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5.8: Problem-Solving Strategy: Use a Proportion; Use Unit Analysis

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Introduction

The Book Tally

After a whole year of reading, the students in Mrs. Henderson’s class are ready to tally up the number of books that were read. The students read a total of 544 books. There were many different types of books that were read. A few of the students took everyone’s lists and organized them into categories. Then they tallied the number of books in each category and composed a list.

History 12 books

Adventure 250 books

Romance 100 books

Mystery 120 books

Nature/Science 62 books

Then the students began comparing the number of books in each category. They compared the three largest categories with each other. They compared romance to adventure, mystery to romance, and mystery to adventure.

Given what you have learned about ratios and proportions answer each of the following questions.

What is the ratio of romance books to adventure books? Write your answer in simplest form too.

What is the ratio of mystery to romance books? Write your answer in simplest form.

What is the ratio of mystery to adventure? Write your answer in simplest form.

Do any of these ratios form a proportion? Why or why not?

Take a few minutes to write down your answers. Now go through the lesson and check your answers at the end.

What You Will Learn

By the end of this lesson, you will be able to demonstrate the following skills:

  • Read and understand given problem situations.
  • Develop and use the strategy: Write a Proportion.
  • Develop and use the strategy: Use Unit Analysis.
  • Plan and compare alternative approaches to solving problems.
  • Solve real-world problems involving rates, proportions and scale using selected strategies as part of a plan.

Teaching Time

I. Read and Understand Given Problem Situations

As you already know, proportions and proportional reasoning can help us solve many different kinds of problems. For example, they can help us solve problems involving ratios, rates, measurement conversions, scale drawings, and scale models.

However, there are a number of different strategies that we can use to solve these types of problems. Sometimes, it makes the most sense to set up a proportion and solve it by cross multiplying. Other times, it may be easier for us to use proportional reasoning or to find a unit rate.

In this lesson, we will explore several different strategies that can be used to solve some of these types of problems.

The first thing that you should do when tackling a new problem is to read the problem carefully. As you read the problem, you can look for key words to help you to select a strategy.

For rates, ratios and proportions, you want to look for words that compare one quantity to another. Whenever you have a comparison problem, ratios and proportions will be a possible way to solve the problem.

Another example that would use proportions for problem solving is when you see a map, scale model or a drawing.

Now let’s look at how we can use the strategy of writing a proportion.

II. Develop and Use the Strategy: Write a Proportion

As you already know, some types of problems can be solved by writing and solving a proportion. However, how you choose to solve the proportion may vary. Sometimes, it may be easier to use proportional reasoning. Other times, cross multiplying may be easier.

Let’s look at two different examples.

Example

On a map, Sonia measured the straight-line distance between Baltimore, Maryland and Washington D.C. to be 2 centimeters. The scale on the map shows that 1 centimeter = 28 kilometers. What is the actual straight-line distance between Baltimore and Washington D.C.?

This problem involves a map, which is a type of scale drawing. It makes sense to use proportions to solve it. The unit scale, 1 centimeter = 28 kilometers, can be represented as a ratio. We can also write a ratio that compares the scale distance, 2 centimeters, to the unknown actual distance, d.

\frac{centimeters}{kilometers} = \frac{1}{28} \qquad \quad \frac{centimeters}{kilometers} = \frac{2}{d}

These are equivalent ratios, so we can use them to write a proportion.

\frac{1}{28} = \frac{2}{d}

Consider which strategy to use. Should we solve for d by using proportional reasoning? Or should we cross multiply?

Either strategy will work, but look at the terms in the numerators. The relationship between those two terms is easy to see––we can multiply 1 by 2 to get 2. So, the computation will probably be simpler if we use proportional reasoning and multiply both terms of the first ratio by 2.

\frac{1}{28} = \frac{1 \times 2}{28 \times 2} = \frac{2}{56} = \frac{2}{d}

From the work above, we can see that when the first term is 2, the second term is 56. So, d = 56.

The actual distance between Baltimore and Washington D.C. is 56 kilometers.

Sometimes, it is easier to use the cross product property of proportions than to use proportional reasoning. This is especially true when the relationship between a pair of terms in a proportion is not immediately obvious.

Example

A baker uses 22 cups of flour to make 4 loaves of bread. How many cups of flour will he need to use to make 31 loaves of bread?

We can write a proportion to help us solve this problem. The first ratio can use the fact that it takes 21 cups of flour to make 4 loaves of bread. The second ratio can compare the unknown number of cups of flour needed, c, to the 31 loaves of bread the baker wants to make.

\frac{cups}{loaves} = \frac{22}{4} \qquad \quad \frac{cups}{loaves} = \frac{c}{31}

These are equivalent ratios, so we can use them to write a proportion.

\frac{22}{4} = \frac{c} {31}

Consider which strategy to use. Should we solve for c by using proportional reasoning? Or should we cross multiply?

The relationship between the terms in the denominators, 4 and 31, is not immediately obvious because 31 is not a multiple of 4. So, cross multiplying is probably easier.

