fan rotates 2000 times in 40 seconds.

3) Find the unit rate. Twelve cows produce 224,988 pounds of milk.

4) Lydia wrote 1 pages of her science report in one 2

hour. What was her writing rate in pages per minute?

4

1) $48

2) 50 times/sec

3) 18749 pounds of milk/cow

4) 3 pages per minute

40

CONFIDENTIAL 23

5) 3 = 1 z 8


6) x = 1 3 5

7) b = 3 4 2

8) f + 3 = 7 12 2

9) -1 = 3 5 2d

10) 3 = s - 2 14 21

5) z = 24

6) x = 0.6

7) b = 6

8) f = 39

9) d = -7.5

10) s = 6.5

CONFIDENTIAL 24

Rates, Ratios, and Proportions

A ratio is a comparison of two quantities by division. The ratio of a to b can be written a:b or a ,

b

where b ≠ 0. Ratios that name the same comparison are said to be equivalent.

A statement that two ratios are equivalent,such as 1 = 2 , is called a proportion.

12 24

Read the proportion

1 = x 15 675

“1 is to 15 as x is to 675.”

Let’s review

CONFIDENTIAL 25

Using Ratiosreview

The ratio of faculty members to students at a college is 1:15. There are 675 students. How many

faculty members are there?

Faculty = 1Students 15

1 = x15 675

1 = x 15 675675 675

x = 45

There are 45 faculty members.

Write a ratio comparing faculty to students.

Write a proportion. Let x be the numberof faculty members.

Since x is divided by 675, multiply both sides of the equation by 675.

CONFIDENTIAL 26

A rate is a ratio of two quantities with different units, such as or 34mi ,

2gal

Rates are usually written as unit rates. A unit rate is a rate with a second quantity of 1 unit, such as

34mi , 2gal

or 17 mi/gal. You can convert any rate to a unit rate.

review

CONFIDENTIAL 27

Garry ate 53.5 hot dogs in 12 minutes to win a contest.

Find the unit rate. Round your answer to the nearest hundredth.

Finding Unit Rates

4.46 ≈ x

Write a proportion to find an equivalent ratio with a second quantity of 1.

Divide on the left side to find x.

53.5 = x 12 1

The unit rate is approximately 4.46 hot dogs per minute.

review

CONFIDENTIAL 28

A rate such as 12in. , 1 ft

in which the two quantities are equal but use different units, is called a conversion factor.

To convert a rate from one set of unitsto another, multiply by a conversion factor.

Conversion factor

review

CONFIDENTIAL 29

Conversion factor

A) As you go deeper underground, the earth’s temperature increases. In some places, it may increase by 25°C per kilometer. What is this rate in degrees per meter?


25°C × 1km1km 1000m

The rate is 0.025°C per meter.

0.025°C 1m

review

CONFIDENTIAL 30

B) The dwarf sea horse Hippocampus zosterae swims at a rate of 52.68 feet per hour. What is this speed in

inches per minute?

To convert the first quantity in a rate, multiply by a conversion factor with that unit in the second quantity.

52.68ft × 12 in 1h 1ft

The speed is 632.16 inches per hour.

632.16 in. 1h

Step1: Convert the speed to inches per hour.

Next page

review

CONFIDENTIAL 31


632.16 in × 1 h 1h 60 min

The speed is 10.536 inches per minute.

10.536 in 1 min

Step2: Convert this speed to inches per minute.

Check that the answer is reasonable. The answer is about 10 in./min.

•There are 60 min in 1 h, so 10 in./min is 60 (10) = 600 in./h.

• There are 12 in. in 1 ft, so 600 in./h is 600 = 50 ft/h. This is close to the rate

12 given in the problem, 52.68 ft/h.

review

CONFIDENTIAL 32

In the proportion a = c, b d

the products a · d and b · c are called cross products.You can solve a proportion for a missing value by using

the Cross Products Property.

Cross Products Property

WORDS NUMBERS ALGEBRA

In a proportion, cross products are equal. 53.5 = x

12 1

2 · 6 = 3 · 4

If 53.5 = x 12 1

and b ≠ 0 and d ≠ 0,

then ad = bc.

review

CONFIDENTIAL 33

Solving Proportions

A) 5 = 3 9 w

Use cross products.

w = 27 5


5w = 27

Divide both sides by 5.

5 = 39 w

5 (w) = 9 (3)

5w = 27 5 5

review

CONFIDENTIAL 34

B) 8 = 1 x + 10 12

Use cross products.

86 = x

96 = x + 10Subtract 10 from both sides.

8 = 1 x + 10 12

8 (12) = 1 (x + 10)

-10 -10

review

CONFIDENTIAL 35

A scale is a ratio between two sets of measurements, such as 1 in : 5 mi.

A scale drawing or scale model uses a scale to represent an object as smaller or larger than the

actual object.

A map is an example of a scale drawing.

A scale written without units, such as 32 : 1, means that 32 units of any measure

correspond to 1 unit of that same measure.

review

CONFIDENTIAL 36

Solution:

x · 1 = 100 (0.8)

Let x be the actual distance.


A) On the map, the distance from Houston to Beaumont is 0.8 in. What is the actual distance?

1 in : 100 mi

x = 80

Scale Drawings and Scale Models



0.8 in = 1 in x 100 mi

review

CONFIDENTIAL 37

Solution:

127 = 100x

Let x be the distance on the map.


B) The actual distance between Bryan- College Station and Galveston is 127 mi. What is this distance on the map?

1 in : 100 mi

1.27 = x



x = 1 in 127 100 mi

127 = 100x100 100

Since x is multiplied by 100, divide both sides by 100 to undo the multiplication.

review

CONFIDENTIAL 38

You did a great job You did a great job today!today!

grade 8 pre-algebra rates, ratios, and proportions

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