The students are still actively reading books. Take a look at this situation.

Manuel was reading all of the medieval books on knights. Well after he finished reading the series, he loaned it to his friend Rafael. Rafael is enjoying the series as much as Manuel did.

Rafael had finished 9 of the 12 books. It took Manuel read three-fourths of the books in the same time. Are Rafael and Manuel reading at the same rate?

**To figure this out, you will need to write two proportions and figure out if they are equal. This Concept will teach you how to do this.**

### Guidance

Remember 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.**

\begin{align*}\frac{1}{2} = \frac{2}{4}\end{align*}

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

In the situation 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 four of them. When this happens, we have to use a variable and solve for the missing number.

Look at this proportion.

\begin{align*}\frac{1}{2} = \frac{n}{12}\end{align*}

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 when we figure out a missing value in a proportion by thinking about the relationship between the numbers in the two ratios.**

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

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

\begin{align*}\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12} = \frac{n}{12}\end{align*}

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

**The value of \begin{align*}n\end{align*} is 6.**

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

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

**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 to get 7?”**

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

\begin{align*}\frac{15}{35} = \frac{15 \div 5}{35 \div 5} = \frac{3}{7} = \frac{x}{7}\end{align*}

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

**The value of \begin{align*}x\end{align*} is 3.**

Use proportional reasoning to find the value of each unknown variable.

#### Example A

\begin{align*}\frac{2}{3} = \frac{x}{6}\end{align*}

**Solution: \begin{align*}x = 4\end{align*}**

#### Example B

\begin{align*}\frac{12}{24} = \frac{24}{z}\end{align*}

**Solution: \begin{align*}z = 48\end{align*}**

#### Example C

\begin{align*}\frac{y}{5} = \frac{14}{20}\end{align*}

**Solution: \begin{align*}3.5\end{align*}**

Here is the original problem once again.

Manuel was reading all of the medieval books on knights. Well after he finished reading the series, he loaned it to his friend Rafael. Rafael is enjoying the series as much as Manuel did.

Rafael had finished 9 of the 12 books. It took Manuel read three-fourths of the books in the same time. Are Rafael and Manuel reading at the same rate?

To determine if they are reading at the same rate, we have to write two ratios to describe each pace.

Rafael \begin{align*}\frac{9}{12}\end{align*}

Manuel \begin{align*}\frac{3}{4}\end{align*}

Now let's write these as a statement of equal ratios.

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

These are equal.

**The boys are reading at the same rate.**

### 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.

Use equal ratios to solve for \begin{align*}z: \ \frac{z}{9} = \frac{32}{36}\end{align*}.

**Answer**

The problem 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?”**

\begin{align*}9 \times 4 &= 36\\ \frac{z}{9} &= \frac{z \times 4}{9 \times 4} = \frac{32}{36}.\end{align*}

From this, we can see that \begin{align*}z \times 4 = 32\end{align*}.

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

\begin{align*}8 \times 4 = 32\end{align*}**, so** \begin{align*}z = 8\end{align*}.

**This is our answer.**

### Video Review

This James Sousa video is an introduction to proportions.

### Practice

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

1. \begin{align*}\frac{1}{2}\end{align*} *and* \begin{align*}\frac{4}{8}\end{align*}

2. \begin{align*}\frac{3}{7}\end{align*} *and* \begin{align*}\frac{6}{14}\end{align*}

3. \begin{align*}\frac{5}{2}\end{align*} *and* \begin{align*}\frac{10}{6}\end{align*}

4. \begin{align*}\frac{3}{1}\end{align*} *and* \begin{align*}\frac{9}{3}\end{align*}

5. \begin{align*}\frac{2}{9}\end{align*} *and* \begin{align*}\frac{1.5}{4.5}\end{align*}

6. \begin{align*}\frac{4}{9}\end{align*} *and* \begin{align*}\frac{8}{10}\end{align*}

7. \begin{align*}\frac{1}{4}\end{align*} *and* \begin{align*}\frac{5}{20}\end{align*}

8. \begin{align*}\frac{3}{4}\end{align*} *and* \begin{align*}\frac{9}{10}\end{align*}

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

9. \begin{align*}\frac{1}{4} = \frac{a}{20}\end{align*}

10. \begin{align*}\frac{15}{30} = \frac{x}{2}\end{align*}

11. \begin{align*}\frac{2}{9} = \frac{n}{63}\end{align*}

12. \begin{align*}\frac{z}{7} = \frac{12}{21}\end{align*}

13. \begin{align*}\frac{3}{5} = \frac{t}{60}\end{align*}

14. \begin{align*}\frac{k}{72} = \frac{5}{12}\end{align*}

15. \begin{align*}\frac{x}{32} = \frac{4}{8}\end{align*}