Grade 8 Pre-Algebra Rates, Ratios, and Proportions
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![Page 1: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/1.jpg)
CONFIDENTIAL 1
Grade 8 Pre-AlgebraGrade 8 Pre-Algebra
Rates, Ratios, and Rates, Ratios, and ProportionsProportions
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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
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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.”
![Page 4: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/4.jpg)
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.
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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
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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.
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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.
![Page 8: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/8.jpg)
CONFIDENTIAL 8
Now you try!
1) Cory earns $52.50 in 7 hours. Find the unit rate.
1) 7.5
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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
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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
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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
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CONFIDENTIAL 12
To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity.
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.
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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
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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.
![Page 15: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/15.jpg)
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
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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
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CONFIDENTIAL 17
Now you try!
1) -5 = y 2 8
2) g + 3 = 7 5 4
Solve each proportion.
1) y = -20
2) g = 5.75
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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.
![Page 19: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/19.jpg)
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
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CONFIDENTIAL 20
Solution:
127 = 100x
Let x be the distance on the map.
The actual distance is 80 mi.
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
Use cross products to solve.
Write the scale as a fraction. map = 1 in actual 100 mi
x = 1 in 127 100 mi
127 = 100x100 100
Since x is multiplied by 100, divide both sides by 100 to undo the multiplication.
![Page 21: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/21.jpg)
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
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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
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CONFIDENTIAL 23
5) 3 = 1 z 8
Solve each proportion.
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
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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
![Page 25: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/25.jpg)
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.
![Page 26: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/26.jpg)
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
![Page 27: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/27.jpg)
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
![Page 28: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/28.jpg)
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
![Page 29: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/29.jpg)
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?
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
review
![Page 30: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/30.jpg)
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
![Page 31: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/31.jpg)
CONFIDENTIAL 31
To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity.
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
![Page 32: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/32.jpg)
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
![Page 33: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/33.jpg)
CONFIDENTIAL 33
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
review
![Page 34: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/34.jpg)
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
![Page 35: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/35.jpg)
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
![Page 36: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/36.jpg)
CONFIDENTIAL 36
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
review
![Page 37: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/37.jpg)
CONFIDENTIAL 37
Solution:
127 = 100x
Let x be the distance on the map.
The actual distance is 80 mi.
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
Use cross products to solve.
Write the scale as a fraction. map = 1 in actual 100 mi
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
![Page 38: Grade 8 Pre-Algebra Rates, Ratios, and Proportions](https://reader036.fdocuments.us/reader036/viewer/2022081501/568137ed550346895d9fa60a/html5/thumbnails/38.jpg)
CONFIDENTIAL 38
You did a great job You did a great job today!today!