Ratios Rates 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 6/2 2/3 and 6/9 12/16 and 15/20

Answer True or False on each of the following Pair of ratios if they form a proportion.

True or False: ― = ―

Answer: False

True or False: ― = ―

Answer: False

True or False: ― = ―

Answer: True

True or False: ― = ―

Answer: True

True or False: ― = ―

Answer: False

True or False: ― = ―

Answer: True

Solve the Following

Solve: ― = ― m 10 3

Answer: m = 33⅓

Solve: ― = ― v

Answer: v = 60

Solve: ― = ― 10 k 8484

Answer: k = 5

Solve: ― = ― 2x2x 7979

Answer: x = 2 ⁴⁄₇

Solve: ― = ― 3x3x 7 10

Answer: x = 4 ²⁄₇

Ratio & Proportions Parts = a & b Whole = (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=14 Lets 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=14 Or lets try… 5/2 = k/14 (5/2)·14 = k 35 = k Answer: 35 kids

Kids:adults=5:2 adults=14 Or lets try… Find the factor: 5x kids 2x adults 2x = 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 kids Lets 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=26 5x-3x=26 2x=26 x = 13 (factor) Therefore: 3x (kids) 3·13 =39 Answer: 39 kids

Kids:adults=3:5 k kids k+26 adults 3/5 = k/(k+26) 5k = 3(k+26) 5k = 3k k = 78 k = 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 = 30 10x = 30 x=3 3·3 3·7 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 girls 70 ∙ ³⁄₁₄ = 15 boys Also: find the factor 11x + 3x = 70 14x=70 x=5 11·5=55 girls 3·5=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=6 X=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 1 st place prize is $20,000 more than the 3 rd place prize. How much is the second place prize?

Answer: 1st=5x, 2nd=2x, 3rd=1x 5x-1x=20000 X= nd place = 2x=2∙5000=$10,000

More Proportion Word 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 = x

If you can buy one can of pineapple chunks for $2, then how many can you buy with $10?

Answer: ― = ― x = 5 cans 1212 x 10

One jar of crushed ginger cost $2. How many jars can you buy for $4?

Answer: ― = ― x = 2 jars 1212 x4x4

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 = $2512 or 4/2 = M/1256 2M = 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. 20x 12 3

Answer: ― = ― x = 5 x

Each pair of figures are similar. Find the missing side. 4 x 16 8

Answer: ― = ― x = 2 x4x4 8 16

Each pair of figures are similar. Find the missing side. 14 x1 2

Answer: ― = ― x = 7 x

Similar Figures Word 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 x2x2

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 x6x

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 = 7 x = 14 miles ½2½2 3.5 x

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 × 10 8 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.

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 years B. 35 months C. 158 weeks D days E. 26,200 hours

Answer: 158 weeks