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Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 1
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 2
Equations, Inequalities, and Applications
Chapter 2
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 3
2.6
Ratio, Proportion, and Percent
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 4
Objectives
1. Write ratios.2. Solve proportions.3. Solve applied problems using proportions.4. Find percentages and percents.
2.6 Ratio, Proportion, and Percent
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 5
Writing Ratios
Ratio
A ratio is a comparison of two quantities using a quotient.
The ratio of the number a to the number b is written
a to b, a : b, or .
ab
2.6 Ratio, Proportion, and Percent
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 6
(a) The ratio of 7 yards to 4 yards is
Writing Ratios
(b) To find the ratio of 8 feet to 6 yards, first convert 6 yards to feet.
74
7 yd4 yd
= .
6 yards = 6 • 3 = 18 ft
49
8 ft18 ft
= .8 ft6 yd
=
The ratio of 8 feet to 6 yards is thus
Example 1 Write a ratio for each word phrase.
2.6 Ratio, Proportion, and Percent
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 7
Size Price
18-oz $1.89
40-oz $4.16
64-oz $7.04
Writing Ratios
Example 2 What size is the best buy? That is, which size has the lowest unit price?
PEANUT BUTTER
2.6 Ratio, Proportion, and Percent
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 8
Writing Ratios
Example 2 (continued)
Size Price Unit Cost (dollars per ounce)
18-oz $1.89
40-oz $4.16
64-oz $7.04
$1.8918
$4.1640
$7.0464
= $0.105
= $0.104
= $0.110
Best Buy!
Because the 40-oz size produces the lowest unit cost, it is the best buy.
2.6 Ratio, Proportion, and Percent
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 9
Solving Proportions
ab
cd
=If , then ad and bc are equal and are called cross
products.
A proportion says that two ratios are equal, so it is a special type of equation. We read the proportion
ab
cd
= (b, d ≠ 0).
as “a is to b as c is to d.” We can also find the products ad and bc by multiplying diagonally.
2.6 Ratio, Proportion, and Percent
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 10
Solving Proportions
Cross products must be equal.
Example 3 Solve the proportion.
23
x51=
23
x51=
2 • 51 = 3 • x
Multiply.102 = 3x
Divide by 3.34 = x
Check by substituting 34 for x in the proportion. The solution is 34.
2.6 Ratio, Proportion, and Percent
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 11
Solving Proportions
Cross products must be equal.6 ( w – 4 ) = 3 ( w + 1 )
3w = 27
Divide by 3.w = 9
Check that the solution is 9.
Example 4Solve the equation. =
w – 43
w + 16
=w – 43
w + 16
6w – 24 = 3w + 3 Distribute.
Add 24.6w = 3w + 27
Subtract 3w.
2.6 Ratio, Proportion, and Percent
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 12
Solving Applied Problems Using Proportions
Cross products must be equal.15.33x = 7.0(35.04)
He pumped a total of 16 gal. Check this answer. Notice that the way the proportion is set up uses the fact that the unit price is the same, no matter how the gallons are purchased.
=$15.337.0
$35.04x
15.33x = 245.28 Multiply.
Divide.x = 16
Example 5 After Edwin pumped 7.0 gal of gasoline, the display showing the price read $15.33. When he finished pumping the gasoline, the display read $35.04. How many gallons did he pump?
Price PriceGallons Gallons
2.6 Ratio, Proportion, and Percent
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 13
Finding Percentages and Percents
We can solve a percent problem by writing it as a proportion
ab
P100
=
The amount, or percentage, is compared to the base (the whole amount). Since percent means per 100, we compare the numerical value of the percent to 100.
amountbase
percent100
= or .
2.6 Ratio, Proportion, and Percent
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 14
Finding Percentages and Percents
Cross products must be equal.100a = 750(16)
Thus, 16% of 750 is 120.
100a = 12,000 Multiply.
Divide.a = 120
Example 6 Find 16% of 750.
ab
P100
=
a750
16100
=
2.6 Ratio, Proportion, and Percent
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 15
Finding Percentages and Percents
Cross products must be equal.100a = 15(26)
The amount of the discount on the CD is $3.90, and the sale price is $15.00 – $3.90 = $11.10.
100a = 390 Multiply.
Divide.a = 3.90
Example 7 A CD with a regular price of $15 is on sale this week at 26% off. Find the amount of the discount and the sale price this week.
ab
P100
=
a15
26100
=
2.6 Ratio, Proportion, and Percent
Copyright © 2010 Pearson Education, Inc. All rights reserved. 2.6 – Slide 16
Finding Percentages and Percents
Cross products must be equal.100(255) = 850P
The sale price represented a 30% savings.
25,500 = 850P Multiply.
Divide.30 = P
Example 8 A computer advertisement was listed in the newspaper for $595. The regular price was $850. What percent of the regular price was the savings?
ab
P100
=
255850
P100
=The savings amounted to $850 – $595 = $255.
2.6 Ratio, Proportion, and Percent