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MAT001 – Chapter 4 – Ratios, Rates, and Proportions
CQ4-01. One day, a veterinary clinic treated 6 male
dogs, 10 female dogs, 3 male cats, and 4 female cats.
What is the ratio of dogs to cats that were seen that
day?
Section 4.1: Ratios and Rates
Ratios
A ratio is the comparison of two quantities that have
the same units.
18 ounces
36 ounces
18 to 36
18:36
We can express this ratio
three different ways:
18
36
1 of 36
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
Simplest form
13
14
15
0%
3.
16
17
4.
18
19
20
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10
0%
0%
1.
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
11
12
0%
2.
13
14
0%
3.
15
16
4.
17
18
19
20
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Rates
Ratios
Example:
Dorothy earns $500 weekly. Out of that $500
gross pay, $125 is withheld for federal taxes and
$60 is withheld for state taxes. What is the ratio of
the amount withheld for state taxes to gross pay?
60
500
12
1
27
3
7
5
14
1
3
1.
2.
3.
4.
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state taxes
gross pay
11
0%
2.
CQ4-02. Write 30:84 as a ratio (in
fraction) in simplest form.
36 ounces
1
2
0%
1.
2
A ratio is in simplest form when the two numbers
do not have a common factor and both numbers are
whole numbers.
18
36
0%
1
Simplest Form of a Ratio
18 ounces
10
1. 7 to 16
2. 16 to 7
3. 16 to 23
4. 10 to 13
A rate is a comparison of two quantities that have
different units.
$1.50
18 ounces
3
25
The rate of dollars to ounces is
$3 of it is withheld for state taxes for every $25
Dorothy makes.
dollars
ounces
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1.50
18
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MAT001 – Chapter 4 – Ratios, Rates, and Proportions
CQ4-03. Write “252 miles per 8 gallons”
as a rate in simplest form.
1.
2.
3.
4.
Unit Rates
A unit rate is the rate for a single unit.
28 miles
1 gallon
10
$1.50
63 miles
2 gallons
18 ounces
125 miles
4 gallons
1
2
3
64 miles
34 gallons
5
6
7
21
22
23
24
25
26
27
The rate of dollars to ounces is
0%
0%
1.
8
9
10
28
29
30
11
12
0%
2.
13
14
3.
15
16
dollars
ounces
0%
4.
17
18
19
Denominator is 1.
20
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CQ4-05. Write “$6,188 for 82 shares of
stock” as a unit rate. Round to the nearest
cent, if necessary.
10
3.43 gallons/ day
3.14 gallons/ day
292 gallons/ day
0%
0%
1.
1
2
3
4
5
6
7
8
9
10
22
23
24
25
26
27
28
29
30
11
12
0%
2.
13
14
0%
3.
15
16
10
1. $13.25 / share
2. $69.24 / share
3. $80.92 / share
4. $75.46 / share
4 gallons/ day
21
The unit rate is
approximately
$0.08 per ounce.
0.083
.
1
0.25
3
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CQ4-04. Write “48 gallons in 14 days”
as a unit rate. Round to the nearest
hundredth, if necessary.
1.
2.
3.
4.
1.50
18
0%
4.
17
18
0%
1.
19
20
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1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
11
12
0%
2.
13
14
0%
3.
15
16
4.
17
18
19
20
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Proportions
Section 4.2
A proportion states that two ratios or rates are equal.
16
12
The Concept of
Proportions
4
3
“sixteen-twelfths equals four-thirds”
or
“sixteen is to twelve as four is to three”
Example:
Write the proportion 5.6 is to 4.4 as 112 is to 88.
5.6
4.4
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112
88
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MAT001 – Chapter 4 – Ratios, Rates, and Proportions
Equality Test for Proportions
Equality Test for Proportions
To determine whether a statement is a proportion,
the equality test for proportions is used. This
method is also called finding cross products.
Example:
Is the rate 75 miles equal to the rate 105 miles ?
5 hours
7 hours
Equality Test for Fractions
For any two fractions where b ≠ 0 and d ≠ 0,
if and only if
cthen a
,
d
a
b
2
1
1=4
?
2 8
4
8
d=b
c.
75 = 105
?
5
7
5 105
75 7
The two rates are
equal. This is a
proportion.
525
525
The products are equal,
1 4
therefore
.