\frac{22}{4} &= \frac{c} {31}\\4 \cdot c &= 22 \cdot 31\\4c &= 682\\\frac{4c}{4} &= \frac{682}{4}\\c &= 170.5 The baker will need 170.5, or 170 \frac{1}{2}, cups of flour to bake 31 loaves of bread.

Sometimes, we may need to convert from one unit of measurement to another. The next section will review how we can write proportions to help us. It will also show us a second strategy which can help us convert units.

III. Develop and Use the Strategy: Use Unit Analysis

Proportions can help us convert from one unit of measure to another. For example, suppose you needed to convert from liters to milliliters.

Example

A pitcher holds 4 liters of water. Determine how many milliliters of water the pitcher holds. Use the unit conversion: 1 liter = 1000 milliliters

First, let's set up a proportion to solve this problem.

The first ratio can use the unit conversion and compare liters to milliliters. The second ratio can compare 4 liters to the unknown number of milliliters, n.

\frac{liters}{milliliters} = \frac{1}{1000} \qquad \quad \frac{liters}{milliliters} = \frac{4}{n}

These are equivalent ratios, so we can use them to write a proportion.

\frac{1}{1000} = \frac{4}{n}

Consider which strategy to use. Should we solve for d by using proportional reasoning? Or should we cross multiply?

The relationship between the terms in the numerators is easy to see––we can multiply 1 by 4 to get 4. So, the computation will probably be simpler if we use proportional reasoning and multiply both terms of the first ratio by 4.

\frac{1}{1000} = \frac{1 \times 4}{1000 \times 4} = \frac{4}{4000} = \frac{4}{n}

From the work above, we can see that when the first term is 4, the second term is 4000. So, n = 4000.

The pitcher holds 4000 milliliters of water.

Another strategy for solving a problem like the one with the pitcher is to use unit analysis. In unit analysis, we write ratios as fractions, just as we do when we write a proportion. However, in unit analysis, we do not want the terms in the fractions to be consistent. Instead, the units in the fractions are written so that certain units cancel one another out.

This will be easier to understand if we consider an example. Let’s go back to the last example with the liters and the pitcher. We can use unit analysis to solve it.

Example

The problem requires us to convert 4 liters to milliliters. Our answer should be in milliliters, so the number of milliliters is unknown.

The measure we are given is 4 liters.

We know that 1 liter (L) = 1000 milliliters (mL). This can be expressed as either \frac{1L}{1000mL} or \frac{1000mL}{1L}. Each of these is a possible conversion factor by which we might multiply 4 liters.

We should start by writing 4 liters as a fraction over 1. We can do this because 4L = \frac{4L}{1}.

\frac{4L}{1}

We want our answer to be in milliliters, not in liters. So, we want the liters to cancel each other out. Since liters is in the numerator of the fraction above, we should make sure that liters is in the denominator of the conversion factor we use, like this:

\frac{4L}{1} \times \frac{1000 mL}{1L}

Since liters appears in the numerator of one factor and in the denominator of another factor, we can cancel them out, like this:

\frac{4 \bcancel{L}}{1} \times \frac{1000mL}{1 \bcancel{L}}

Now we can multiply what is left as we would multiply any fractions.

\frac{4}{1} \times \frac{1000mL}{1} = \frac{4 \times 1000 mL}{1 \times 1} = \frac{4000mL}{1} = 4000 mL

The pitcher holds 4000 milliliters of water.

IV. Plan and Compare Alternative Approaches to Solving Problems

We can also use other approaches to solving problems. Combining strategies can help us with some real-world problems. For example, you may asked to compare two ratios to determine which of two purchases is the better buy. One way to solve this type of problem is to find the unit rate for each purchase and then compare them. Let's review how to do that.

Example

Arnaldo needs to buy olive oil. He could buy a 15-ounce bottle of Brand A olive oil for $3, or he could buy a 20-ounce bottle of Brand B olive oil for $5. Which is the better buy?

One way to solve this problem is to find the unit price for each bottle.

Find the unit price for the 15-ounce bottle. Remember, you can find the unit price by dividing the first term by the second term.

&\$ 3 \ \text{for} \ 15 \ oz = \frac{\$3}{15oz} && \overset{ \ \ \$0.20}{15 \overline{ ) {\$3.00 \;}}}\\&&& \quad \underline{-30\;\;\;\;}\\&&& \qquad \ \ \ \ 0\\&&& \qquad  \ \ \underline{-0\;\;}\\&&& \qquad \quad \ \  0

Find the unit price for the 20-ounce bottle.

&\$ 5 \ \text{for} \ 20 \ oz = \frac{\$5}{20oz} && \overset{ \ \ \$0.25}{20 \overline{ ) {\$5.00 \;}}}\\&&& \quad \ \underline{-40\;\;\;\;}\\&&& \qquad \ 100\\&&& \quad  \ \ \underline{-100\;\;}\\&&& \qquad \quad \ 0

Since $0.20 < $0.25, the 15-ounce bottle of Brand A olive oil has the cheaper unit price and is the better buy.

