4-1 ratios & proportions
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4-1 Ratios & Proportions. Notes. A ratio is a comparison of two quantities. . Ratios can be written in several ways . 7 to 5, 7:5, and name the same ratio. 15 ÷ 3 9 ÷ 3. bikes skateboards. Example 1: Writing Ratios in Simplest Form. - PowerPoint PPT PresentationTRANSCRIPT
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4-1 RATIOS & PROPORTIONS
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A ratio is a comparison of two quantities. Ratios can be written in several ways. 7 to 5, 7:5, and name the same ratio.7
5
Notes
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Example 1: Writing Ratios in Simplest Form
Write the ratio 15 bikes to 9 skateboards in simplest form.
159
53
The ratio of bikes to skateboards is , 5:3, or 5 to 3.
=
15 ÷ 39 ÷ 3
Write the ratio as a fraction.
= = Simplify.
53
bikesskateboards
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Check It Out! Example 2
Write the ratio 24 shirts to 9 jeans in simplest form.
249
83
The ratio of shirts to jeans is , 8:3, or 8 to 3.
=shirtsjeans
24 ÷ 39 ÷ 3
Write the ratio as a fraction.
= = Simplify.
83
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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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Write the ratio 3 yards to 12 feet in simplest form.First convert yards to feet.
9 feet12 feet=3 yards
12 feet34=9 ÷ 3
12 ÷ 3=
There are 3 feet in each yard.
Example 3: Writing Ratios Based on Measurement
3 yards = 3 ● 3 feet= 9 feet Multiply.
Now write the ratio.
Simplify.
The ratio is , 3:4, or 3 to 4.34
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Write the ratio 36 inches to 4 feet in simplest form.First convert feet to inches.
36 inches48 inches=36 inches
4 feet34 =36 ÷ 12
48 ÷ 12=
There are 12 inches in each foot.
Check It Out! Example 3
4 feet = 4 ● 12 inches
= 48 inches Multiply.Now write the ratio.
Simplify.
The ratio is , 3:4, or 3 to 4.34
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Practice 4 feet to 24 inches
3 yards to 12 feet
2 yards to 20 inches
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Ratios that make the same comparison are equivalent ratios.
To check whether two ratios are equivalent, you can write both in simplest form.
Notes
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Example 4: Determining Whether Two Ratios Are Equivalent
Simplify to tell whether the ratios are equivalent.
1215
B. and 2736
327
A. and 218 Since ,
the ratios are equivalent.
19= 1
919=3 ÷ 3
27 ÷ 3327 =
19=2 ÷ 2
18 ÷ 2218 =
45=12 ÷ 3
15 ÷ 31215=
34=27 ÷ 9
36 ÷ 92736=
Since ,the ratios are not equivalent.
45
34
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Practice
5628
4921
4816
3913
and
and
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Lesson Quiz: Part IWrite each ratio in simplest form.1. 22 tigers to 44 lions
2. 5 feet to 14 inches
415
3.
721
4.
830
1245Possible answer: ,
13
1442Possible answer: ,
Find a ratio that is equivalent to each given ratio.
12
307
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Lesson Quiz: Part II
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?864
16128
and ; yes, both equal 1 8
85
85= ; yes16
105.
3624
6.
Simplify to tell whether the ratios are equivalent.
and 32 20
and 28 18
32
149 ; no
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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?
710
, 2130
x 3
x 3
Yes, these two ratios DO form a proportion, because the same relationship exists in both the numerators and denominators.
89
, 23
÷ 4
÷ 3
No, these ratios do NOT form a proportion, because the ratios are not equal.
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Example
340
=7
÷ 5
÷ 5
8
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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.
412
, 39
12 x 3 = 369 x 4 = 36
These two ratios DO form a proportion because their cross products are the same.
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Example 2
58
, 23
8 x 2 = 163 x 5 = 15
No, these two ratios DO NOT form a proportion, because their cross products are different.
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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
k17 = 20
68
Start with the variable.
=68k 17(20)
Simplify.
68k = 340
Now we have an equation. To get the k by itself, divide both sides by 68.
68 68k = 5
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Example 2: Solve the ProportionStart with the variable.
=2x(30) 5(3)
Simplify.
60x = 15
Now we have an equation. Solve for x.
60 60x = ¼
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Example 3: Solve the ProportionStart with the variable.
=(2x +1)3 5(4)
Simplify.
6x + 3 = 20
Now we have an equation. Solve for x.
x =
=
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Example 4: Solve the ProportionCross Multiply.
=3x 4(x+2)
Simplify.
3x = 4x + 8
Now we have an equation with variables on both sides. Solve for x.
x = -8
=