2 8
8
8
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CQ4-06. Determine which equation is a
true statement?
12 ? 10
42 35
1?3
2 4
10 ? 11
9 10
1.
2.
3.
4.
CQ4-07. Determine which equation is a
true statement?
1.
2.
3.
4.
10
48 ? 40
56 48
0%
0%
1.
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
11
12
0%
2.
13
14
0%
3.
15
16
4.
17
18
19
15 ? 22
8 12
10
11 13.75
12 15
7 ? 15.5
8 18
?
2.5 ? 1.6
6
4
0%
0%
1.
20
1
2
3
4
5
6
7
8
9
10
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21
22
23
24
25
26
27
28
29
30
11
12
0%
2.
13
14
0%
3.
15
16
4.
17
18
19
20
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Variable & Equation
Section 4.3
A variable is a letter used to represent a number we
do not yet know.
8
n
72
An equation has an equal sign. This indicates that
the values on each side are equivalent.
Solving Proportions
This will not change
the value of n in the
equation.
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8
n
72
8
n
72
8
8
8
n
9
1
n
9
8
We divide both sides of the
equation of the form a n =
b by the number that is
multiplied by n.
Therefore, n = 9.
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MAT001 – Chapter 4 – Ratios, Rates, and Proportions
Solving for a Variable
Finding Missing Numbers in a Proportion
Sometimes one of the pieces of a proportion is unknown.
We can use an equation of the form a n = b and solve
for n to find the unknown quantity.
Example: Solve for n.
n 11.4 = 57
n
11.4 = 57
15
4
n
11.4
57
=
11.4
11.4
11.4
n
= 5
11.4
n = 5
Check: 5
n
6
To Solve for a Missing Number in a Proportion
1. Find the cross products.
2. Divide each side of the equation by the
number multiplied by n.
3. Simplify the result.
4. Check your answer.
11.4 = 57
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CQ4-08. Solve for x. Round to the nearest
tenth, if necessary.
8 33
Solving for a Variable
Example: Find the value of n.
15
4
n
6
4
n
15
4
n
90
n
90
4
22.5
4
4
n
6
1.
2.
3.
4.
Find the cross products.
Divide each side by 4.
Check your answer: 15 ? 22.5
4
6
4 22.5
15 6
90 
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1.
2.
3.
4.
x 7.8
x
15
0%
x 10.9
1
2
3
4
5
6
7
8
9
10
22
23
24
25
26
27
28
29
30
x 19.8
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
11
12
0%
2.
13
14
0%
3.
15
16
12
13
14
0%
3.
15
16
4.
17
18
19
20
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4.
17
18
19
20
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87 quarters
n dollars
87
12
n
261
12
n
261
12
12 n
12
Divide each side by 12.
n
n = $21.75
21.75
1.
11
0%
2.
Solving for a Variable
12 quarters
3 dollars
10 Seconds
Remaining
0%
0%
1.
21
3
0%
10 Seconds
Remaining
x 11
Example: Find the value of n.
x 20.2
x 9.9
45
x 9.5
CQ4-09. Solve for x. Round to the nearest
tenth, if necessary.
14
21.2
x
x 12
Find the cross products.
Check your answer: 12 ? 87
3
21.75
3 87
12 21.75
261

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MAT001 – Chapter 4 – Ratios, Rates, and Proportions
CQ4-11. Solve for x. Round to the nearest
tenth, if necessary.
CQ4-10. Solve for x. Round to the nearest
tenth, if necessary.
$9.75
24 ounces
1. x
2. x
3. x
4. x
$5.85
x
10.6 ounces
14 .2 cups of flour
6 loaves of bread
1.
2.
3.
4.
14.4 ounces
10 Seconds
Remaining
9.8 ounces
0%
1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
0%
1.
15.2 ounces
11
12
0%
2.
13
14
0%
3.
15
16
4.
17
18
19
20 cups of flour
x loaves of bread
x 47.3 loaves
x 11.8 loaves
10 Seconds
Remaining
x 8.5 loaves
0%
x 10.7 loaves
20
1
2
3
4
5
6
7
8
9
10
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21
22
23
24
25
26
27
28
29
30
0%
1.
11
12
0%
2.
13
14
0%
3.
15
16
4.
