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Introduction

The Longest Books

Candice loves to read really long books. The first one that she selected was over 800 pages. The next one was 825 pages. She is not a very fast reader, but she enjoys all of the details that are put in really long books. Often, she walks through the bookstore looking at books that have tons of pages, then when she finds one that seems to be the right size, she takes it off the shelf to see if she is interested in the topic.

After 12 weeks, Candice had finished reading two books. She was very proud of herself given that she had read over 1600 pages in all. She had finished the first and second books in about the same length of time.

Given this, how long did it take Candice to read 1 book?

Because Candice read the two books in the same amount of time, you can use a proportion to solve this problem. Proportions are formed by equal ratios. Use this lesson to learn about proportions then you will be able figure out the time that it took Candice to read 1 book.

What You Will Learn

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

  • Recognize Proportions as a statement of equivalent ratios.
  • Solve proportions with the variable in the numerator using algebra.
  • Recognize equivalent proportions representing the same situation with the variable in the numerator.
  • Model and solve real-world problems using proportions.

Teaching Time

I. Recognize Proportions as a Statement of Equivalent Ratios

Think back to what you have already learned about ratios.

A ratio represents a comparison between two quantities. We can write ratios in fraction form, using a colon or using the word “to”.

We also learned that equivalent ratios are two ratios that are equal. The numbers in the ratios may not be the same, but the comparison of quantities is the same.

Equivalent ratios are directly related to proportions.

What is a proportion?

A proportion states that two ratios are equivalent. Here is an example of a proportion.

\frac{1}{2} = \frac{2}{4}

This proportion shows that the ratios \frac{1}{2} and \frac{2}{4} are equivalent. In other words, a proportion is made up of two equivalent ratios.

In the example above, we knew all of the parts of the two ratios that made up the proportion. Sometimes, we will know three of the numbers, but not all four of them. In this instance a variable is in the place of the missing number.

Look at this proportion.

\frac{1}{2} = \frac{n}{12}

Notice that the first term of the second ratio––its numerator––is a variable. Suppose we wanted to find the value of this variable. We could do that by using proportional reasoning.

Proportional reasoning is figuring out a missing value in a proportion by thinking about the relationship between the numbers in the two ratios.

Take a look at this example.

Example

Use proportional reasoning to solve for n: \ \frac{1}{2} = \frac{n}{12}.

To figure this out, we need to figure out a relationship between either numerators or denominators. The proportion does not show the relationship between the first terms in the ratios––the numerators of the fractions. However, we can determine the relationship between the second terms in the ratios––the denominators of the fractions.

We can ask ourselves: “what number, when multiplied by 2, results in 12?”

Since 2 \times 6 = 12, we can multiply both the numerator and the denominator of \frac{1}{2} by 6 to find the value of n.

\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12} = \frac{n}{12}

This shows that in this ratio the second term (the denominator) of the ratio is 12, the first term (the numerator) is 6.

The value of n is 6.

Good for you! Mental math is very helpful when using proportional reasoning. When you can figure out the relationship between numbers, then you can solve for the missing value of the variable.

Let’s look at another example.

Example

Use proportional reasoning to solve for x: \ \frac{15}{35} = \frac{x}{7}.

Which relationship can we use to figure out the variable? This proportion does not show the relationship between the first terms in the ratios––the numerators of the fractions. We need to find the relationship between the second terms in the ratios––the denominators of the fractions.

We can ask ourselves, “what number can we divide 35 by to get 7?”

Since 35 \div 5 = 7, we can divide both the numerator and the denominator of \frac{15}{35} by 5 to find the value of x.

\frac{15}{35} = \frac{15 \div 5}{35 \div 5} = \frac{3}{7} = \frac{x}{7}

This shows that in this ratio the second term (the denominator) of the ratio is 7, the first term (the numerator) is 3.

The value of x is 3.

Example

Use proportional reasoning to solve for z: \ \frac{z}{9} = \frac{32}{36}.

The proportion does not show the relationship between the first terms in the ratios––the numerators of the fractions. We need to find the relationship between the second terms in the ratios––the denominators of the fractions.

We can ask ourselves “what number, when multiplied by 9, results in 36?” The answer is 4.

9 \times 4 &= 36\\\frac{z}{9} &=  \frac{z \times 4}{9 \times 4} = \frac{32}{36}. From this, we can see that z \times 4 = 32.

We must ask ourselves, “what number, when multiplied by 4, results in 32?”

