Decimals and Fractions

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KEEP IN MIND

When comparing decimals, compare the place value where the numbers are different.

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When comparing decimals, compare the numbers in the same place value. For example, determine the greatest decimal from the group 0.4, 0.41, 0.39, and 0.37. In these numbers, there is a value to the right of the decimal point. Comparing the tenths places, the numbers with 4 tenths (0.4 and 0.41) are greater than the numbers with three tenths (0.39 and 0.37).

Then, compare the hundredths in the 4 tenths numbers. The value of 0.41 is greater because there is a 1 in the hundredths place versus a 0 in the hundredths place.

Here is another example: determine the least decimal of the group 5.23, 5.32, 5.13, and 5.31. In this group, the ones value is 5 for all numbers. Then, comparing the tenths values, 5.13 is the smallest number because it is the only value with 1 tenth.

HELPFUL TIPS

The multiplication and division algorithms can be used, even with decimal numbers. With the multiplication algorithm, count the total number of place values to the right the decimal must move in the multiplier numbers. Simply place the decimal the total number of place values to the left in the sum. For example, 0.5 × 0.5, the decimal must move one place value to the right for each number and a total of two place values. Place the decimal two place values to the left in the answer. After completing the algorithm, 25 becomes 0.25 from the decimal.

For the division algorithm, move the decimal to the right as many place values as needed to make the divisor and dividend whole numbers. The place value should be moved the same amount of times in both numbers. For example, 180 ÷ 0.2, the divisor must have the place value moved one place value to the right to become 2. Move the decimal from 180 (180.0) the same amount, to complete long division with whole numbers: 1800 ÷ 2 = 900, which is equal to 180 ÷ 0.2 = 900.

Introduction to Decimals Review

Changing Decimals and Fractions

Decimal to a Fraction

Three steps change a decimal to a fraction.

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STEP BY STEP

Step 1. Write the decimal divided by 1 with the decimal as the numerator and 1 as the denominator.

Step 2. Multiply the numerator and denominator by 10 for every number after the decimal point. (For example, if there is 1 decimal place, multiply by 10. If there are 2 decimal places, multiply by 100).

Step 3. Reduce the fraction completely.

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To change the decimal 0.37 to a fraction, start by writing the decimal as a fraction with a denominator of one, \(\frac{0.37}{1}\). Because there are two decimal places, multiply the numerator and denominator by 100, \(\frac{0.37\times100}{1\times100}=\frac{37}{100}\). The fraction does not reduce, so \(\frac{37}{100}\) is 0.37 in fraction form.

Similarly, to change the decimal 2.4 to a fraction start by writing the decimal as a fraction with a denominator of one, \(\frac{0.4}{1}\), and ignore the whole number. Because there is one decimal place, multiply the numerator and denominator by 10, \(\frac{0.4\times10}{1\times10}=\frac{4}{10}\). The fraction does reduce: \(2\frac{4}{10}=2\frac{2}{5}\) is 2.4 in fraction form.

The decimal \(0.\overline{3}\) as a fraction is \(\frac{0.\overline{3}}{1}\). In the case of a repeating decimal, let \(n=0.\overline{3}\) and \(10n=3.\overline{3}\). Let \(10n-n=3.\overline{3}-0.\overline{3}\), resulting in 9n = 3 and solution of \(n=\frac{3}{9}=\frac{1}{3}\). The decimal \(0.\overline{3}\) is \(\frac{1}{3}\) as a fraction.

Fraction to a Decimal

Two steps change a fraction to a decimal.

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STEP BY STEP

Step 1. Divide the numerator by the denominator . Add zeros after the decimal point as needed.

Step 2. Complete the process when there is no remainder or the decimal is repeating.

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To convert \(\frac{1}{5}\) to a decimal, rewrite \(\frac{1}{5}\) as a long division problem and add zeros after the decimal point, 1.0 ÷ 5. Complete the long division and \(\frac{1}{5}\) as a decimal is 0.2. The division is complete
because there is no remainder.

