4-1 Ratios

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4-1 Ratios Proportions
2
Notes
A ratio is a comparison of two quantities.
Ratios can be written in several ways. 7 to 5,
75, and name the same ratio.
 
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Example 1 Writing Ratios in Simplest Form
Write the ratio 15 bikes to 9 skateboards in
simplest form.
15 9
Write the ratio as a fraction.

5 3
Simplify.


5 3
The ratio of bikes to skateboards is , 53, or
5 to 3.
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Check It Out! Example 2
Write the ratio 24 shirts to 9 jeans in simplest
form.
Write the ratio as a fraction.
24 9

8 3
Simplify.


8 3
The ratio of shirts to jeans is , 83, or 8 to
3.
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Practice

• 15 cows to 25 sheep
• 24 cars to 18 trucks
• 30 Knives to 27 spoons

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When simplifying ratios based on measurements,
write the quantities with the same units, if
possible.
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Example 3 Writing Ratios Based on Measurement
Write the ratio 3 yards to 12 feet in simplest
form.
First convert yards to feet.
There are 3 feet in each yard.
3 yards 3 ? 3 feet
Multiply.
9 feet
Now write the ratio.
9 feet 12 feet
Simplify.
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Check It Out! Example 3
Write the ratio 36 inches to 4 feet in simplest
form.
First convert feet to inches.
There are 12 inches in each foot.
4 feet 4 ? 12 inches
48 inches
Multiply.
Now write the ratio.
36 inches 48 inches
Simplify.
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Practice

• 4 feet to 24 inches
• 3 yards to 12 feet
• 2 yards to 20 inches

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Notes
Ratios that make the same comparison are
equivalent ratios. To check whether two ratios
are equivalent, you can write both in simplest
form.
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Example 4 Determining Whether Two Ratios Are
Equivalent
Simplify to tell whether the ratios are
equivalent.
1 9
1 9
4 5
3 4
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Practice
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Lesson Quiz Part I
Write each ratio in simplest form. 1. 22 tigers
to 44 lions 2. 5 feet to 14 inches
Find a ratio that is equivalent to each given
ratio.
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Lesson Quiz Part II
Simplify to tell whether the ratios are
equivalent.
and
and
7. Kate poured 8 oz of juice from a 64 oz bottle.
Brian poured 16 oz of juice from a 128 oz
bottle. Are the ratios of poured juice to
starting amount of juice equivalent?
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Vocabulary

• A proportion is an equation stating that two
  ratios are equal.

To prove that two ratios form a proportion, you
must prove that they are equivalent. To do this,
you must demonstrate that the relationship
between numerators is the same as the
relationship between denominators.
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Examples Do the ratios form a proportion?
x 3
Yes, these two ratios DO form a proportion,
because the same relationship exists in both the
numerators and denominators.
7
21
,
10
30
x 3
4
8
2
,
No, these ratios do NOT form a proportion,
because the ratios are not equal.
9
3
3
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Example
5
3
7

8
40
5
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Cross Products

• When you have a proportion (two equal ratios),
  then you have equivalent cross products.
• Find the cross product by multiplying the
  denominator of each ratio by the numerator of the
  other ratio.

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Example Do the ratios form a proportion? Check
using cross products.
4
3
,
12
9
These two ratios DO form a proportion because
their cross products are the same.
12 x 3 36
9 x 4 36
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Example 2
5
2
,
8
3
No, these two ratios DO NOT form a proportion,
because their cross products are different.
8 x 2 16
3 x 5 15
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Solving a Proportion Using Cross Products

• Use the cross products to create an equation.
• Solve the equation for the variable using the
  inverse operation.

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Example 1 Solve the Proportion
Start with the variable.
20
k

17
68
Simplify.
Now we have an equation. To get the k by itself,
divide both sides by 68.
68k
17(20)

68k

340
68
68
k
5

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Example 2 Solve the Proportion
 
Start with the variable.
Simplify.
Now we have an equation. Solve for x.
2x(30)
5(3)

60x

15
60
60
x
¼

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Example 3 Solve the Proportion
 
Start with the variable.
Simplify.
Now we have an equation. Solve for x.
(2x 1)3
5(4)

6x 3

20
x
 

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Example 4 Solve the Proportion
 
Cross Multiply.
Simplify.
Now we have an equation with variables on both
sides. Solve for x.
3x
4(x2)

3x

4x 8
x
-8