On the first day of school, the students in the seventh grade Language arts class started a reading contest to see how many books they could read during the year. Kayla and Torrey have been participating in the challenge since day one. By October, eight weeks into school, Kayla had already finished 6 books. Torrey said she was reading 3 books every four weeks. The girls concluded that at the present point in time, they had both read the same number of books. Are they correct?

In this concept, you will learn how to work with equivalent ratios.

### Working with Equivalent Ratios

A **ratio** is the relationship, or comparison, of a part to the whole.

A ratio can be expressed:

in words, 3 to 4, or 3 of 4

with a colon, 3:4

as a fraction,\begin{align*}\frac{3}{4}\end{align*}

The easiest way to work with ratios is by thinking of them as fractions.

For example, think about the three lights on a traffic signal.

One of three is red, \begin{align*}\frac{1}{3}\end{align*}

2:3 are primary colors,

\begin{align*}\frac{3}{3}\end{align*}are colors of the rainbow

**Equivalent means equal.**

**Equivalent ratios** show the same, or equal, relationship between two quantities.

Using the traffic lights again, suppose there are two signals on one street and one on another.

The first street has 2 of 6 lights that are red. The second has 1 of 3.

The ratio of red lights to the total lights per signal is still the same.

Writing ratios in fractional form and simplifying can determine whether or not they are equivalent.

Here is an example:

Determine if the two ratios are equivalent.

\begin{align*}\frac{2}{3}\end{align*} and \begin{align*}\frac{10}{15}\end{align*}

First, recognize that \begin{align*}\frac{2}{3}\end{align*} is in its simplest form.

Next, simplify \begin{align*}\frac{10}{15}\end{align*}.

\begin{align*}\frac{10\div 5}{15\div 5}=\frac{2}{3}\end{align*}

Then, compare the two fractions, or ratios.

\begin{align*}\frac{2}{3}\end{align*}=\begin{align*}\frac{2}{3}\end{align*}

The answer is that the two ratios are equal, or equivalent.

Another way to check is to change \begin{align*}\frac{2}{3}\end{align*}into a fraction with 15 as the denominator.

First, write an equation.

\begin{align*}\frac{2}{3}\end{align*}=\begin{align*}\frac{x}{15}\end{align*}

Next, recognize that \begin{align*}3\times 5=15\end{align*} and multiply both the numerator and denominator times 3. Remember that whatever is done to one term must also be done to the other.

\begin{align*}\frac{2\times 5}{3\times 5}=\frac{10}{15}\end{align*}

Then, compare the ratios.

\begin{align*}\frac{10}{15}\end{align*}=\begin{align*}\frac{10}{15}\end{align*}

The answer is that the two ratios are equivalent.

The last way to determine whether or not two ratios are equivalent is to cross multiply.

First, write an equation that assumes the fractions are equal.

\begin{align*}\frac{2}{3}=\frac{10}{15}\end{align*}

Next, multiply the numerator of one fraction times the denominator of the other and vice versa.

\begin{align*}2\times 15=3\times 10\end{align*}

Then, compare the two products.

30=30

The products are equal.

The answer is that the ratios are equivalent.

### Examples

#### Example 1

Earlier, you were given a problem about Kayla and Torrey and the reading challenge.

Kayla had read 6 books in 8 weeks and Torrey had read 3 books every four weeks. Had they read the same number of books? Are the ratios equivalent?

First, write the ratios as fractions.

Kayla \begin{align*}\frac{6 \ \rm{books}}{8 \ \rm{weeks}}\end{align*} or \begin{align*}\frac{6}{8}\end{align*}

Torrey \begin{align*}\frac{3 \ \rm{books}}{4 \ \rm{weeks}}\end{align*} or \begin{align*}\frac{3}{4}\end{align*}

Next, determine a method, cross multiplication, and write an equation.

\begin{align*}\frac{6}{8}=\frac{3}{4}\end{align*}

Then cross multiply and compare.

24 = 24

The answer is that the girls have read the same number of books. The ratios are equivalent.

