What you should learn

To solve proportions

NCTM Curriculm Standards 2, 4, 6 - 10

 

In doing this the teacher wants to make sure that the following words are incorporated into the introductory lesson:

Ratio

Proportion

Extremes

Means

Scale

Rate

Introduction: How would you like to eat chocolate for a living? That's exactly what Carl Wong does every day as associiate director of product development for a chocolate company. His job is to create new candy bars as well as taste test existing candy barsin order to improve them.

The chocolate company keeps their candy recipes confidential, but the basic ingredients found in a batch of chocolate bars are sugar, cocoa beans, milk, and flavorings. The ingredients for a batch of candy are shown in the table below.

A ratio is a comparison of two numbers by division. The ratio of x to y can be expressed in the following ways.

Ratios are often expressed as fractions in simplest form. A ratio that is equivalent to a whole number is written with a denominator of 1.

The table above shows that for every 10 parts of sugar in a batch of chocolate bars, there are 5 parts cocoa beans. The ratio of sugar to cocoa beans is 10/5. Suppose the company uses 30 pounds of sugar and 15 pounds of cocoa beans in a batch of chocolate bars. This ratio is 30/15. Is this ratio different than the first ratio of 10/5? When simplified, both ratios are equivalent to 2/1.

An equation stating that two ratios are equal is called a proportion. So, (10/5) = (30/15) is a proportion.

One way to determine if two ratios form a proportion is to check their cross products. In the proportion above, the cross products are 10 * 15 and 5 * 30. In this proportion, 10 and 15 are called the extremes, and 5 and 30 are called the means.

The cross products of a proportion are equal.

Means-Extremes Property of Proportions: In a proportion, the product of the extremes is equal to the product of the means. If (a/b) = (c/d), then ad = bc

 

Exercise 1: Use cross products to determine whether each pair of ratios forms a proportion.

a. 2/3, 12/18

2 * 18 = 3 * 12

36 = 36

So, 2/3 = 12/18 and this is a proportion.

b. 2.5/6, 3.4/5.2

 

 

You can write proportions that involve a variable and then use cross products to solve the proportion.

Exercise 2: Refer to the application at the beginning of the lesson. Suppose the chocolate company makes a batch of chocolate with 75 pounds of cocoa beans. How many gallons of milk will they use?

You know that the ratio of cocoa beans to milk is 5:4. Let m represent the gallons of milk.

In a batch of chocolate with 75 pounds of cocoa beans, the company needs to use 60 gallons of milk.

 

 

A ratio called a scale is used when making a model to represent something that is too large or small to be conveniently drawn at actual size. The scale compares the size of the model to the actual size of the object being modeled.

Exercise 3: In a recent movie about dinosaurs, the dinosaurs were scale models and so was the sport utility vehicle that the T-Rex overturned. The vehicle was made to the scale of 1 inch to 8 inches. The actual vehicle is about 14 feet long. What was the length of the model sport utility vehicle?

First, change 14 feet to 168 inches. Then let l represent the length of the model vehicle.

1/8 = l/168

168 = 8l

21 = l

The model sport utility vehicle was 21 inches long.

 

 

You can solve proportions by using a calculator.

Exercise 4: Solve each proportion.

a. 5/4.25 = 11.32/m

b. x/3 = (x + 5)/15

 

 

The ratio of two measurements having different units of measure is called a rate. For example, 30 miles per gallon is a rate. Proportions are often used to solve problems involving rates.

Exercise 5: In the first 30 minutes of the opening day of the Texas State Fair, 1252 people entered the gates. If this atttendence rate continued, how many people visited the fair during the operating hours of 8AM and 12 Midnight the first day?

Explore

Let p represent the number of people attending the fair on opening day.

Plan

Write a proportion for the problem.

1252/0.5 = p/16

Solve

1252/0.5 = p/16

1252(16) = 0.5p

40,064 = p

If the attendance rate continued, 40,064 people visited the fair on opening day.

Examine

Use estimation to check your answer. About 1250 people entered the fair every 30 minutes. This means that aobut 2500 people entered every hour. The fair was open for 16 hours each day. Therefore, about 16 * 2.5 thousand or 40,000 people entered, and the answer is reasonable.

Closing Activity: Check for understanding by using this as a quick review before class is over. It should take about the last five to ten minutes. I would use it for my students as their 'ticket out the door'. Click Here.

Homework: The homework to be assigned for tonight would be: 15 - 37 odd, 39 - 46

Alternative Homework: Enriched: 14 - 34 even, 35 - 46

Extra Practice: Students book page 764 Lesson 4-1

Extra Practice Worksheet: Click Here.

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