Ratios, Rates and Proportions

Write the ratios:

Shaded to Unshaded,

Unshaded to total,

Total to Shaded,

Shaded to Total

Write equivalent ratios

with 100 in the denominator

Ratio: Comparison of 2 quantities - fraction form.

Apples to apples or oranges to oranges. It could be Granny Smith to Red delicious or Temple oranges to Navel oranges

Example: 5/6 (no units)

Write each ratio in simplest form. REDUCE!

20

154

3= 30

365

6

270

220 220 270

10 2710 22

Rate: Comparison of two numbers with different units

Example: Dollars to ounces

Example: Miles to gallons

Reduce fraction

Unit Rate

Rate Comparison of the numeratorto 1 in the denominator

Divide by the denominator

Example: A car traveled 600 miles in 10 hours.

600/10 = 60 mph

Unit Rate Examples:

You purchased 5 pounds of cat food for $2.75.

What was the per pound cost?

$2.75 ÷ 5 pounds

75.25

55.

025

250

25

75.25$ per pound

The Concept of Proportions

The Proportion has a fraction on each side of an equal sign. The fractions have the same value.

6448

43

Write a proportion to express the following:

5 is to 7 as 15 is to 21 21

15

7

5

Write one for: 6 is to 8 as 9 is to 12

12

9

8

6

Determine if a Proportion Is True or False byMultiplying diagonally across the equal sign

21

6

14

4 )6(14)21(4

84=84 True

Products are equal

4/14 could have been reduced first.

If the cross-products are not equal, statement is false.

231

174

21

16 16(231)?21(174) 16(231)?21(174)

3 3

16(77) ? 7 ( 174) =16(11) ? 174 176=174

Or 16 * 231 = 3696 21*174=3654

3696 is not the same as 3654 = false

7 7

Solving Proportions

14412

5 n 144*512 n

12

144*5

12

12

n60

12*5

n

n

n

150

2

125 150*2125 n

125

150*2

125

125

n

25

25

125

150*2N

4.25225/12

5

6

1

25

6*2

n

Applied Proportion Problems

A car assembly line made 243 engines on Monday. 27 were defective.

If that rate holds true for each day, how many of the 4131 engines made in a month are defective?

Method: 1. Determine unit rate - How many engines per defective unit. Divide that into the total engines. Divide engines by engines.

Method: 2. Write proportion, cross multiply and divide.You need to keep units corresponding

APPLIED PROPORTION PROBLEMS

Write the proportion keeping units corresponding.

Cross multiply.

Divide by the coefficient of the variable

Create a table to keep units corresponding:

4131

24327

total

x

defective

month

yesterday27*4131/243= x

11537/243=459 defective

Or reduce first: 27*4131/243(27/27)=4131/9

27*4131=243x

Never multiply the same units together. If you have, you have set up your equation wrong.

A 10-speed bike has a gear ratio of 3 revolutions of the pedal for every two revolutions of the bike wheel. If the bike wheel turns at 65 revolutions per minute,how many times must she pedal per minute.

wheel pedalgear ratio 2 3 65 x

2x=65*3

2

195

2

2x

X = 97.5 rpm

When there are 8500 passengers a day, Tim rents 340 bikes.The next projection is for 10,300 passengers. How many bikes should Tim plan on renting?

passengers bikes 8500 340 10300 x

8500x = 340*10300

85x = 340 * 103

x = 340* 103/ 85

X = 412 bikes

A biologist caught 42 fish in a lake and tagged them.She returned them to the lake. After a few days, she caught 50 fish and 7 were tagged. How many fish are in the lake?

caught tagged 50 7 x 42

7x = 50 * 42

x = 50 * 6 = 300 fish

A park ranger an Alaska forest captures and tags 50 bears. He releases them to range through the forest. Later,he recaptures 50. Four have tags. Estimate the numberof bears in the forest.

caught tagged 50 4 x 50

4x = 50*50

X = 2500 / 4 = 625 bears