Transcript cross product

LESSON
6.3
Solving Proportions Using Cross Products
Every pair of ratios has two cross products. A cross product of two ratios is
the product of the numerator of one ratio and the denominator of the other
ratio.
3 6
,
5 10
Ratios:
Cross products:
3 • 10
3
2 6
,
3 11
5•6
2 • 11
3•6
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Notice that for the ratios and , the ratios are equal and their cross products
5
2 10 6
are also equal. For the ratios and , the ratios are not equal, and neither are
3
11
their cross products.
You can use cross products to tell whether two ratios form a proportion. If the cross
products are equal, then the ratios form a proportion.
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LESSON
6.3
Solving Proportions Using Cross Products
EXAMPLE
1
Determining if Ratios Form a Proportion
Tell whether the ratios form a proportion.
12 32
,
20 50
12 ? 32
=
20 50
Write proportion.
12 • 50 =? 20 • 32
600 = 640
ANSWER
Form cross products.
Multiply.
The ratios do not form a proportion.
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LESSON
6.3
Solving Proportions Using Cross Products
You can use the multiplication property of equality to demonstrate an important
property about the cross products of a proportion.
a
= c
b
d
1
1
Given.
1
a
bd
c
bd
•
=
•
b
1
d1 1
ad = cb
Multiply each side by bd.
Divide out common factors.
Simplify.
This result proves the cross products property.
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LESSON
6.3
Solving Proportions Using Cross Products
Cross Products Property
Words
The cross products of a proportion are equal.
2
6
Numbers Given that 5 = 15 , you know that 2 • 15 = 5 • 6.
Algebra
If
a c
= , where b = 0 and d = 0, then ad = bc.
b d
You can use the cross products property to solve proportions.
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LESSON
6.3
Solving Proportions Using Cross Products
EXAMPLE
2
Writing and Solving a Proportion
Hair Growth Human hair grows about 0.7 centimeter in 2
weeks. How long does hair take to grow 14 centimeters?
SOLUTION
0.7 14
= x
2
0.7 • x = 2 • 14
Length of hair grown
Number of weeks
Cross products property
0.7x = 28
Multiply.
0.7x = 28
= 0.7
0.7
Divide each side by 0.7.
x = 40
ANSWER
Simplify.
Hair takes about 40 weeks to grow 14 centimeters.
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LESSON
6.3
Solving Proportions Using Cross Products
Methods for Solving a Proportion
SUMMARY
To solve the proportion
5
x
= , use one of the following:
12 36
Equivalent ratios
5  3 15
12
36
5
x
12  3 36
Algebra
5
x
36 • 12 = 36 • 36
15 = x
Multiply each side by 36.
Simplify.
Cross products
5 • 36 = 12x
15 = x
Cross products property
Divide each side by 12.
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