Download Proportions Using Cross Products

Proportions Using Cross
Products
Jen Kershaw
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Printed: September 2, 2013
AUTHOR
Jen Kershaw
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Concept 1. Proportions Using Cross Products
C ONCEPT
1
Proportions Using Cross
Products
Here you’ll learn to solve proportions using cross products.
Remember Manuel who was reading all of the medieval books on knights in the Recognize Proportions as a
Statement of Equivalent Ratios Concept? Well after he finished reading the series, he loaned it to his friend Rafael.
Rafael is enjoying the series as much as Manuel did.
In five weeks, Rafael had already finished 8 of the 12 books. It took Manuel 7.5 weeks to read all 12 books. Will
Rafael and Manuel finish the series in the same amount of time? Are they reading at the same rate? How can you
figure this out?
To figure this out, you will need to know how to determine if two ratios form a proportion. If the reading rate
of the boys is the same, then the ratios will form a proportion.
Use this Concept to learn how to solve proportions using cross products. Then you will know how to figure
out this dilemma.
Guidance
Previously we learned that a proportion states that two ratios are equivalent. Here are two proportions.
a
b
=
c
d
or
a:b=c:d
In a proportion, the means are the two terms that are closest together when the proportion is written with colons. So,
in a : b = c : d, the means are b and c.
The extremes are the terms in the proportion that are furthest apart when the proportion is written with colons. So,
in a : b = c : d, the extremes are a and d.
The diagram below shows how to identify the means and the extremes in a proportion.
1
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In the last lesson, you learned how to solve proportions by using proportional reasoning. We can also solve a
proportion for a variable in another way. This is where the cross products property of proportions comes in.
What is the Cross Products Property of Proportions?
The Cross Products Property of Proportions states that the product of the means is equal to the product of the
extremes. You can find these cross products by cross multiplying, as shown below.
a c
=
b d
b·c = a·d
a
4
=
6
8
To solve this, we can multiply the means and the extremes.
a·8 = 4·6
8a = 24
Next, we solve the equation for the missing variable. To do this, we use the inverse operation. Multiplication is
in the problem, so we use division to solve it. We divide both sides by 8.
8a 24
=
8
8
a=3
Our answer is 3.
Solve for each variable in the numerator by using cross products.
Example A
x
5
=
6
10
Solution:x = 3
Example B
a
9
=
15
27
Solution:a = 5
Example C
b
4
=
12
16
Solution:b = 3
2
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Concept 1. Proportions Using Cross Products
Here is the original problem once again.
Remember Manuel who was reading all of the medieval books on knights? Well after he finished reading the series,
he loaned it to his friend Rafael. Rafael is enjoying the series as much as Manuel did.
In five weeks, Rafael had already finished 8 of the 12 books. It took Manuel 7.5 weeks to read all 12 books. Will
Rafael and Manuel finish the series in the same amount of time? Are they reading at the same rate? How can you
figure this out?
Let’s write a proportion to solve this problem.
8 books
5 weeks
=
12 books
7.5 weeks
Next, we can use cross products to see if the two ratios form a proportion. If they do, then the two boys will
finish the series in the same amount of time.
8 × 7.5 = 60
5 × 12 = 60
The two cross products are equal so the two ratios form a proportion. The two boys will finish the series of
books in the same amount of time.
Vocabulary
Proportion
two equal ratios form a proportion.
Means
the values of a proportion that are close to each other when written in ratio form using a colon.
Extremes
the values of a proportion that are farther apart from each other when written in ratio form using a colon.
Cross Product Property of Proportions
states that the cross products of two ratios will be equal if the two ratios form a proportion.
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Guided Practice
Here is one for you to try on your own.
The ratio of boys to girls in the school chorus is 4 to 5. There are a total of 20 boys in the chorus. How many total
students are in the chorus?
Answer
The ratio given, 4 to 5, compares boys to girls. However, the question asks for the total number of students in the
chorus.
One way to set up a proportion for this problem would be to write two equivalent ratios, each comparing boys to
total students.
The ratio of boys to girls is 4 to 5. We can use this ratio to find the ratio of boys to total students.
boys
total
=
boys
boys+girls
=
4
4+5
=
4
9
You know that there are 20 boys in the chorus. The total number of students is unknown, so represent that as x.
boys
total
=
20
x
Get those ratios equal to form a proportion. Then cross multiply to solve for x.
4 20
=
9
x
9 · 20 = 4 · x
180 = 4x
180 4x
=
4
4
45 = x
So, there are a total of 45 students in the school chorus.
Video Review
MEDIA
Click image to the left for more content.
- This is a James Sousa video onsolvingproportions.
Practice
Directions: Use cross products to find the value of the variable in each proportion.
1.
2.
4
6
x
10 = 5
2
x
3 = 9
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3.
4
9
=
a
45
4.
7
8
=
a
4
5.
b
8
=
5
16
6.
6
3
=
x
9
7.
4
x
=
8
10
8.
1.5
y
=
3
9
9.
4
11
=
c
33
10.
2
6
=
5
y
11.
2
10
=
5
x
12.
4
12
=
6
n
13.
5
70
r = 126
4
14
14 = k
14.
15.
16.
Concept 1. Proportions Using Cross Products
8
6
w = 3
2
17
5 = a
5