1. Grade 8 Pre-Algebra Rates, Ratios, and Proportions
CONFIDENTIAL

2. Factor each trinomial by guess and check:
Warm Up
Factor each trinomial by guess and check:
1) 2c - 5 = c + 4
1) c = 9
2) r = 1
2) 8r + 4 = r
3) x = 12
3) 2x -1 = x + 11
4) y = 0.7y - 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.
Read the proportion
1 = x
“1 is to 15 as x is to 675.”
CONFIDENTIAL

4. There are 45 faculty members.
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 = 1
Students
Write a ratio comparing faculty to students.
1 = x
Write a proportion. Let x be the number
of faculty members.
1 = x
675
Since x is divided by 675, multiply both sides of the equation by 675.
x = 45
There are 45 faculty members.
CONFIDENTIAL

5. 1) The ratio of games won to games lost for a baseball team is 3 : 2.
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. or 17 mi/gal. You can convert any rate to a unit rate.
A rate is a ratio of two quantities with different units, such as or 34mi , gal
Rates are usually written as unit rates. A unit rate is a rate with a second quantity of 1 unit, such as
34mi , gal
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.
Finding Unit Rates
Garry ate 53.5 hot dogs in 12 minutes to win a contest.
Find the unit rate. Round your answer to the nearest hundredth.
= x
Write a proportion to find an equivalent ratio with a second quantity of 1.
4.46 ≈ x
Divide on the left side to find x.
The unit rate is approximately 4.46 hot dogs per minute.
CONFIDENTIAL

8. 1) Cory earns $52.50 in 7 hours. Find the unit rate.
Now you try!
1) Cory earns $52.50 in 7 hours. Find the unit rate.
1) 7.5
CONFIDENTIAL

9. To convert a rate from one set of units
Conversion factor
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 units
to another, multiply by a 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?
25°C × 1km
1km m
To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity.
0.025°C 1m
The rate is 0.025°C per meter.
CONFIDENTIAL

11. Step1: Convert the speed to inches per hour.
B) The dwarf sea horse Hippocampus zosterae swims at a rate of feet per hour. What is this speed in inches per minute?
Step1: Convert the speed to inches per hour.
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 ft
in. 1h
The speed is inches per hour.
Next page 
CONFIDENTIAL

12. Step2: Convert this speed to inches per minute.
in × h
1h min
To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity. in
1 min
The speed is 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, ft/h.
CONFIDENTIAL

13. Now you try!
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) ft/sec
CONFIDENTIAL

14. the products a · d and b · c are called cross products.
Cross Products Property
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.
WORDS
NUMBERS
ALGEBRA
In a proportion, cross products are equal.
= x
2 · 6 = 3 · 4
If = x
and b ≠ 0
and d ≠ 0,
then ad = bc.
CONFIDENTIAL

15. Solving Proportions Solve each proportion. A) 5 = 3 9 w 5 = 3 9 w
5 = 3 w
5 (w) = 9 (3)
Use cross products.
5w = 27
5w = 27
Divide both sides by 5.
w = 27 5
CONFIDENTIAL

16. B) = 1 x
= 1 x
8 (12) = 1 (x + 10)
Use cross products.
96 = x + 10
Subtract 10 from both sides.
86 = x
CONFIDENTIAL

17. Now you try! Solve each proportion. 1) -5 = y 2 8 1) y = -20
1) y = -20
2) g = 7
2) g = 5.75
CONFIDENTIAL

18. A map is an example of a scale drawing.
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. The actual distance is 80 mi.
Scale Drawings and Scale Models
A) On the map, the distance from Houston to Beaumont is 0.8 in. What is the actual distance?
1 in : 100 mi
Solution:
map = in
actual mi
Write the scale as a fraction.
0.8 in = in
x mi
Let x be the actual distance.
x · 1 = 100 (0.8)
Use cross products to solve.
x = 80
The actual distance is 80 mi.
CONFIDENTIAL

20. 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
Solution:
map = in
actual mi
Write the scale as a fraction.
x = 1 in mi
Let x be the distance on the map.
127 = 100x
Use cross products to solve.
127 = 100x
Since x is multiplied by 100, divide both sides by 100 to undo the multiplication.
1.27 = x
The actual distance is 80 mi.
CONFIDENTIAL

21. Now you try!
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?
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?
1) $48
2) 50 times/sec
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.
3) pounds of milk/cow
4) Lydia wrote pages of her science report in one 2
hour. What was her writing rate in pages per minute? 4
4) 3 pages per minute 40
CONFIDENTIAL