#### Example 2

Determine if 7:6 and 13:12 are equivalent ratios.

First, rewrite the ratios as fractions.

\begin{align*}\frac{7}{6}\end{align*} and \begin{align*}\frac{13}{12}\end{align*}.

Next, recognize that 6 must be multiplied by 2 in order to equalize the denominators and perform the same operation to both terms.

\begin{align*}\frac{7}{6} = \frac{7 \times 2}{6 \times 2} = \frac{14}{12}\end{align*}

Then, compare the ratios.

** \begin{align*}\frac{13}{12}\neq \frac{14}{12}\end{align*}**

The answer is that the two ratios are not equivalent.

#### Example 3

Determine if the ratios are equivalent.

\begin{align*}\frac{3}{4}\end{align*} and \begin{align*}\frac{9}{12}\end{align*}

First, since a method is not specified, choose a method one. Determine if it is easiest to work with fourths, twelfths, or cross multiply. In this problem, it is easy to cross multiply.

Next, write an equation assuming the ratios are equal.

\begin{align*}\frac{3}{4}=\frac{9}{12}\end{align*}

Then, cross multiply.

\begin{align*}3\times 12=9\times 4\end{align*}

Next, compare.

36 = 36

The answer is that the ratios are equivalent.

#### Example 4

Change both fractions to 30ths to determine whether or not the ratios \begin{align*}\frac{5}{6}\end{align*} and \begin{align*}\frac{20}{30}\end{align*} are equivalent.

First, write an equation.

\begin{align*}\frac{5}{6}=\frac{x}{30}\end{align*}

Next, recognize that 6 must be multiplied times 5 to get 30, and multiply both terms times 5.

\begin{align*}\frac{5\times 5}{6\times 5}=\frac{25}{30}\end{align*}

Then, compare the ratios expressed as 30ths.

\begin{align*}\frac{25}{30}\neq \frac{20}{30}\end{align*}

The answer is that the ratios are not equivalent.

#### Example 5

Are the ratios 4 to 5 and 8 to 10 equivalent?

First, write the ratios as fractions.

\begin{align*}\frac{4}{5}\end{align*} and \begin{align*}\frac{8}{10}\end{align*}

Next, determine a method. In this case, cross multiplication.

\begin{align*}4\times 10=5\times 8\end{align*}

Then, compare.

40 = 40

The answer is that the ratios are equivalent.

### Review

Determine whether each of the following ratio pairs is equal. Write yes if they are equal and no if they are not equal.

- \begin{align*}\frac{1}{2} \text{ and } \frac{6}{12}\end{align*}
- \begin{align*}\frac{3}{8} \text{ and } \frac{1}{4}\end{align*}
- \begin{align*}\frac{6}{7} \text{ and } \frac{2}{3}\end{align*}
- \begin{align*}\frac{6}{7} \text{ and } \frac{12}{14}\end{align*}
- \begin{align*}\frac{2}{3} \text{ and } \frac{10}{15}\end{align*}
- \begin{align*}\frac{17}{21} \text{ and } \frac{6}{7}\end{align*}
- \begin{align*}\frac{24}{48} \text{ and } \frac{12}{24}\end{align*}
- \begin{align*}\frac{16}{18} \text{ and } \frac{32}{38}\end{align*}
- \begin{align*}\frac{9}{45} \text{ and } \frac{1}{9}\end{align*}
- \begin{align*}\frac{4}{6} \text{ and } \frac{44}{66}\end{align*}
- \begin{align*}\frac{6}{9} \text{ and } \frac{4}{6}\end{align*}
- \begin{align*}\frac{14}{16} \text{ and } \frac{20}{24}\end{align*}
- \begin{align*}\frac{12}{16} \text{ and } \frac{24}{32}\end{align*}
- \begin{align*}\frac{24}{48} \text{ and } \frac{1}{2}\end{align*}
- \begin{align*}\frac{84}{96} \text{ and } \frac{3}{4}\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 5.1.