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Chapter 0: Problem 147

Write each fraction as a decimal. For repeating decimals, write the answer by first using bar notation and then rounding to the nearest thousandth. See Example 11. $$ \frac{5}{9} $$

Short Answer

Expert verified
0.556

Step by step solution

01

- Divide Numerator by Denominator

To convert the fraction into a decimal, divide the numerator (5) by the denominator (9): \( \frac{5}{9} = 5 \div 9 = 0.5555... \).
02

- Identify Repeating Decimal

Notice the pattern of the division. The decimal 0.5555... repeats the digit 5. Therefore, we can write it using bar notation as \( 0.\bar{5} \).
03

- Round to Nearest Thousandth

To round the repeating decimal \( 0.\bar{5} \) to the nearest thousandth, observe that the number remains 5 in every place beyond the third decimal place. Thus, the rounded decimal is 0.556.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Repeating Decimals
A repeating decimal is a decimal number that has digits that repeat infinitely. This usually happens when you divide two numbers, and the division doesn't resolve to a finite number of digits. For example, when you divide 5 by 9, you get 0.555..., where the digit '5' repeats forever. To denote that a decimal is repeating, we need a special method, which brings us to our next concept: bar notation.
Bar Notation
Bar notation is a way to easily identify and write repeating decimals. In bar notation, a horizontal line (bar) is placed over the digit or group of digits that repeat. For instance, in the fraction \frac{5}{9}, our division results in 0.5555..., where '5' repeats infinitely. Using bar notation, we write this as \(0.\bar{5}\). This tells anyone reading it that the digit '5' will repeat forever. Sometimes, more than one digit will repeat, for example, \frac{7}{6} results in 1.16666..., which in bar notation is written as \(1.1\bar{6}\).
Rounding Decimals
Rounding is the process of reducing the number of digits in a decimal while trying to keep its value close to the original number. For repeating decimals, we often round to a specific place value, such as the nearest thousandth. Let's use our fraction \frac{5}{9} again, which we know results in the repeating decimal \(0.\bar{5}\). To round \(0.\bar{5}\) to the nearest thousandth, look at the first three digits after the decimal, which are 0.555. Because the digit following our thousandth place is also a '5', we round up the last digit, making it 0.556. Remember: If the digit you are rounding is followed by a 5 or higher, round up. If it's less than 5, round down.

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Most popular questions from this chapter

Simplify each expression. $$ -14\left(-\frac{2}{7}\right) \div(2 \cdot 6-10) $$

Evaluate each expression for \(w=4, x=-\frac{3}{4}, y=\frac{1}{2},\) and \(z=1.25 .\) $$ -w+2 x+3 y+z $$

The distributive property can be used to mentally perform calculations. $$\begin{array}{l}38 \cdot 17+38 \cdot 3 \\\=38(17+3) \\\=38(20) \\\=760\end{array} $$Use the distributive property to calculate each value mentally. $$ 27 \cdot 60+27 \cdot 40 $$

The distributive property can be used to mentally perform calculations. $$\begin{array}{l}38 \cdot 17+38 \cdot 3 \\\=38(17+3) \\\=38(20) \\\=760\end{array} $$Use the distributive property to calculate each value mentally. $$ 8.75(15)-8.75(5) $$

Complete the table of fraction, decimal, and percent equivalents. $$\begin{array}{|c|c|c|}\hline \begin{array}{c}\text { Fraction in Lowest Terms } \\\\\text { (or Whole Number) }\end{array} & \text { Decimal } & \text { Percent } \\\\\hline \frac{1}{50} & & 2 \%\\\\\hline\end{array}$$
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