Download Module - Ratio, Proportion, and Percent

Pre-Algebra
Ratio, Proportion,
and Percent
Copyright
This publication © The Northern
Alberta Institute of Technology
2002. All Rights Reserved.
LAST REVISED June, 2007
Ratio, Proportion, and Percent
Statement of Prerequisite Skills
Complete all previous TLM modules before beginning this module.
Required Supporting Materials
Access to the World Wide Web.
Internet Explorer 5.5 or greater.
Macromedia Flash Player.
Rationale
Why is it important for you to learn this material?
Ratio, proportion, and percent are basic math skills that the student will encounter in
many applied situations. These skills are also essential to a beginning algebra student.
Learning Outcome
When you complete this module you will be able to…
Solve problems using ratio, proportion, and percent.
Learning Objectives
1.
2.
3.
4.
5.
6.
7.
Determine equivalent ratios and solve.
Change percent to fractions.
Change fractions to percent.
Change percent to decimals.
Change decimals to percent.
Solve percent questions.
Solve percent error in measurement problems.
Connection Activity
Consider the many times you have encountered fractions or percentages in daily life:
• 1/3 off regular cost
• 7% gst
• Top 5% of the class
• What percentage of your paycheck do you spend in rent?
Can you think of other applications of ratio, proportion, and percent?
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Ratio, Proportion, and Percent
OBJECTIVE ONE
When you complete this objective you will be able to…
Determine equivalent ratios and solve.
Exploration Activity
A ratio is a comparison of two quantities.
The ratio of one number to another is the first number divided by the second number.
That is, the ratio of a to b is:
a
b
Therefore, a ratio is a comparison of numbers by division.
EXAMPLE 1
2
9
7
b) The ratio of 7 to 3 is
3
a) The ratio of 2 to 9 is
NOTE:
A PROPORTION is a statement of equality between two ratios;
i.e.
2
4
=
is a proportion.
3
6
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Ratio, Proportion, and Percent
EXAMPLE 2
If a car travels 80 km in 2 hours, the ratio of distance to time is:
80km
2h
reducing this gives us;
40km
1h
and
80km 40km
=
2h
1h
The ratios are equal.
CHECK:
To see if the ratios are equal, perform the cross products.
80 40
=
2
1
If this is true, then:
80 × 1 = 40 × 2
80 = 80
The cross products are equal, therefore the ratios are equal.
The general statement for the equality of 2 ratios is:
a c
=
b d
then a ⋅ d = b ⋅ c
if
Notice the proportion has 4 components which are a, b, c, and d.
We use ratios to solve problems when we are given 3 of these 4 components.
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Ratio, Proportion, and Percent
EXAMPLE 3
a x
=
b d
If we are given the values for a, b, d, then we could solve for x.
x ⋅b = a ⋅d
x=
a⋅d
b
EXAMPLE 4
The ratio of a given number to 3 is the same as the ratio of 16 to 6. Find the given
number.
1. Maintain proper order; i.e. use given number to 3 and 16 to 6
given number 16
=
3
6
2. Let x = given number
x 16
=
3 6
3. If these ratios are equal then
6(x ) = 16(3)
x=
16(3)
6
x =8
4. Check by using cross products in original proportion
8 16
=
3 6
6(8) = 3(16)
48 = 48
The cross products are equal, therefore:
x = 8 is correct.
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Ratio, Proportion, and Percent
EXAMPLE 5
On a blueprint the scale is 1 km to 25 cm. What is the actual distance between 2 points, if
they are 5 cm apart?
1. Maintain order i.e. km to cm
1
x
=
25 5
2. let x = actual distance and write ratios
1(5) = 25(x )
x=
5
25
x=
1
km
5
x = 0.2 km
CHECK:
1 0.2
=
25
5
5(1) = 25(0.2)
5=5
x = 0.2 km is correct
5
Ratio, Proportion, and Percent
EXAMPLE 6
A cedar board 8 m long is cut into two pieces that are in the ratio 1:4. Find the length of
each piece.