17
18
19
20
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Problem Solving Steps
1. Understand the problem.
Section 4.4
Solving Applied Problems
Involving Proportions
a) Read the problem carefully.
b) Draw a picture if this is helpful.
c) Fill in the Mathematics Blueprint so that you have the
facts and a method of proceeding in this situation.
2. Solve and state the answer.
a) Perform the calculations.
b) State the answer, including the unit of measure.
3. Check.
a) Estimate the answer.
b) Compare the exact answer with the estimate to see if your
answer is reasonable.
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Mathematics Blueprint
Mathematics Blueprint
The Mathematical Blueprint is simply a sheet of paper with
four columns. Each column tells you something to do.
Mathematics Blueprint for Problem Solving
Gather the
Facts
What Am I
Asked to Do?
How Do I
Proceed?
Key Points to
Remember
Example:
A baseball pitcher gave up 52 earned runs in 260 innings of pitching.
At this rate, how many runs would he give up in a 9-inning game?
(This decimal is called the pitcher’s earned run average, ERA.)
Mathematics Blueprint for Problem Solving
Gather the
Facts
What Am I
Asked to Do?
How Do I
Proceed?
Key Points to
Remember
52 runs were
given up in 260
innings.
Find the number
of runs in 9
innings.
Set up a
proportion
comparing runs to
innings
One fraction
represents the
total innings and
one represents the
9 innings.
Example continues.
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MAT001 – Chapter 4 – Ratios, Rates, and Proportions
Mathematics Blueprint
Mathematics Blueprint
Example:
A baseball pitcher gave up 52 earned runs in 260 innings of
pitching. At this rate, how many runs would he give up in a 9inning game? (This decimal is called the pitcher’s earned run
average, ERA.)
earned runs
innings
260
52
260
n
52
260 n
260
n
It is recommended that 2 gallons of paint are used for every 750
square feet of wall. A painter is going to paint 7,875 square feet of
wall with a paint that costs $8.50 per gallon. How much will the
painter spend for paint?
n
9
Mathematics Blueprint for Problem Solving
9
468
260
1.8
The pitcher will
give up 1.8 runs
in a 9-inning
game.
21 gallons of
paint will be
needed.
15750
750
21
Find the total cost Set up a
for the paint.
proportion
comparing gals to
ft.; multiply the
answer by $8.50.
1.
2.
3.
4.
85
1700
178.5
$178.50.
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1
2
3
4
5
6
7
8
9
10
21
22
23
24
25
26
27
28
29
30
11
12
0%
2.
13
14
15
16
18
1137.5 miles
0%
650 miles
0%
1.
2
3
4
5
6
7
8
9
10
22
23
24
25
26
27
28
29
30
11
12
0%
2.
13
14
0%
3.
15
16
4.
17
18
19
20
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1. 48 cakes
2. 51 cakes
3. 156 cakes
4. 85 cakes
4.
17
10
1
0%
3.
728 miles
21
10
0%
682.5 miles
CQ4-14. At a bakery, for every 55 cakes baked, 3
have unacceptable texture. If the bakery makes
935 cakes per month, how many have
unacceptable texture?
CQ4-13. If 2 centimeters on a map
represents 86 miles, what distance does 5
centimeters represent?
1.
One fraction
represents the
recommended
paint and one
represents the
needed paint.
CQ4-12. Mark traveled 455 miles in 5
hours. At this rate, how far could he travel
in 7.5 hours?
The total cost for the
0%
Key Points to
Remember
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21
paint is
1. 92 miles
2. 34.4 miles
3. 258 miles
4. 215 miles
2 gal per 750 sq.
ft.; 7875 sq. ft.
total to be
painted; cost is
8.50 per gallon
8.5
2
n
750
7875
2 7875
How Do I
Proceed?
Example continues.
It is recommended that 2 gallons of paint is used for every 750
square feet of wall. A painter is going to paint 7,875 square
feet of wall with a paint that costs $8.50 per gallon. How
much will the painter spend for paint?
750 n
750
n
What Am I
Asked to Do?
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Mathematics Blueprint
gallons of paint
square feet
750 n
Gather the
Facts
19
10
0%
0%
1.
20
1
2
3
4
5
6
7
8
9
10
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21
22
23
24
25
26
27
28
29
30
11
12
0%
2.
13
14
0%
3.
15
16
4.
17
18
19
20
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