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Proportions Using Cross Products

Cross-multiply to solve proportions with one variable

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Proportions Using Cross Products

Have you ever been on a swim team? Do you know how to relate proportions to real - world situations? Take a look at this one.

Tony also works at the supermarket, but at school, he is on the swim team.

Tony swims 10 laps in 30 minutes. How long does it take him to swim 15 laps?

Solving this problem involves proportions and cross- products. You will know how to figure this out by the end of the Concept.

Guidance

Proportions are everywhere in the world around us. Proportions are comparisons that we make between different things. You will often hear the words “in proportion” meaning that there is a relationship between things.

What is the relationship of a proportion? That is exactly what this Concept is going to work on.

What is a proportion?

A proportion is two equal ratios. Remember that a ratio compares two quantities; well, a proportion compares two equal ratios.

While ratios can be written in three different ways, often you will see proportions written as equal fractions. Let’s look at a proportion.

412=13

Here we have two ratios. We have four compared to twelve and one compared to three. These two ratios form a proportion. Simplified, they equal the same thing. You can simplify four-twelfths and it equals one-third.

Sometimes one of the trickiest things is figuring out if two ratios form a proportion. In the example above, we can see the equals sign letting us know that the ratios form a proportion.

How can we tell if two ratios form a proportion?

There are two different ways to figure this out. The first has already been mentioned and that is to simplify the two ratios and see if they are equal.

14\begin{align*}\frac{1}{4}\end{align*} and 520\begin{align*}\frac{5}{20}\end{align*}

One-fourth is already in simplest form, we leave that one alone. If we simplify five-twentieths, we get one-fourth as an answer. One-fourth is equal to one-fourth, so these two ratios do form a proportion.

28\begin{align*}\frac{2}{8}\end{align*} and 36\begin{align*}\frac{3}{6}\end{align*}

If we simplify these two fractions we get two different answers. Two-eighths simplifies to one-fourth. Three-sixths simplifies to one-half. The two ratios are not equal. Therefore, they DO NOT form a proportion.

The second way of figuring out if two ratios form a proportion is to cross multiply or to use cross products.

What is a cross product?

A cross product is when you multiply the numerator of one ratio with the denominator of another. Essentially you multiply on the diagonals. If the product is the same, then the two ratios form a proportion.

23\begin{align*}\frac{2}{3}\end{align*} and 46\begin{align*}\frac{4}{6}\end{align*}

Let’s use cross products here.

2×63×412=12=12=12
The two ratios form a proportion.

We can use cross products to figure out whether or not two ratios form a proportion.

Try a few of these on your own. Use cross products to determine if the two ratios form a proportion. Write yes if they form a proportion and no if they do not.

Example A

25\begin{align*}\frac{2}{5}\end{align*} and 59\begin{align*}\frac{5}{9}\end{align*}

Solution: Not a proportion

Example B

36\begin{align*}\frac{3}{6}\end{align*} and 510\begin{align*}\frac{5}{10}\end{align*}

Solution: Yes

Example C

47\begin{align*}\frac{4}{7}\end{align*} and 1228\begin{align*}\frac{12}{28}\end{align*}

Solution: Not a proportion

Now back to Tony and the swim team. Here is the original problem once again.

Tony also works at the supermarket, but at school, he is on the swim team.

Tony swims 10 laps in 30 minutes. How long does it take him to swim 15 laps?

Our first step is to write two ratios.

1030\begin{align*}\frac{10}{30}\end{align*} This is our known information.

15x\begin{align*}\frac{15}{x}\end{align*} This is what we are trying to figure out.

Notice that we put the same unit in the numerator of both ratios and the same unit in the denominator of both units.

lapsmin utes=lapsmin utes

Now we can write a proportion.

1030=15x

Our answer is not obvious in this problem. Because of this, we need to use cross - products. We multiply 10 times x\begin{align*}x\end{align*} and get 10x\begin{align*}10x\end{align*} and then we multiply 15 times 30 and get 450.

10x=450

We can ask ourselves, “what times ten will give me 450?” or we can simplify the zeros and solve.

1x=45\begin{align*}1x = 45\end{align*} Here is our answer if we simplify the zeros. 1 times 45 equals 45.

Or we can think “10 times 45 equals 450.”

Vocabulary

Proportion
two equal ratios.
Ratio
a comparison of two quantities can be written in fraction form, with a colon or with the word “to”.
Cross Products
to multiply the diagonals of each ratio of a proportion.

Guided Practice

Here is one for you to try on your own.

Do these two ratios form a proportion? Why or why not?

69and34.5\begin{align*} \frac{6}{9} and \frac{3}{4.5}\end{align*}

To figure this out, we use cross - products.

6 x 4.5 = 27

9 x 3 = 27

The cross - products are equal.

These two ratios form a proportion.

Practice

Directions: Use cross products or simplifying to identify whether each pair of ratios form a proportion. If they do, write yes. If not, write no.

1. 12=612\begin{align*}\frac{1}{2} = \frac{6}{12}\end{align*}

2. 13=412\begin{align*}\frac{1}{3} = \frac{4}{12}\end{align*}

3. 14=315\begin{align*}\frac{1}{4} = \frac{3}{15}\end{align*}

4. 56=1012\begin{align*}\frac{5}{6} = \frac{10}{12}\end{align*}

5. 34=610\begin{align*}\frac{3}{4} = \frac{6}{10}\end{align*}

6. 25=615\begin{align*}\frac{2}{5} = \frac{6}{15}\end{align*}

7. 27=421\begin{align*}\frac{2}{7} = \frac{4}{21}\end{align*}

8. 47=1221\begin{align*}\frac{4}{7} = \frac{12}{21}\end{align*}

9. 78=1416\begin{align*}\frac{7}{8} = \frac{14}{16}\end{align*}

10. 2575=13\begin{align*}\frac{25}{75} = \frac{1}{3}\end{align*}

11. 1133=13\begin{align*}\frac{11}{33} = \frac{1}{3}\end{align*}

12. 1533=23\begin{align*}\frac{15}{33} = \frac{2}{3}\end{align*}

13. 1830=3660\begin{align*}\frac{18}{30} = \frac{36}{60}\end{align*}

14. 13=612\begin{align*}\frac{1}{3} = \frac{6}{12}\end{align*}

15. 85100=43.550\begin{align*}\frac{85}{100} = \frac{43.5}{50}\end{align*}

Vocabulary Language: English

Cross Products

Cross Products

To simplify a proportion using cross products, multiply the diagonals of each ratio.
Proportion

Proportion

A proportion is an equation that shows two equivalent ratios.