SOLUTION:
Total number of units is 1 + 4 = 5
1
4
Total = 5
Therefore:
5 - total number of parts
8 - total length of board
Therefore the ratio is either:
larger piece
smaller piece
or
total
total
=
1
4
or
5
5
let x = the length of the shorter piece.
Therefore:
1 x
=
5 8
x=
8
5
= 1 .6
Shorter piece = 1.6 m
Longer piece = 6.4 m
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Ratio, Proportion, and Percent
Experiential Activity One
I. Solve the given proportions for x.
1.
2.
3.
x 5
=
2 8
x 7
=
3 12
8
x
=
15 9
II. Solve the given problems by setting up the proper proportion.
4. The ratio of a number to 15 is the same as the ratio of 17 to 60. Find the number.
5. The ratio of a number to 40 is the same as the ratio of 7 to 16. Find the number
6. 908 g = 2 lb; what weight in grams is 10 lbs?
7. Medication contains 2 substances, A and B, in the ratio of 3 to 5 respectively. If
there is 200 mg of substance B, how many mg of substance A is there?
8. A 6 m length of pipe is cut into 2 parts that are in the ratio 8 to 1. Find the length
of each part. Show Me.
9. A 5 m length of 2 by 10 planking is to be cut into 2 parts that are in the ratio of 4
to 3. Find the length of each part.
Experiential Activity One Answers
1. 1.25
2. 1.75
3. 4.8
4. 4.25
5. 17.5
6. 4540
7. 120
8. 0.67 m, 5.33 m
9. 2.86 m, 2.14 m
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Ratio, Proportion, and Percent
OBJECTIVE TWO
When you complete this objective you will be able to…
Change percent to fractions.
Exploration Activity
Percent
To this point we have used fractions and decimals for representing parts of a unit or
quantity. Now we will consider the concept of percent and shall find that percentages are
useful in numerous applications. The word percent means by the hundred. Therefore,
percent represents a decimal fraction with a denominator of 100. The symbol % is used to
denote percent.
EXAMPLE 1
For 5% the denominator is 100
Write the fraction with a numerator 5 and get
Reduce the fraction and get =
5
100
1
20
EXAMPLE 2
3
% : the denominator is 100.
4
3
3
= 4.
Numerator is
4 100
Write fraction =
3 1
×
.
4 100
Reduce and apply rules for dividing fractions
3
.
400
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Ratio, Proportion, and Percent
EXAMPLE 3
5
1
%: the denominator is 100.
2
1
5
1
Numerator is 5 = 2
2 100
11 2
Write fraction =
100
Reduce =
11 1
11
×
=
2 100 200
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Ratio, Proportion, and Percent
Experiential Activity Two
Change the following percent to fractions.
1
%
3
1. 50%
2.
3. 6%
4. 12%
4
%
5
1
7. 4 %
2
5.
9. 30%
1
5
1
8. 2 % Show Me.
2
1
10. %
8
6. 25 %
Experiential Activity Two Answers
1.
3.
5.
7.
9.
1
2
3
50
1
125
9
200
3
10
2.
4.
6.
8.
10.
1
300
3
25
63
250
1
40
1
800
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Ratio, Proportion, and Percent
OBJECTIVE THREE
When you complete this objective you will be able to…
Change fractions to percent.
Exploration Activity
Fractions
EXAMPLE 1
Change
3
to a percent. Use ratio and proportion.
5
3 is to 5 as a number is to 100 (% means per hundred)
Let x = a number
3
x
=
5 100
5 ⋅ x = 3 ⋅100
x=
3 ⋅100
5
Solve for x
so,
60
= 60%
100
and
3
= 60%
5
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Ratio, Proportion, and Percent
EXAMPLE 2
Change
5
. So 5 is to 6 as a number is to 100.
6
Let x = a number so we get,
5
x
=
6 100
Write ratios 6 ⋅ x = 5 ⋅100
x=
5 ⋅100
6
Solve for x
x = 83.3
so,
83.3
= 83.3%
100
and
5
= 83.3%
6
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Ratio, Proportion, and Percent
Experiential Activity Three
Change the following to percent.
1. 3/4
2. 1/100
3. 1/8
4. 4/5 Show Me.
