RatiosRates

Proportions

A Ratio is a comparison of two quantities by division.

Equivalent Ratios: two ratios that name the same number.

A rate is a ratio that compares two quantities measured in different ways.

Unit rate is the rate for one unit of a given quantity.

Unit cost is a unit rate that gives the cost per unit.

A proportion is an equation stating that 2 ratios are equal.

The cross products are the two products found by multiplying the denominator of each ratio by the numerator of the other ratio.

Ratio: 3:1, 3/1, 3 to 1, 1:3, 1/3,1 to 3, etc…

Conversions Factors &Rates: 55mph, 25mpg, 5ft/s, etc…

Proportions: 3/1 and 6/22/3 and 6/912/16 and 15/20

Answer True or False on each of the following Pair of

ratios if they form a proportion.

True or False:

― = ―42

206

Answer:

False

True or False:

― = ―32

188

Answer:

False

True or False:

― = ―43

1612

Answer:

True

True or False:

― = ―43

86

Answer:

True

True or False:

― = ―1224

34

Answer:

False

True or False:

― = ―69

23

Answer:

True

Solve the Following

Solve:

― = ―m10

103

Answer:

m = 33⅓

Solve:

― = ―v12

102

Answer:

v = 60

Solve:

― = ―10k

84

Answer:

k = 5

Solve:

― = ―2x

79

Answer:

x = 2 ⁴⁄₇

Solve:

― = ―3x

710

Answer:

x = 4 ²⁄₇

Ratio & ProportionsParts = a & bWhole = (a+b)

Part(a):Part(b) = a:b Proportions a:b = x·a:x·b

Part:Whole = a:(a+b) = b:(a+b)

The ratio of kids to adults is 5 to 2. If there is 14 adults, how many

kids are there?

Kids:adults=5:2 adults=14Lets try… 5:2 = 2·5:2·2 = 10:4 (no) 3·5:3·2 = 15:6 (no) 4·5:4·2 = 20:8 (no)… 7·5:7·2 = 35:14 35kids:14adults

Kids:adults=5:2 kids=k adults=14Or lets try…

5/2 = k/14

(5/2)·14 = k35 = k

Answer: 35 kids

Kids:adults=5:2 adults=14Or lets try…

Find the factor: 5x kids

2x adults2x = 14

x = 7 (factor)Therefore: 5x (kids)

5·7 =35

Answer: 35 kids

The ratio of kids to adults is 3 to 5. If there are 26 more adults than kids, how many kids are there?

Kids:adults=3:5 26 more adults than kidsLets try… 3:5 = 10·3:10·5 = 30:50 (no) 15·3:15·5 = 45:75 (no) 13·3:13·5 = 39:65

39kids:65adults

Kids:adults=3:5

Find the factor: 3x kids & 5x adults

adults – kids=265x-3x=26

2x=26x = 13 (factor)

Therefore: 3x (kids)3·13 =39

Answer: 39 kids

Kids:adults=3:5

k kidsk+26 adults

3/5 = k/(k+26)5k = 3(k+26)5k = 3k + 78

2k = 78k = 39

Answer: 39 kids

30 boys and girls are planning for a picnic. There are a ratio of 3 girls to 7 boys. How many boys

are there?

Answer:30 ⁷⁄₁₀ = ∙ 21

3x + 7x = 3010x = 30

x=33·33·7

9/21 (girls/boys)

I have 70 friends. Among them are 11 girls for every 3 boys. How

many friends are boys? How many are girls?

Answer:70 ¹¹⁄₁₄ = ∙ 55 girls70 ³⁄₁₄ = ∙ 15 boys

Also: find the factor11x + 3x = 70

14x=70x=5

11·5=55 girls3·5=15 boys

The ratio of dogs to cats is 9 to 2. If there are 72 dogs, how many total dogs and cats are there?

Answer:

88

To get a shade of orange paint, we mix white, red, & yellow in the

ratio 2:5:4. If we need 6 gallons of orange paint, then how many

gallons of each color do we need?

Answer:White=2x, red=5x, yellow=4x

2x+5x+4x=6X=6/11

White=12/11, red=30/11, yellow=24/11

The top 3 cash prizes in the math tournament are in the ratio 5:2:1. The 1st place prize is $20,000 more

than the 3rd place prize. How much is the second place prize?

Answer:1st=5x, 2nd=2x, 3rd=1x

5x-1x=20000X=5000

2nd place = 2x=2 5000=∙ $10,000

More ProportionWord Problems

5 pizzas cost $60. What will 9 pizzas cost?

Answer:

⁶⁰⁄₅ ∙ 9 = 108

6 stuffed peppers cost $36. What will 12 stuffed peppers cost?

Answer:

³⁶⁄₆ 12 = ∙ 72

10 glasses cost $40. How much do 20 glasses cost?

Answer:

⁴⁰⁄₁₀ ∙ 20 = $80

Twenty-eight is to five as eight hundred ninety-six is

to _?_.

Answer:

― = ―

x = 160

285

896x

If you can buy one can of pineapple chunks for $2, then how many can you buy with

$10?

Answer:

― = ―

x = 5 cans

12

x10

One jar of crushed ginger cost $2. How many jars can you

buy for $4?

Answer:

― = ―

x = 2 jars

12

x4

Lauren divided her money in the ratio 4:2 between Haley and

Makenzie. Haley got the smaller amount of $1256. How much did

Makenzie receive?

²⁄₆ T = 1256∙T=1256 ⁶⁄₂∙Total= 3768

Makenzie received 3768 – 1256 = $2512or

4/2 = M/12562M = 4·1256

M =2512

12 miles is approximately equal to 6 km. How many miles are equal

to 42 km?

Answer:

¹²⁄₆ 42 = ∙ 84

12 miles is approximately equal to 6 km. How many km are equal to

18 miles?