To convert \(\frac{8}{9}\) to a decimal, rewrite \(\frac{8}{9}\) as a long division problem and add zeros after the decimal point, 8.00 ÷ 9. Complete the long division and \(\frac{8}{9}\) as a decimal is \(0.\overline{8}\). The process is complete because the decimal is complete.

To rewrite the mixed number \(2\frac{3}{4}\) as a decimal, the fraction needs changed to a decimal. Rewrite \(\frac{3}{4}\) as a long division problem and add zeros after the decimal point, 3.00 ÷ 4. The whole number is needed for the answer and is not included in the long division. Complete the long division, and \(2\frac{3}{4}\) as a decimal is 2.75.

Changing Decimals and Fractions Review

Convert among Fractions, Decimals, and Percentages

Fractions, decimals, and percentages can change forms, but they are equivalent values.

Decimal to a Percent

There are two ways to change a decimal to a percent. One way is to multiply the decimal by 100 and add a percent sign. 0.24 as a percent is 24%.

Another way is to move the decimal point two places to the right. The decimal 0.635 is 63.5% as a percent when moving the decimal point two places to the right.

Any decimal, including repeating decimals, can change to a percent. \(0.\overline{3}\) as a percent is \(0.\overline{3}\times100=33.\overline{3}%\)

Percent to a Decimal

There are two ways to change a percent to a decimal. One way is to remove the percent sign and divide the decimal by 100. For example, 73% as a decimal is 0.73.

Another way is to move the decimal point two places to the left. For example, 27.8% is 0.278 as a decimal when moving the decimal point two places to the left.

Any percent, including repeating percents, can change to a decimal. For example, \(44.\overline{4}%\) as a decimal is \(44.\overline{4}\div100=0.\overline{4}\).

Example

Fraction to a Percent

Two steps change a fraction to a percent.

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STEP BY STEP

Step 1. Divide the numerator and denominator.

Step 2. Multiply by 100 and add a percent sign.

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To change the fraction \(\frac{3}{5}\) to a decimal, perform long division to get 0.6. Then, multiply 0.6 by 100 and \(\frac{3}{5}\) is the same as 60%.

To change the fraction \(\frac{7}{8}\) to a decimal, perform long division to get 0.875. Then, multiply 0.875 by 100 and \(\frac{7}{8}\) is the same as 87.5%.

Fractions that are repeating decimals can also be converted to a percent. To change the fraction \(\frac{2}{3}\) to a decimal, perform long division to get \(0 .\overline{6}\). Then, multiply \(0 .\overline{6}\) by 100 and the percent is \(66 .\overline{6}\)%.

Percent to a Fraction

Remove the percent sign from 45% and write as a fraction with a denominator of 100, \(\frac{45}{100}\). The fraction reduces to \(\frac{9}{20}\).

Remove the percent sign from 22.8% and write as a fraction with a denominator of 100, \(\frac{22.8}{100}\). The fraction reduces to \(\frac{228}{1000}=\frac{57}{250}\)

Repeating percentages can change to a fraction. Remove the percent sign from \(016.\overline{6}\) and write as a fraction with a denominator of 100, \(\frac{16.\overline{6}}{100}\). The fraction simplifies to \(\frac{0.1\overline{6}}{1}=\frac{1}{6}\).

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STEP BY STEP

Step 1. Remove the percent sign and write the value as the numerator with a denominator of 100.

Step 2. Simplify the fraction.

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It is useful to see how these numbers compare on a number line when working between fractions, decimals, and percentages. Use the table and number line below as a quick reference to draw comparisons.

Convert Among Fractions, Decimals and Percents Review

Let’s Review!

• A fraction is a number with a numerator and a denominator. A fraction can be written as a proper fraction, an improper fraction, or a mixed number. Changing fractions to a common denominator enables you to determine the least or greatest fraction in a group of fractions.
• A decimal is a number that expresses part of a whole. By comparing the same place values, you can find the least or greatest decimal in a group of decimals.
• A number can be written as a fraction, a decimal, and a percent. These are equivalent values. Numbers can be converted between fractions, decimals, and percents by following a series of steps.