Transcript Chapter 5: Rates and Proportions

Chapter 2.4
Rates, Ratios,
and
Proportions
A ratio compares two quantities.
Slide 5.1- 2
Parallel
Example 1
Writing Ratios
Tamara spent $13 on fish, $8 on salad and $7 on
bread. Write each ratio as a fraction.
a. Ratio of amount spent on salad to amount
spent on bread.
$8 8

The ratio of salad to bread is
$7
Numerator
Denominator
(mentioned first)
(mentioned second)
7
b. Ratio of fish to bread.
$13 13

7
$7
Slide 5.1- 3
Parallel
Example 2
Writing Ratios in Lowest Terms
Write each ratio in lowest terms.
a. 80 days to 20 days.
Divide the numerator and denominator by 20.
80 80  20 4


20 20  20 1
b. 30 ounces of medicine to 140 ounces of
medicine
30  10 3
30


140 140  10 14
Slide 5.1- 4
Parallel
Example 3
Using Decimal Numbers in a Ratio
The price of a bag of dog food increased from
$22.95 to $25.50. Find the ratio of the increase in
price to the original price.
new price – original price = increase
$25.50  $22.95 = $2.55
Find the ratio of the increase in price to the original
price.
2.55  increase in price
22.95
 original price
Now write the ratio as a ratio of whole numbers.
255  255
1
2.55
2.55  100
255




22.95 22.95  100 2295 2295  255 9
Slide 5.1- 5
Parallel
Example 4
Using Mixed Numbers in Ratios
Write each ratio as a comparison of whole
numbers in lowest terms.
1
a. 4 days to 4 days
4
Write the ratio and divide out common units.
4 days
4 41
4
 1
days 4 4
Write as improper fractions.
4
4
4 17 4 4 16
1

17  1  4  1  17  17
4 41
4
Reciprocals
Slide 5.1- 6
Parallel
Example 4
Using Mixed Numbers in Ratios
b. 4 3 pounds to 2 1 pounds
8
4
3 35
4 
8
8
4 38
2 41
1 9
2 
4 4
35
 8
9
4
1
35 9 35 4 35 4
35

 
 


8 4
8 9 82 9
18
Slide 5.1- 7
Parallel
Example 1
Writing Rates in Lowest Terms
Write each rate as a fraction in lowest terms.
a. 8 gallons of antifreeze for $40.
8 gallons  8
1 gallon

40 dollars  8
5 dollars
b. 192 calories in 6 ounces of yogurt
192 calories  6
32 calories

6 ounces  6
1 ounce
Slide 5.2- 8
Parallel
Example 1
continued
Writing Rates in Lowest Terms
Write each rate as a fraction in lowest terms.
c. 84 hamburgers on 7 grills.
84 hamburgers  7
12 hamburgers

7 grills  7
1 grill
Slide 5.2- 9
When the denominator of a rate is 1, it is called
a unit rate. For example, you earn $16.25 for
1 hour of work.
This unit rate is written:
$16.25 per hour
Use per or a slash mark (/) when writing unit rates.
Slide 5.2- 10
Parallel
Example 2
Finding Unit Rates
Find each unit rate.
a. 445.5 miles on 16.5 gallons of gas
445.5 miles
16.5 gallons
Divide to find the unit rate.
27
16.5 445.5
445.5 miles  16.5 27 miles

