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Proportions with Variable in the Numerator

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Proportions with Variable in the Numerator

Do you like to read short books or long 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. Thinking back, Candice thought about her reading. 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 Concept to learn about proportions then you will be able figure out the time that it took Candice to read one big book.

Guidance

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.

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.

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.

\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.

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 isn’t crystal clear, but they are still equivalent.

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

Example A

\frac{2}{5} = \frac{6}{x}

Solution: x = 15

Example B

\frac{a}{9} = \frac{20}{36}

Solution: a = 5

Example C

\frac{4}{b} = \frac{24}{36}

Solution: b = 6

Here is the original problem once again.

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. Thinking back, Candice thought about her reading. 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 one 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 one book.

Vocabulary

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 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.

Guided Practice

Here is one for you to try on your own.

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, set up a proportion that could be used to find c, the number of cats in the waiting room.

Answer

One way to set up a proportion for this problem would be to write two equivalent ratios, each comparing cats to dogs. Then we can solve it for the number of cats.

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.

Video Review

Here is a video for review.

- This James Sousa video is about solving proportions.

Practice

Directions : Solve the following proportions.

1. \frac{1}{2} = \frac{x}{8}

2. \frac{1}{2} = \frac{5}{x}

3. \frac{1}{3} = \frac{4}{x}

4. \frac{2}{3} = \frac{x}{6}

5. \frac{1}{2} = \frac{x}{16}

6. \frac{5}{6} = \frac{x}{12}

7. \frac{14}{16} = \frac{x}{8}

8. \frac{1}{2} = \frac{x}{18}

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

10. \frac{1}{4} = \frac{x}{24}

11. \frac{3}{6} = \frac{x}{18}

12. \frac{4}{5} = \frac{x}{2.5}

13. \frac{3}{7} = \frac{x}{21}

14. \frac{3}{8} = \frac{x}{48}

15. \frac{1}{8} = \frac{x}{18}

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