Answer:

⁶⁄₁₂ ∙ 18 = 9

Conversion Word Problems

How many inches are equal to 8 yards?

Answer:

8 yds 3ft/1yd ·12in/1ft = ∙ 288 inches

How many inches are equal to 2 miles?

Answer:

2 miles 5280ft/1mi ·12in/1ft =∙126,720 inches

How many seconds are equal to 2 years?

Answer:

2 yr (365days/1yr)·(24hr/1day)·(60min/1hr)·∙(60sec/1min)=

63,072,000 seconds

How many cups are in 5 gallons?

Answer:

5 gal (4qt/1gal)·(2pint/1qt)·(2cups/1pint)=∙

80 cups

I am making some math videos. I shoot 10 “takes” for each video and expect to make 144 videos per book. I have 11 books. Each “take”

averages 5 minutes. If I shoot continuously, how many days will it take me to make a full

set of videos for all the books?

Answer:

11bk (144vid/1bk)·(10takes/1vid)·(5min/∙1take)·

(1hr/60min)·(1day/24hr)

55 days

Similar Figures

Polygon is a closed plane figure formed by 3 or more line segments that do not cross.

Two polygons are similar polygons if: • corresponding angles have the same measure.• the lengths of the corresponding sides form equivalent ratios.

Example: BC/GH = AC/FH (Text Book) 50/75 = 40/60 or BC/AC = GH/FH 50/40 = 75/60

Each pair of figures are similar. Find the missing side.

20 x

123

Answer:

― = ―

x = 5

x20

312

Each pair of figures are similar. Find the missing side.

4x168

Answer:

― = ―

x = 2

x4

816

Each pair of figures are similar. Find the missing side.

14

x 1

2

Answer:

― = ―

x = 7

x14

12

Similar FiguresWord Problems

Brayden reduced the size of a rectangle to a height of 2

inches. What is the new width if it was originally 24 inches

wide and 12 inches tall?

Answer:

― = ―

x = 4 inches

2412

x2

You can use indirect measurement to measure distances that are difficult to measure directly by using proportions and similar figures.

A 6 ft tall person standing near a flagpole cast a shadow 4.5 ft

long. The flagpole casts a shadow 15 ft long. What is the

height of the flagpole?

Answer:

― = ―

4.5 x = 6 15∙ ∙x = (6 15)/4.5∙x = 20

x6

154.5

Scale Drawings

A scale drawing is an enlarged or reduced drawing of an object that is similar to the actual object.

A scale is the ratio that compares a length in a drawing or model to the corresponding length in the actual object.

The distance of the Rio Grande is 3.5 inches on a map. If the map scale is ½ inch = 2 miles,

what is the distance of the Rio Grande?

STOP

Answer:

― = ―

½ (x) = 2 (3.5)∙ ∙½ x = 7x = 14 miles

½2

3.5x

Competition Problems

Write 4 inches to 2 yards as a ratio in simplest form.

STOP

Answer:

1/18

Solve the proportion:

Answer:

3

The scale of a map is 1 in. = 20 mi. Find the actual distance , in miles, represented by 2 ¾ inches on the

map?

Answer:

55

Ford makes 7 red cars for every 4 white cars made. If it makes 16 white cars a day, how many red

cars will it make a day?

Answer:

28

A telephone pole is 30 feet tall. It casts a shadow that is 24 feet long. A tree that is next to the

telephone pole is 5 feet tall. How long is the shadow of the tree?

Answer:

4

If , solve for n.

Answer:

4 ⅖

A college football stadium holds 45,000 fans. In a random sample of 30 fans, 26 were wearing the

colors of the home team. Predict the total number of fans who are wearing the colors of the home

team.

Answer:

39,000

The distance between Earth and Planet Xeron is 5.274 × 108 miles. The Xerons make the trip in 3000

days. On average, how many miles per day do they travel?

Answer:

175,800

In a coastal region, there are 104 marked dolphins. In a sampling, biologists count 45 dolphins, of

which 33 are marked. About how many dolphins are in the region?

Answer:

142

The ratio of girls to boys in a sixth grade art class is 6:5. If there are

121 students in the class, how many are boys?

Answer:

55

A recipe uses 1 ¾ cups of chocolate chips for every 24 cookies. How many cups of

chocolate chips will be needed to make five dozen cookies?

Answer:

4 3/8

Triangle EFG is similar to triangle XYZ. Find the area of triangle XYZ.

15

1624

X

Y Z

E

F G

Answer:

80

John is building a bird house. The scale is ¼ ft. to 1 ft. One wall is 2

½ inches tall as drawn on the blueprint. What is the actual wall

height?

Answer:

10 inches

The distance from Birmingham to Atlanta is 2.25 inches on a map. If the map scale is 1/4 inch=2 miles,

what is the distance in miles?

Answer:

18 miles

Rebecca takes 20 minutes to cut a stick of bamboo into 5 pieces.

How long does it take her to cut another stick into 9 pieces?

Answer:

40 minutes

If a quart is ¼ of a gallon and a pint is 1/8 of a gallon, how much

of a quart is a pint?

Answer:

1/2

The clock in the bell tower rings every half hour. How many times

will it ring in one week?

Answer:

336

If a worker makes 45 boxes of labels in 2 ½ hours, then at the same rate, how many boxes of

labels can be made in 50 minutes?

Answer:

15

If 15 blips are equal to 7 bloops, 5 blumps are equal to 7 blatts, 4 bloops are equal to 10 blatts,

then how many blips are equal to 30 blumps?

Answer:

36

Which of these represents the greatest amount of time?

A. 3 yearsB. 35 monthsC. 158 weeksD. 1096 daysE. 26,200 hours

Answer:

158 weeks