Finding the unit rate is not the only strategy we could have used to solve a problem like the one in the last example. Instead of determining the unit rate, we could have imagined buying several bottles of each brand until we had the same number of ounces of Brand A oil as Brand B oil. Then we could have compared those costs. This strategy is known as the scaling strategy. Take a look at the next problem.

Example

Find a common number of ounces for both brands.

The first few multiples of 15 are: 15, 30, 45, 60, and 75.

The first few multiples of 20 are: 20, 40, 60, 80, and 100.

The least common multiple of 15 and 20 is 60. So, we can find the cost of buying 60 ounces of each brand of oil.

A 15-ounce bottle of Brand A oil costs $3.

15 \times 4 = 60, so \frac{ounces}{price} = \frac{15}{3} = \frac{15 \times 4}{3 \times 4} = \frac{60}{12}.

The cost of 60 ounces of Brand A oil (four 15-ounce bottles of oil) is $12.

A 20-ounce bottle of Brand B oil costs $5.

20 \times 3 = 60, so \frac{ounces}{price} = \frac{20}{5} = \frac{20 \times 3}{5 \times 3} = \frac{60}{15}.

The cost of 60 ounces of Brand B oil (three 20-ounce bottles of oil) is $15.

Since $12 < $15, it would cost less to buy 60 ounces of Brand A olive oil than to buy 60 ounces of Brand B olive oil. So, the 15-ounce bottle of Brand A olive oil is the better buy.

Now let’s return to our introductory problem and apply all that we have learned to solve it.

Real Life Example Completed

The Book Tally

Here is the original problem again. It is time to check your answers.

After a whole year of reading, the students in Mrs. Henderson’s class are ready to tally up the number of books that were read. The students read a total of 544 books. There were many different types of books that were read. A few of the students took everyone’s lists and organized them into categories. Then they tallied the number of books in each category and composed a list.

History 12 books

Adventure 250 books

Romance 100 books

Mystery 120 books

Nature/Science 62 books

Then the students began comparing the number of books in each category. They compared the three largest categories with each other. They compared romance to adventure, mystery to romance, and mystery to adventure.

Given what you have learned about ratios and proportions answer each of the following questions.

What is the ratio of romance books to adventure books? Write your answer in simplest form too.

\frac{100}{250} = \frac{2}{5}

What is the ratio of mystery to romance books? Write your answer in simplest form.

\frac{120}{100} = \frac{6}{5}

What is the ratio of mystery to adventure? Write your answer in simplest form.

\frac{120}{250} = \frac{12}{25}

Do any of these ratios form a proportion? Why or why not?

None of these ratios form a proportion because none of them are equal.

Take a few minutes to go over your answers with a friend.

Vocabulary

Here are the vocabulary words that are found in this lesson.

Unit Analysis
comparing the number of units by using fractions and canceling out common values.
Scaling Strategy
looking at a common number of units to figure out a best buy. You may compare the amount in two different objects to a common unit and then figure out the best value based on the comparison.

Technology Integration

James Sousa, Example of Applications Using Proportions

Time to Practice

Directions: Use what you have learned to solve each problem. Consider more than one strategy for solving each problem. Then choose the strategy you think will work best and use it to solve the problem.

1. A jar contains only pennies and nickels. The ratio of pennies to nickels in the jar is 2 to 7. If there are 14 nickels in the jar, how many pennies are in the jar?

2. Anya charges $40 for 5 hours of babysitting. Lionel charges $14 for 2 hours of babysitting. Which babysitter charges the cheapest rate?

3. On a map, Derek measured the straight-line distance between Toronto, Canada and Niagara Falls, New York to be 2 inches. The scale on the map shows that \frac{1}{2} \ inch = 11 \ miles. What is the actual straight-line distance between Toronto and Niagara Falls?

4. A desk is 120 centimeters long. What is the length of the desk in meters? Use this unit conversion: 1 meter = 100 centimeters.

5. On a field trip, the ratio of teachers to students is 1 : 25. If there are 5 teachers on the field trip, how many students are on the trip?

6. Kara bought 5 pounds of Brand X roast beef for $43. Cameron bought 3 pounds of Brand Y roast beef for $27. Which brand of roast beef is the better buy?

7. Fred needs to buy vanilla extract to bake a cake. He could buy a 4-ounce bottle of vanilla extract for $8, or a 6-ounce bottle of vanilla extra for $15. Which bottle is the better buy?

8. A rope is 3 yards long. How many inches long is the rope? Use these unit conversions: 1 yard = 3 feet and 1 foot = 12 inches.

9. At the farmer's market, Maureen can buy 6 ears of corn for $3. At that price, how much would it cost to buy 9 ears of corn?

10. James bought a 128-ounce bottle of apple juice. How many pints of apple juice did James buy? Use these unit conversions: 1 cup = 8 fluid ounces and 1 pint = 2 cups.

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Feb 22, 2012

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Sep 23, 2014
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CK.MAT.ENG.SE.1.Math-Grade-7.5.8

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