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Problem Solving Plan, Proportions

Write proportions with one variable to solve word problems

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Practice Problem Solving Plan, Proportions
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Problem Solving Plan, Proportions

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

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.

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

Guidance

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 situations where you can use a proportion.

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\begin{align*}d\end{align*}.

centimeterskilometers=128centimeterskilometers=2d\begin{align*}\frac{centimeters}{kilometers} = \frac{1}{28} \qquad \quad \frac{centimeters}{kilometers} = \frac{2}{d}\end{align*}

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

128=2d\begin{align*}\frac{1}{28} = \frac{2}{d}\end{align*}

Consider which strategy to use. Should we solve for d\begin{align*}d\end{align*} 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.

128=1×228×2=256=2d\begin{align*}\frac{1}{28} = \frac{1 \times 2}{28 \times 2} = \frac{2}{56} = \frac{2}{d}\end{align*}

From the work above, we can see that when the first term is 2, the second term is 56. So, d=56\begin{align*}d = 56\end{align*}.

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.

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\begin{align*}c\end{align*}, to the 31 loaves of bread the baker wants to make.

cupsloaves=224cupsloaves=c31\begin{align*}\frac{cups}{loaves} = \frac{22}{4} \qquad \quad \frac{cups}{loaves} = \frac{c}{31}\end{align*}

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

224=c31\begin{align*}\frac{22}{4} = \frac{c} {31}\end{align*}

Consider which strategy to use. Should we solve for c\begin{align*}c\end{align*} 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.

2244c4c4c4c=c31=2231=682=6824=170.5\begin{align*}\frac{22}{4} &= \frac{c} {31}\\ 4 \cdot c &= 22 \cdot 31\\ 4c &= 682\\ \frac{4c}{4} &= \frac{682}{4}\\ c &= 170.5\end{align*}

The baker will need 170.5, or 17012\begin{align*}170 \frac{1}{2}\end{align*}, cups of flour to bake 31 loaves of bread.

Now it is time for you to try a few on your own. Use the information from the baker above.

Example A

How many cups for 6 loaves?

Solution: 33 cups

Example B

How many loaves for 55 cups?

Solution: 10 loaves

Example C

If the baker made half as many loaves to start with, how many cups would he have needed?

Solution: 11 cups

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

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.

100250=25\begin{align*}\frac{100}{250} = \frac{2}{5}\end{align*}

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

120100=65\begin{align*}\frac{120}{100} = \frac{6}{5}\end{align*}

120250=1225\begin{align*}\frac{120}{250} = \frac{12}{25}\end{align*}

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

Proportion
two equal ratios.
Ratio
a comparison between quantities.

Guided Practice

Here is one for you to try on your own.

Use a proportion to solve the following problem.

If a person can run 3 miles in 18 minutes, how long will it take the same person to run 21 miles if it is at the same rate?

In this problem, we are comparing miles and time. That is our ratio. Let's set it up.

milestime=milestime\begin{align*}\frac{miles}{time} = \frac{miles}{time}\end{align*}

Next we fill in the given information.

318=21x\begin{align*}\frac{3}{18} = \frac{21}{x}\end{align*}

Now we cross multiply and solve.

3x=378\begin{align*}3x = 378\end{align*}

x=126\begin{align*}x = 126\end{align*}

The person would run 21 miles in 126 minutes a little over two hours.

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 12 inch=11 miles\begin{align*}\frac{1}{2} \ inch = 11 \ miles\end{align*}. 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. If two inches on a map are equal to three miles, how many miles are represented by four inches?

8. If eight inches on a map are equal to ten miles, how many miles are 16 inches equal to?

9. Casey drew a design for bedroom. On the picture, she used one inch to represent five feet. If her bedroom wall is ten feet long, how many inches will Casey draw on her diagram to represent this measurement?

10. If two inches are equal to twelve feet, how many inches would be equal to 36 feet?

11. If four inches are equal to sixteen feet, how many feet are two inches equal to?

12. The carpenter chose a scale of 6” for every twelve feet. Given this measurement, how many feet would be represented by 3”?

13. If 9 inches are equal to 27 feet, how many feet are equal to three inches?

14. If four inches are equal to 8 feet, how many feet are equal to two inches?

15. If six inches are equal to ten feet, how many inches are five feet equal to?

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