Today:Today:

Fast 15Fast 15Warm-Up (5 Questions)Warm-Up (5 Questions)

VocabularyVocabularyProportionsProportions

Ratios/Rates/Unit RatesRatios/Rates/Unit RatesMake-Up TestsMake-Up Tests

November 29, 2012November 29, 2012

Warm-Up Questions

1. A student scored 96% on a test with 50 questions. How many questions were missed?

2. Ramon spent 40% of his savings on new shoes. He has saved $275. How much were the shoes?

3. In a survey, people were asked to name their favorite sport. 30 said Baseball, 42 football, and 8 soccer. What percent like baseball best?

4. Minimum wage in the CNMI has increased from $4.60 to $5.55 in the past two years. What is the percent increase?5. You are 15 years old and have lived 18% of your life. At what age will you die?

Ratios, Proportions, & Rates

A Proportion is an equation that two ratios are equal. To determine if ratios are equal, cross-multiply and check for equality. 2/5 and 6/15 are proportional ratios.

We have used the Percent Proportion to solve percent problems, but there are other problems which do not involve percents.

A Ratio is a comparison of two numbers by division. Ratios can be expressed by: 4:3, or 4 to 3, or 4/3.

A Rate is the comparison (ratio) of two different units of measure. Ex: miles per hour, gallons an hour, dollars a pound

Solving Proportions

Example 1: Ben runs 4 miles in 45 minutes. If he only has 30 minutes, how far can he run?

Set up a proportion and solve: 4 (miles) = x (miles) 45 (min.) 30 (min.)

Example 2: Jill can jump rope 420 times in 2.5 minutes. At this rate, how many can she do in 30 minutes? Set up a proportion and solve: 402(jumps) = x (jumps) 2.5 (min.) 30 (min.)

Proportions and Similar Figures.

You can use proportions to find dimensions of objects that are difficult to measure directly…

In the Figure below, ABC ~ (is similar) DFE. Find DE.

C

A

21 cm18 cm

15 cmB

x

D F10 cm

ESet up the Set up the proportion:proportion: 1515 = = 2121 10 10 xx

Example 1: Writing Ratios in Simplest Form

Write the ratio 15 bikes to 9 skateboards in simplest form.

159

53

The ratio of bikes to skateboards is , 5:3, or 5 to 3.

=

15 ÷ 39 ÷ 3

Write the ratio Write the ratio as a fraction.as a fraction.

= = Simplify.Simplify.

53

bikesskateboards

Example 1: Using Ratios

The ratio of the number of bones in a human’s ears to the number of bones in the skull is 3:11. There are 22 bones in the skull. How many bones are in the ears?

Write a ratio comparing bones in ears to bones in skull.

Write a proportion. Let x be the number of bones in ears.

Since x is divided by 22, multiply both sides of the equation by 22.

There are 6 bones in the ears.

The ratio of games lost to games won for a baseball team is 2:3. The team has won 18 games. How many games did the team lose?

The team lost 12 games.

Write a ratio comparing games lost to games won.

Write a proportion. Let x be the number of games lost.

Since x is divided by 18, multiply both sides of the equation by 18.

The ratio of left handed to right handed students at North High is 2:21. North High has an enrollment of 1058 students. How many left handers are at the school?

A A raterate is a ratio of two quantities with different units, such as is a ratio of two quantities with different units, such as

Rates are usually written as Rates are usually written as unit rates.unit rates. A A unit rateunit rate is a rate with a is a rate with a

second second quantityquantity of 1 unit, such as or 17 mi/gal. You can of 1 unit, such as or 17 mi/gal. You can

convert any rate to a unit rate.convert any rate to a unit rate.

Solving Rate Problems

Class Work: Problem 16; side Z should be 9.

Class Work:Class Work:

(1) Mindy has 72 candy bars. If the ratio of Mars to Snickers is 8:4, Find the number of each type of candy.

(2) Explain what this ratio tell us.

Rate Problems: Example 1

Cory earns $52.50 in 7 hours. Find the unit rate.

The unit rate is $7.50.

Write a proportion to find an equivalent ratio with a second quantity of 1.

Divide on the left side to find x.

Converting Rates of Different Units

A cheetah can run at a rate of 60 miles per hour in short bursts. What is this speed in feet per minute?

Step 1 Convert the speed to feet per hour.

The speed is 316,800 feet per hour.

Step 2 Convert the speed to feet per minute.

The speed is 5280 feet per minute.

To convert the first quantity in a rate, multiply by a conversion factor with that unit in the first quantity.

To convert the first quantity in a rate, multiply by a conversion factor with that unit in the second quantity.

Step 1 Convert the speed to feet per hour.

Example 2

A cyclist travels 56 miles in 4 hours. What is the cyclist’s speed in feet per second? Round your answer to the nearest tenth, and show that your answer is reasonable.

Change to miles in 1 hour.

The speed is 73,920 feet per hour.

(1) You have 150 different shirts. The ratio of blue to black shirts is 20 . How many black shirts do you have? 30