8 \times 4 = 32, so z = 8.

5H. Lesson Exercises

Use proportional reasoning to find each unknown.

  1. \frac{2}{3} = \frac{x}{6}
  2. \frac{12}{24} = \frac{24}{z}
  3. \frac{y}{5} = \frac{14}{20}

Take a few minutes to check your answers with a partner.

II. Solve Proportions with the Variable in the Numerator Using Algebra

Sometimes, it is difficult to use proportional reasoning alone to figure out the missing variable in a proportion. When this happens, we can use algebra to figure out the missing variable.

Using algebra involves thinking of multiplication or division in relationship to the values in the proportion. Sometimes, you will be multiplying as we did in the last section and sometimes, you will be dividing. Either way, we will be using our algebraic thinking to figure out the variable.

Example

Find the value of x: \ \frac{x}{10} = \frac{24}{30}

Here we can begin by looking at the relationship between the denominators. You can see that 10 \times 3 = 30.

Now we aren’t looking for the multiple of the first numerator, but we have the second numerator. We have to do the inverse operation of multiplication to figure out the value of the variable.

We have been given the value 24 as a numerator. The inverse operation for multiplication is division.

We can divide 24 by 3 to find the other numerator.

24 \div 3 = 8

The value of x is 8.

5I. Lesson Exercises

Use inverse operations to find the value of each unknown variable.

  1. \frac{2}{5} = \frac{6}{x}
  2. \frac{a}{9} = \frac{20}{36}
  3. \frac{4}{b} = \frac{24}{36}

Now check your answers with a partner.

III. Recognize Equivalent Proportions Representing the Same Situation with the Variable in the Numerator

Sometimes, we will have equivalent proportions. This means that two proportions are equal to each other. When this happens, you will have a total of four ratios, but each of them will be equivalent. Let’s look at an example.

Example

\frac{1}{2} &= \frac{2}{4}\\\frac{3}{6} &= \frac{6}{12}

Each of these ratios is the same quantity. Therefore, they are all equivalent.

They have been set up as two proportions, but they are all equivalent proportions.

Why would we have equivalent proportions?

Well, you can think of real-life situations that would represent equivalent proportions. Real-life situations that might have equivalent proportions are found in situations like cooking.

Think about this example.

Example

Tania and her mother are making cookies for a party. They aren’t sure if how many people are coming to the party. They aren’t sure if they should double, triple or quadruple the ingredients. The ratio of sugar to flour is \frac{1}{3}. How will this change if they double, triple or quadruple the recipe?

Now think about this. We can figure out that the ratios will all be equivalent because we are starting with the same measurement. We are beginning with \frac{1}{3}. To double it we multiply each value by 2. To triple we multiply by 3, to quadruple by four.

\frac{1}{3} &= \frac{2}{6}\\ \frac{3}{9} &= \frac{4}{12}

Notice that the relationships between the ratios in the second proportion are not multiples of each other or divisible by each other, but if viewed as a fraction reduce to the same number in simplest form.

IV. Model and Solve Real-World Problems Using Proportions

Sometimes, the best way to solve a real-world problem is to set up and solve a proportion. Remember that because a proportion uses ratios, they are helpful when comparing quantities.

Example

At the vet's office, the ratio of cats to dogs in the waiting room is 2 to 3. If there are 6 dogs in the waiting room, what is the number of cats in the waiting room?

One way to set up a proportion for this problem would be to write two equivalent ratios, comparing cats to dogs. We will need to use a variable to represent the number of cats in the second ratio.

The ratio of cats to dogs is 2 to 3. So, we can express this ratio as a fraction.

\frac{cats}{dogs} = \frac{2}{3}

Let's write a second ratio comparing cats to dogs in the waiting room. We know that there are a total of 6 dogs in the waiting room. We don't know the total number of cats. So, we can use the variable c, to represent the unknown number of cats and set up a second equivalent ratio.

\frac{cats}{dogs} = \frac{c}{6}

Since these two ratios are equivalent, we can put them together to form a proportion.

\frac{2}{3} = \frac{c}{6}.

To find the total number of cats in the vet's office, solve this proportion for c. The proportion does not show the relationship between the first terms in the ratios––the numerators of the fractions.

We need to find the relationship between the second terms in the ratios––the denominators of the fractions.

We can ask ourselves: what number, when multiplied by 3, results in 6?

Since 3 \times 2 = 6, we can multiply by 2 to find the value of c.