5. 1/50
6. 1/4
Experiential Activity Three Answers
1. 75%
2. 1%
3. 12.5%
4. 80%
5. 2%
6. 25%
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Ratio, Proportion, and Percent
OBJECTIVE FOUR
When you complete this objective you will be able to…
Change percent to decimals.
Exploration Activity
EXAMPLE 1
Change 25% to a decimal. Write it as a fraction with denominator = 100
Divide by 100
25
= 0.25
100
EXAMPLE 2
Write
3
% as a decimal. Write it as a fraction with denominator = 100.
4
34
100
Simplify the fraction
3 1
×
4 100
3
=
400
=
= 0.0075
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Ratio, Proportion, and Percent
Experiential Activity Four
Change the following percents to decimals.
1. 75%
2.
3.
4.
5.
6.
3
%
5
1
4 %
4
1
2 %
2
1
5 %
3
1
6 %
5
Show Me.
Experiential Activity Four Answers
1.
2.
3.
4.
5.
6.
0.75
0.006
0.0425
0.025
0.0533
0.062
15
Ratio, Proportion, and Percent
OBJECTIVE FIVE
When you complete this objective you will be able to…
Change decimals to percent.
Exploration Activity
EXAMPLE 1
0.5 = _______________ % (multiply by 100)
0.5 = 50%
CHECK:
Change percent to decimal by dividing by 100
50% =
50
= 0.5
100
EXAMPLE 2
1.1 = _______________ % (multiply by 100)
1.1 = 110%
CHECK:
Change percent to decimal by dividing by 100
110% =
110
= 1.1
100
EXAMPLE 3
0.062 = _______________ % (multiply by 100)
0.062 = 6.2%
CHECK:
Change percent to decimal by dividing by 100
6.2% =
6.2
= 0.062
100
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Ratio, Proportion, and Percent
Experiential Activity Five
Change the following decimals to a percent.
1.
2.
3.
4.
5.
6.
0.1
0.8
0.25
4.5
0.125
0.05
Show Me.
Experiential Activity Five Answers
1.
2.
3.
4.
5.
6.
10%
80%
25%
450%
12.5%
5%
17
Ratio, Proportion, and Percent
OBJECTIVE SIX
When you complete this objective you will be able to…
Solve percent questions.
Exploration Activity
All problems using percent will be done using ratios. Therefore all percent problems can
be grouped into 3 types.
TYPE I:
Calculate the percent of
a quantity.
Example:
Find 30% of 75
TYPE II:
Determine what percent
one quantity is of
another.
Example:
25 is what percent of
60?
TYPE III:
Determine the quantity
from percentage and
percent.
Example:
25 is 50% of what
number?
The following model will be used to solve all percentage problems:
(1)
(2)
=
(3)
(4)
These are 4 positions; one of them is always taken up by the number 100 because percent
is always based out of 100.
(1)
100
=
(3)
(4)
Percent always goes over 100.
%
(3)
=
100 (4)
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Ratio, Proportion, and Percent
Finding the percent of a number Æ the number always goes in position 4.
%
100
=
(3)
of a number
In position (3) we find the answer.
%
100
=
answer
of a number
This is the model we will use to solve percent problems.
TYPE I Problems
EXAMPLE 1
Find 30% of 75.
One position is taken up by 100.
Percent always goes over 100.
30 ?
=
100 ?
We are finding 30% of the number 75. Answer in last position.
30 answer
=
100
75
Replace the word answer with the variable x,
30
x
=
100 75
Solve:
100(x ) = (30 )(75)
100 x = 2250
2250
x=
100
x = 22.5
Therefore: 30% of 75 = 22.5
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Ratio, Proportion, and Percent
EXAMPLE 2
Find 60% of 35.
Place the 100.
Then 60% goes over 100.
60 ?
=
100 ?
60% of the number 35 goes where?
Answer = x, goes where?
60
x
=
100 35
Solve
100 x = (60 )(35)
x=
(60)(35)
100
Fill in the blank.
Therefore: 60% of 35 = ____________
All problems finding the percent of a certain number are called TYPE 1.
TYPE II Problems
%
100
=
answer
of a number
EXAMPLE 1
36 is what percent of 48?
Fill in the 100.