16.5 gallons  16.5
1 gallon
The unit rate is 27 miles per gallon or 27 miles/gallon.
Slide 5.2- 11
Parallel
Example 2
continued
Finding Unit Rates
Find each unit rate.
b. 413 feet in 14 seconds
413 feet
14 seconds
Divide to find the unit rate.
29.5
14 413.0
The unit rate is 29.5 feet/second.
Slide 5.2- 12
Parallel
Example 3
Determining the Best Buy
A local store charges the following prices for
jars of jelly.
$3.69
$3.09
$2.39
18 oz.
24 oz.
28 oz.
The best buy is the container with the lowest cost per
unit. All the jars are measured in ounces. Find the
cost per ounce for each one by dividing the price of
the jar by the number of ounces in it. Round to the
nearest thousandth if necessary.
Slide 5.2- 13
Parallel
Example 3
continued
Determining the Best Buy
Size
Cost per Unit (rounded)
18 ounces
$2.39
 $0.133 per ounce
18 ounces
$3.09
 $0.129 per ounce
24 ounces
24 ounces
28 ounces
highest
lowest
$3.69
 $0.132 per ounce
28 ounces
The lowest cost per ounce is $0.129, so the 24-ounce
jar is the best buy.
Slide 5.2- 14
Parallel
Example 4
Solving Best Buy Applications
Juice is sold as a concentrated can as well as
in a ready to serve carton. Which of the choices
below is the best buy?
12 oz can makes 48
ounces of juice for $1.69
60 oz carton for $2.59
To determine the best buy, divide the cost by the
number of ounces.
Slide 5.2- 15
Parallel
Example 4
continued
Solving Best Buy Applications
12 oz can makes 48
ounces of juice for $1.69
60 oz carton for $2.59
Concentrate
$1.69
48 ounces
$0.0352 per ounce
Carton
$2.59
60 ounces
$0.0432 per ounce
Although, you must mix it yourself, the concentrated can of
juice is the better buy.
Slide 5.2- 16
Four numbers are used in a proportion. If any
three of these numbers are known, the fourth can
be found.
Slide 5.4- 17
Parallel
Example 1
Solving Proportions for Unknown
Numbers
Find the unknown number in each proportion.
Round answers to the nearest hundredth when
necessary.
a. 30  48
x
40
Ratios can be written in lowest terms. You can do
that before finding the cross products.
6
48
can be written in lowest terms as ,
5
40
which gives the proportion
30 6
 .
x
5
Slide 5.4- 18
Parallel
Example 1
continued
Solving Proportions for Unknown
Numbers
30 6

x
5
Step 1
Step 2
x  6  30  5
x 6
Find the cross products
30  5
Show that the cross products are equivalent.
x  6  150
1
Step 3
x  6 150

6
6
1
x  25
Slide 5.4- 19
Parallel
Example 1
continued
Solving Proportions for Unknown
Numbers
3 20
b.

7
x
3 20

7
x
Step 1
3  x  140
Step 2
7  20
Find the cross products
3x
Show that the cross products are equivalent.
1
Step 3 3  x  140
3
3
1
x  46.67
Rounded to the nearest hundredth.
Slide 5.4- 20
Parallel
Example 2
Solving Proportions with Mixed
Numbers and Decimals
Find the unknown number in each proportion.
6 32
x
8x
a.

6 32
x

8 36
Find the cross products
8 36
2
6  36
3
2
Find 6  36.
3
12
2
20 36
240  240
6  36 


3
3 1
1
1
Show the cross products are equivalent. 8  x  240
Divide both sides by 8.
8  x 240

8
8
x  30
Slide 5.4- 21
Parallel
Example 2
continued
2
3
6
30

8 36
Solving Proportions with Mixed
Numbers and Decimals
8  30  240
Equal
2
6  36  240
3
The cross products are equal, so 30 is the correct
solution.
Slide 5.4- 22
Parallel
Example 2
continued
Solving Proportions with Mixed
Numbers and Decimals
Find the unknown number in each proportion.
10.4 6.76

b.
12.4
x
Show that cross products are equivalent.
10.4 ( x)  (12.4)(6.76)
10.4 ( x)  83.824
10.4 x   83.824
10.4
Divide both sides by 10.4.
10.4
x  8.06
Slide 5.4- 23
Parallel
Example 2
continued
Solving Proportions with Mixed
Numbers and Decimals
10.4 6.76

12.4 8.06
10.4 ∙ 8.06 = 83.824
Equal
12.4 ∙ 6.76 = 83.824
The cross products are equal, so 8.06 is the
correct solution.
Slide 5.4- 24
Similar Triangles
• Similar Triangles whose angles have the same
measure, but their sides have different lengths.
• The triangles will look identical, but one will be
smaller than the other.
Slide 1- 25
•
𝑟
𝑐
=
𝑝
𝑏
=
𝑞
𝑎
x
y
Slide 1- 26
Slide 1- 27
Slide 1- 28
Solution
𝑌𝑜𝑢𝑟 𝐻𝑒𝑖𝑔ℎ𝑡
𝑌𝑜𝑢𝑟 𝑆ℎ𝑎𝑑𝑜𝑤 𝐿𝑒𝑛𝑔𝑡ℎ
=
𝑇𝑟𝑒𝑒 𝐻𝑒𝑖𝑔ℎ𝑡
𝑇𝑟𝑒𝑒 𝑠ℎ𝑎𝑑𝑜𝑤 𝑙𝑒𝑛𝑔𝑡ℎ
Slide 1- 29
Hw section 2.4
1-20
Try 21
Slide 1- 30