Remember Joanna from the Write Ratios in Simplest Form Concept?

Joanna read the following fraction of books.

She read \begin{align*}\frac{1}{4}\end{align*} books that she had intended to read. We can write that a decimal.

\begin{align*}\frac{1}{4} = .25\end{align*}

Kara read \begin{align*}\frac{1}{3}\end{align*} books that she intended to read.

Can you compare these two ratios?

**Pay attention and you will learn how to do this by the time this Concept is complete.**

### Guidance

Previously we worked on writing fractions in decimal form. Just like fractions can be written in decimal form, well, ratios can be written in fraction form, so they can also be written as decimals. We can look at how to write a ratio as a decimal too.

**How do we write a ratio as a decimal?**

**To convert a ratio to decimal form, write the ratio as a fraction. Then divide the term above the fraction bar by the term below the fraction bar.**

Let’s look at how to do this.

Rewrite the ratio 1 to 4 in decimal form.

**The ratio 1 to 4 can be expressed as the fraction \begin{align*}\frac{1}{4}\end{align*}. This is our first step.**

**Next, divide the term above the fraction bar, 1, by the term below the fraction bar, 4.**

\begin{align*}\frac{1}{4} = {4 \overline{ ) 1 \;\;\;}}\end{align*}

Since 1 cannot be evenly divided by 4, rewrite 1 as a decimal with a zero after the decimal point You can do this because \begin{align*}1 = 1.0 = 1.00 = 1.000\end{align*}. Before you divide, write a decimal point in the quotient directly above the decimal point in the dividend. Then divide.

\begin{align*}& \overset{ \ \ 0.2}{4 \overline{ ) {1.0 \;}}}\\ & \quad \underline{-8}\\ & \quad \ \ 2 \end{align*}

Continue adding zeroes after the decimal point and diving until the quotient has no remainder.

\begin{align*}& \overset{ \ \ 0.25}{4 \overline{ ) {1.00 \;}}}\\ & \quad \underline{-8 \;\;}\\ & \quad \ \ 20\\ & \ \ \ \underline{-20}\\ & \qquad \ 0 \end{align*}

**The decimal form of the ratio \begin{align*}\frac{1}{4}\end{align*} is 0.25.**

Rewrite the ratio 9:5 in decimal form.

**The ratio 9:5 can be expressed as the fraction \begin{align*}\frac{9}{5}\end{align*}.**

**Next, divide the term above the fraction bar, 9, by the term below the fraction bar, 5.**

\begin{align*}& \overset{ \ \ \ 1}{14 \overline{ ) { \ 9 \;}}}\\ & \ \underline{-5\;}\\ & \ \ \ 4 \end{align*}

There is a remainder. So, add zeroes after the decimal point in 9 to continue dividing.

\begin{align*}& \overset{ \ \ 1.8}{5 \overline{ ) {9.0 \;}}}\\ & \quad \underline{-5\;}\\ & \quad \ 40\\ & \ \ \underline{-40}\\ & \qquad 0 \end{align*}

**The decimal form of 9:5 is 1.8.**

**What about comparing? Can we use decimals to compare ratios?**

**Sometimes, you may want to compare two ratios and determine if they are equivalent or not. Rewriting both ratios in decimal form is one way to do this.**

Compare these two ratios and determine if they are equivalent \begin{align*}\frac{7}{14}\end{align*} and \begin{align*}\frac{11}{20}\end{align*}.

**Rewrite \begin{align*}\frac{7}{14}\end{align*} in decimal form.**

\begin{align*}& \overset{ \ \ 0.5}{5 \overline{ ) {\ 7.0 \;}}}\\ & \ \ \underline{-70\;}\\ & \qquad 0\\ & \quad \ \underline{-0}\\ & \qquad 0 \end{align*}

**Rewrite \begin{align*}\frac{11}{20}\end{align*} in decimal form.**

\begin{align*}& \overset{ \ \ 0.55}{20 \overline{ ) { 11.00 \;}}}\\ & \quad \ \underline{-100\;}\\ & \qquad 100\\ & \quad \ \underline{-100}\\ & \qquad \quad 0 \end{align*}

**To compare the ratios in decimal form, give each decimal the same number of decimal places. In other words, give 0.5 two decimal places:** 0.5 = 0.50.

**Now compare. Since both decimals have a 0 in the ones place and a 5 in the tenths place, compare the digits in the hundredths place.**

\begin{align*}&0.5\underline{0}\\ &0.5\underline{5}\end{align*}

Since \begin{align*}0 < 5, 0.50 < 0.55\end{align*}. So, the ratios, \begin{align*}\frac{7}{14}\end{align*} *and* \begin{align*}\frac{11}{20}\end{align*}, are not equivalent.

**In fact,** \begin{align*}\frac{7}{14} < \frac{11}{20}\end{align*}.

Write each ratio as a decimal.

#### Example A

5 to 10

**Solution: \begin{align*}0.5\end{align*}**

#### Example B

4 to 10

**Solution: \begin{align*}0.4\end{align*}**

#### Example C

Compare 6 to 10 and 1 to 4

**Solution: >**

Now back to the book comparison. Here is the original problem once again.

Joanna read the following fraction of books.

She read \begin{align*}\frac{1}{4}\end{align*} books that she had intended to read. We can write that a decimal.

\begin{align*}\frac{1}{4} = .25\end{align*}

Kara read \begin{align*}\frac{1}{3}\end{align*} books that she intended to read.

Can you compare these two ratios?

First, let's write a statement so that we can compare the decimals.

\begin{align*}0.25\end{align*} and \begin{align*}\frac{1}{3} = .33\end{align*}

One-third becomes a repeating decimal, but for our purposes we can round to the hundredths place.

\begin{align*}.25<.33\end{align*}

**This is our answer.**

### Vocabulary

- Ratio
- a comparison between two quantities. Ratios can be written as a fraction, with a colon or by using the word "to".

- Simplify
- to write in a simpler form by using the greatest common factor to divide the numerator and the denominator of a fraction by the same number.

- Decimal
- a part of a whole written using a decimal point and the place value system.

### Guided Practice

Here is one for you to try on your own.

Write this ratio as a decimal.

\begin{align*}\frac{3}{5}\end{align*}

**Answer**

To do this, we divide 3 by 5.

\begin{align*}3 \div 5 = .6\end{align*}

**This is our answer.**

### Video Review

### Practice

Directions: Write each ratio as a decimal. Round to the nearest hundredth when necessary.

1. 1 to 4

2. 3 to 6

3. 3:4

4. 8 to 5

5. 7 to 28

6. 8 to 10

7. 9 to 100

8. 15:20

9. 18:50

10. 3 to 10

11. 6 to 8

12. 15 to 35

Directions: Compare the following ratios using <, > or =.

13. .55 ____1 to 2

14. 3:8 _____ 4 to 9

15. 1 to 2 _____ 4:8