23. Solve each proportion. 5) 3 = 1 z 8 8) f + 3 = 7 12 2 5) z = 24
5) 3 = 1 z
8) f = 7
5) z = 24
8) f = 39
6) x = 1
9) -1 = 3 d
6) x = 0.6
9) d = -7.5
7) b = 3
10) 3 = s - 2
7) b = 6
10) s = 6.5
CONFIDENTIAL

24. Rates, Ratios, and Proportions
Let’s review
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.
Read the proportion
1 = x
“1 is to 15 as x is to 675.”
CONFIDENTIAL

25. There are 45 faculty members.
review
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 = 1
Students
Write a ratio comparing faculty to students.
1 = x
Write a proportion. Let x be the number
of faculty members.
1 = x
675
Since x is divided by 675, multiply both sides of the equation by 675.
x = 45
There are 45 faculty members.
CONFIDENTIAL

26. or 17 mi/gal. You can convert any rate to a unit rate.
review
A rate is a ratio of two quantities with different units, such as or 34mi , gal
Rates are usually written as unit rates. A unit rate is a rate with a second quantity of 1 unit, such as
34mi , gal
or 17 mi/gal. You can convert any rate to a unit rate.
CONFIDENTIAL

27. Garry ate 53.5 hot dogs in 12 minutes to win a contest.
review
Finding Unit Rates
Garry ate 53.5 hot dogs in 12 minutes to win a contest.
Find the unit rate. Round your answer to the nearest hundredth.
= x
Write a proportion to find an equivalent ratio with a second quantity of 1.
4.46 ≈ x
Divide on the left side to find x.
The unit rate is approximately 4.46 hot dogs per minute.
CONFIDENTIAL

28. To convert a rate from one set of units
review
Conversion factor
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 units
to another, multiply by a conversion factor.
CONFIDENTIAL

29. review 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 × 1km
1km m
To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity.
0.025°C 1m
The rate is 0.025°C per meter.
CONFIDENTIAL

30. Step1: Convert the speed to inches per hour.
review
B) The dwarf sea horse Hippocampus zosterae swims at a rate of feet per hour. What is this speed in inches per minute?
Step1: Convert the speed to inches per hour.
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 ft
in. 1h
The speed is inches per hour.
Next page 
CONFIDENTIAL

31. Step2: Convert this speed to inches per minute.
review
Step2: Convert this speed to inches per minute.
in × h
1h min
To convert the second quantity in a rate, multiply by a conversion factor with that unit in the first quantity. in
1 min
The speed is 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, ft/h.
CONFIDENTIAL

32. the products a · d and b · c are called cross products.
review
Cross Products Property
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.
WORDS
NUMBERS
ALGEBRA
In a proportion, cross products are equal.
= x
2 · 6 = 3 · 4
If = x
and b ≠ 0
and d ≠ 0,
then ad = bc.
CONFIDENTIAL

33. review Solving Proportions Solve each proportion. A) 5 = 3 9 w 5 = 3
5 = 3 w
5 (w) = 9 (3)
Use cross products.
5w = 27
5w = 27
Divide both sides by 5.
w = 27 5
CONFIDENTIAL

34. review B) 8 = 1 x + 10 12 8 = 1 x + 10 12 8 (12) = 1 (x + 10)
= 1 x
8 (12) = 1 (x + 10)
Use cross products.
96 = x + 10
Subtract 10 from both sides.
86 = x
CONFIDENTIAL

35. A map is an example of a scale drawing.
review
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

36. The actual distance is 80 mi.
review
Scale Drawings and Scale Models
A) On the map, the distance from Houston to Beaumont is 0.8 in. What is the actual distance?
1 in : 100 mi
Solution:
map = in
actual mi
Write the scale as a fraction.
0.8 in = in
x mi
Let x be the actual distance.
x · 1 = 100 (0.8)
Use cross products to solve.
x = 80
The actual distance is 80 mi.
CONFIDENTIAL

37. The actual distance is 80 mi.
review
B) The actual distance between Bryan- College Station and Galveston is 127 mi. What is this distance on the map?
1 in : 100 mi
Solution:
map = in
actual mi
Write the scale as a fraction.
x = 1 in mi
Let x be the distance on the map.
127 = 100x
Use cross products to solve.
127 = 100x
Since x is multiplied by 100, divide both sides by 100 to undo the multiplication.
1.27 = x
The actual distance is 80 mi.
CONFIDENTIAL

38. You did a great job today!
CONFIDENTIAL