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# 5.3: Comparison of Ratios in Decimal Form

Difficulty Level: At Grade Created by: CK-12
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Practice Comparison of Ratios in Decimal Form

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Remember Joanna?

Joanna read the following fraction of books.

She read 14\begin{align*}\frac{1}{4}\end{align*} books that she had intended to read. We can write that a decimal.

14=.25\begin{align*}\frac{1}{4} = .25\end{align*}

Kara read 13\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

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 14\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.

14=4)1¯¯¯¯¯¯¯¯¯\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 1=1.0=1.00=1.000\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.

4)1.0¯¯¯¯¯¯¯¯¯¯  0.28  2\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.

4)1.00¯¯¯¯¯¯¯¯¯¯¯¯  0.258  20   20 0\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 14\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 95\begin{align*}\frac{9}{5}\end{align*}.

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

14) 9¯¯¯¯¯¯¯   1 5   4\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.

5)9.0¯¯¯¯¯¯¯¯¯¯  1.85 40  400\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 714\begin{align*}\frac{7}{14}\end{align*} and 1120\begin{align*}\frac{11}{20}\end{align*}.

Rewrite 714\begin{align*}\frac{7}{14}\end{align*} in decimal form.

5) 7.0¯¯¯¯¯¯¯¯¯¯¯  0.5  700 00\begin{align*}& \overset{ \ \ 0.5}{5 \overline{ ) {\ 7.0 \;}}}\\ & \ \ \underline{-70\;}\\ & \qquad 0\\ & \quad \ \underline{-0}\\ & \qquad 0 \end{align*}

Rewrite 1120\begin{align*}\frac{11}{20}\end{align*} in decimal form.

20)11.00¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯  0.55 100100 1000\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.

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

Since 0<5,0.50<0.55\begin{align*}0 < 5, 0.50 < 0.55\end{align*}. So, the ratios, 714\begin{align*}\frac{7}{14}\end{align*} and 1120\begin{align*}\frac{11}{20}\end{align*}, are not equivalent.

In fact, 714<1120\begin{align*}\frac{7}{14} < \frac{11}{20}\end{align*}.

Write each ratio as a decimal.

#### Example A

5 to 10

Solution: .5\begin{align*}.5\end{align*}

#### Example B

4 to 10

Solution: .4\begin{align*}.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 14\begin{align*}\frac{1}{4}\end{align*} books that she had intended to read. We can write that a decimal.

14=.25\begin{align*}\frac{1}{4} = .25\end{align*}

Kara read 13\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.

.25\begin{align*}.25\end{align*} and 13=.33\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.

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

### Vocabulary

Here are the vocabulary words in this Concept.

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.

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

To do this, we divide 3 by 5.

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

### Video Review

Here are videos for 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

### Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes

### Vocabulary Language: English

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).

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