# Comparison of Ratios in Decimal Form

## Rewrite ratios as decimals.

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Comparison of Ratios in Decimal Form

Erinda went with her dad to the toy store. She was looking for stuffed animals to give to children in need during the holidays. Luckily for Erinda, the store was having a big sale, and all the stuffed animals were marked down one third. Some of the toys that Erinda would like to buy were originally priced at 99 cents. How can she convert \begin{align*}\frac{1}{3}\end{align*} to a decimal to calculate how much money she can save on each stuffed animal?

In this concept, you will learn how to convert and compare ratios in decimal form.

### Comparing Ratios in Decimal Form

A ratio is a relationship between two numbers by division.

Fractions are ratios.

Decimals are ratios.

Remember that decimals are based on multiples of 10, so if possible, convert fractions to one of these multiples.

To convert a ratio to its decimal form:

First, write the ratio as a fraction.

Next, divide the numerator by the denominator. "Denominator" and "down" both start with the letter "d."

Look at this example.

Rewrite the ratio 1 to 4 in decimal form.

First, write the ratio as a fraction.

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

Next, divide the numerator by the denominator.

\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. Any number of zeroes can be annexed because \begin{align*}1 = 1.0 = 1.00 = 1.000\end{align*}, etc.

Before you divide, remember to put 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 dividing 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 answer is 0.25. The decimal form of the ratio \begin{align*}\frac{1}{4}\end{align*} is 0.25.

### Examples

#### Example 1

Earlier, you were given a problem about Erinda and the one-third-off sale at the toy store.

She found some stuffed animals that were originally 99 cents.How much will she save on each toy?

First, write the ratio as a fraction.

\begin{align*}\frac{1}{3}\end{align*}

Next, convert the fraction to a decimal by dividing the numerator by the denominator.

\begin{align*}& \overset{ \ \ 0.333}{3 \overline{ ) { 1.000 \;}}}\\ & \quad \ \underline{-9\;}\\ & \qquad 10\\ & \quad \ \underline{-9}\\ & \qquad \quad 1 \end{align*}

Then, recognize that .333 is a repeating decimal and round to two places.

\begin{align*}.333\approx .33\end{align*}

The answer is \begin{align*}\frac{1}{3}\end{align*}off the price of each toy is 33 cents. Erinda will save 33 cents on each toy.

#### Example 2

Rewrite the ratio 9:5 in decimal form.

First, rewrite the ratio as a fraction.

\begin{align*}\frac{9}{5}\end{align*}

Next, divide the numerator by the denominator, remembering to annex zeroes and continue until there is no remainder.

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

The answer is 1.8 The decimal form of 9:5 is 1.8

#### Example 3

Compare the two ratios by converting them to decimals. Are they equivalent?

\begin{align*}\frac{7}{14}\end{align*}and \begin{align*}\frac{11}{20}\end{align*}

First, rewrite \begin{align*}\frac{7}{14}\end{align*} in decimal form by dividing the numerator by the denominator.

\begin{align*}& \overset{ \ \ 0.5}{14 \overline{ ) {\ 7.0 \;}}}\\ & \ \ \underline{-70\;}\\ & \qquad 0\\ & \quad \ \underline{-0}\\ & \qquad 0 \end{align*}

Next, 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*}

Then, compare the two values by writing each one to the same number of decimal places.

\begin{align*}\frac{7}{14}=0.50\end{align*}

\begin{align*}\frac{11}{20}=0.55\end{align*}

\begin{align*}0.50<0.55\end{align*}

The answer is that the ratios are not equivalent. \begin{align*}\frac{7}{14}<\frac{11}{20}\end{align*}

#### Example 4

Compare 5 to 10 and  \begin{align*}\frac{2}{5}\end{align*} and tell if they are equivalent.

First, write 5 to 10 as a fraction.

\begin{align*}\frac{5}{10}\end{align*}

Next, write each fraction as a decimal.

\begin{align*}\frac{5}{10}=0.5\end{align*}

Then, remember that decimals are based on multiples of 10, and convert \begin{align*}\frac{2}{5}\end{align*} to tenths by multiplying both the numerator and denominator times 2.

\begin{align*}\frac{2\times 2}{5\times 2}=\frac{4}{10}\end{align*}= 0.4

Next, compare the decimals.

\begin{align*}0.5>0.4\end{align*}

The answer is that the ratios are not equivalent.

#### Example 5

Compare 6 to 10 and 1:4

First, write the ratios as fractions.

\begin{align*}\frac{6}{10}\end{align*}and \begin{align*}\frac{1}{4}\end{align*}

Next, convert the fractions to decimals.

\begin{align*}\frac{6}{10}=0.60\end{align*}

\begin{align*}\frac{1}{4}=0.25\end{align*}

Then, compare the decimals.

\begin{align*}0.60>0.25\end{align*}

The answer is that the ratios are not equivalent.

### Review

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

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

1. .55 ____1 to 2
2. 3:8 _____ 4 to 9
3. 1 to 2 _____ 4:8

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### Vocabulary Language: English

TermDefinition
Decimal In common use, a decimal refers to part of a whole number. The numbers to the left of a decimal point represent whole numbers, and each number to the right of a decimal point represents a fractional part of a power of one-tenth. For instance: The decimal value 1.24 indicates 1 whole unit, 2 tenths, and 4 hundredths (commonly described as 24 hundredths).