\frac{2}{3} = \frac{2 \times 2}{3 \times 2} = \frac{4}{6} = \frac{c}{6}

So, the value of c is 4. That means that there are 4 cats in the waiting room.

Real-Life Example Completed

The Longest Books

Here is the original problem once again. Reread it, underline the important information and figure out the proportion.

Candice loves to read really long books. The first one that she selected was over 800 pages. The next one was 825 pages. She is not a very fast reader, but she enjoys all of the details that are put in really long books. Often, she walks through the bookstore looking at books that have tons of pages, then when she finds one that seems to be the right size, she takes it off the shelf to see if she is interested in the topic.

After 12 weeks, Candice had finished reading two books. She was very proud of herself given that she had read over 1600 pages in all. She had finished the first and second books in about the same length of time.

Given this, how long did it take Candice to read 1 book?

First, let’s write a proportion to show that it took Candice 12 weeks to read two books.

\frac{12 \ weeks}{2 \ books}

Next, she read the two books in the same amount of time, so we can set up a proportion to show that these ratios are equal.

\frac{12 \ weeks}{2 \ books} = \frac{x} {1 \ book}

Now we can look at the relationship between the denominators and determine the missing numerator.

2 \div 2 = 1

We can do that to the numerator.

12 \div 2 = 6

It took Candice 6 weeks to read 1 book.

Vocabulary

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

Ratio
a comparison between two quantities. Ratios can be written in fraction form, using a colon or with the word “to”.
Equivalent Ratios
two ratios that are equal
Proportion
When two ratios are equal, they form a proportion.
Proportional Reasoning
deducing the relationship between the numerators or the denominators of a proportion. Anytime you have a proportion, there is some kind of relationship between the values.

Technology Integration

Khan Academy Ratio and Proportion

James Sousa, Introduction to Proportions

James Sousa, Proportions

Time to Practice

Directions: Look at each pair of ratios. Tell whether or not these ratios form a proportion.

1. \frac{1}{2} and \frac{4}{8}

2. \frac{3}{7} and \frac{6}{14}

3. \frac{5}{2} and \frac{10}{6}

4. \frac{3}{1} and \frac{9}{3}

5. \frac{2}{9} and \frac{1.5}{4.5}

6. \frac{4}{9} and \frac{8}{10}

7. \frac{1}{4} and \frac{5}{20}

8. \frac{3}{4} and \frac{9}{10}

Directions: Use proportional reasoning to find the value of the variable in each proportion.

9. \frac{1}{4} = \frac{a}{20}

10. \frac{15}{30} = \frac{x}{2}

11. \frac{2}{9} = \frac{n}{63}

12. \frac{z}{7} = \frac{12}{21}

13. \frac{3}{5} = \frac{t}{60}

14. \frac{k}{72} = \frac{5}{12}

Directions: Use proportional reasoning to solve each series of problems.

15. There are 12 quarts of water in a sink. Cliff wants to know how many gallons of water are in the sink. Using this unit conversion 1 gallon = 4 quarts, Cliff writes two ratios that compare gallons to quarts. Then he writes this proportion to represent g, the number of gallons of water in the sink.

\frac{g}{12} = \frac{1}{4}

His sister Nikki says he could also have used this proportion to represent g, the number of gallons of water in the sink.

\frac{g}{1} = \frac{4}{12}

Do the two proportions both have equal ratios? If not, which proportion could be used to find the actual number of gallons of water in the sink?

16. A machine that produces binder clips has been operating for 5 minutes. Mr. Wilson knows that the machine can produce 15 binder clips in 1 minute. So, Mr. Wilson writes two proportions that can be used to represent n, the number of binder clips that the machine produced in 5 minutes.

a) \frac{n}{5} = \frac{15}{1} and b) \frac{n}{15} = \frac{5}{1}

Do the two proportions have equal ratios? If not, which proportion––a or b––could be used to find n, the number of binder clips produced in 5 minutes?

Directions: The ratio of green tiles to orange tiles in a bag is 3 to 8. There are 48 orange tiles in the bag.

17. Set up a proportion that could be used to find g, the number of green tiles in the bag.

18. How many green tiles are in the bag?

Directions: Marty bought 2 pounds of ham for $11 at a deli.

19. Set up a proportion that could be used to find p, the number of pounds of ham Marty could have bought if he had spent $33.

20. How many pounds of ham could he have bought if he had spent $33?  

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Feb 22, 2012

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