Percent over 100. We do not know this value. Let it = x.
36
x
=
100 48
(48)(x) = (36)(100)
x=
(36)(100)
(48)
x = 75
36 is 75% of 48.
20
Ratio, Proportion, and Percent
EXAMPLE 2
18 is what percent of 72?
Place 100.
Percent over 100.
18
x
=
100 72
72 x = (18)(100)
x = 25
Fill in the blank.
Therefore: 18 is ___________ % of 72.
TYPE III Problems
%
100
=
answer
of a number
EXAMPLE 1
30 is 50% of what number?
Place the 100.
Percent over 100.
50% of a number. We do not know the number, therefore let x = the number.
50 30
=
x
100
(50)(x ) = (30)(100)
Fill in the blanks.
x = ______________
Therefore: 30 is 50% of ______________ ?
21
Ratio, Proportion, and Percent
EXAMPLE 2
18 is 25% of what number?
Place the 100.
Fill in the blanks.
% over 100.
=
Let x = the number
(25)(x ) = (18)(100)
x = ___________
Therefore: 18 is 25% of ______________ ?
22
Ratio, Proportion, and Percent
Experiential Activity Six
Solve for the following:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
52% of 30
37% of 125
65% of 75
3.5% of 17.2
5 is what percent of 250?
3.6 is what percent of 48? Show Me.
0.14 is what percent of 3.5?
6.5 is what percent of 30?
7 is 30% of what number?
18 is 75% of what number? Show Me.
240 is 10% of what number?
65 is 90% of what number?
The following exercise is a review of the 3 types of percent problems just presented in
this module.
Complete the table as shown in number 1 and solve for the indicated unknown.
Problem
1. 14 is what % of 52
Type
I, II, III
Model
Solution
II
14
x
=
100 52
x=?
2. 30% of 10
3. 65 is 30% of a number
4. 1.75% of a number is 7
5. 20% of 65
6. 12 is what % of 175?
7. 75 is 95% of what number?
8. 95% of 6000
9. 55 is what % of 90?
10. 3% of a number is 17
23
Ratio, Proportion, and Percent
Experiential Activity Six Answers
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
15.6
46.25
48.75
0.602
2%
7.5%
4%
21.6667%
23.3333
24
2400
72.2222
ANSWERS for review exercise.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
26.9231%
3
216.6667
400
13
6.8571%
78.9474
5700
61.1111%
566.6667
24
Ratio, Proportion, and Percent
OBJECTIVE SEVEN
When you complete this objective you will be able to…
Solve percent error in measurement problems.
Exploration Activity
The percent error in a measurement is calculated from:
% error =
measured value − true value
× 100
true value
EXAMPLE 1
In a laboratory experiment a student determined the velocity of sound to be 352 m/s. The
true value under the same conditions is 343 m/s. Determine the percent error in the
measurement.
Solution:
True value = 343 m/s
Measured value = 352 m/s
Substituting into the above equation, we get
% error =
352 − 343
× 100
343
= 2.62%
Notice the answer is positive. If the measurement were less than the true value the answer
would have been negative.
25
Ratio, Proportion, and Percent
Experiential Activity Seven
1.
An airport runway is measured to be 5362 m in length. Its true value was
supposed to be 5400 m. Find the percent error in the length of the runway.
2.
A surveyor's tape reads 100 m. However on a particular day it is actually
100.12 m in length. Find the percent error in its length.
3.
A grocer's scale reads 3 kg on an item that is actually 2.85 kg. Find the
percent error in the measurement. Is the customer getting a deal?
4.
If the present length of a steel rail is 13.0 m, what will be its length after a 0.5
percent expansion caused by heating?
5.
The volume of a gas is measured to be 58.5 ml. If this is 6% lower than the
true volume, what is the true volume? Show Me.
Experiential Activity Seven Answers
1. −0.7037%
2. 0.1199%
3. 5.2632%; no
4. 13.065 m
5. 62.2340 ml
Practical Application Activity
Complete the ratio, proportion, and percent module assignment in TLM.
Summary
This module introduced the student to the basic concepts of ratio, proportion, and percent.
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Ratio, Proportion, and Percent