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Chapter 6: Problem 43

Convert each fraction to a decimal. Round to five decimal places. \(\frac{7}{9}\)

Short Answer

Expert verified
The decimal representation of \( \frac{7}{9} \) rounded to five decimal places is 0.77778.

Step by step solution

01

Understand the Problem

You are given the fraction \( \frac{7}{9} \) and asked to convert it into a decimal. This means you need to divide 7 by 9 and round the result to five decimal places.
02

Perform the Division

Divide 7 by 9 using long division or a calculator to find the decimal representation of the fraction. When you divide 7 by 9, you get approximately 0.777777... which is a repeating decimal.
03

Round the Decimal

Since the problem asks for the decimal rounded to five decimal places, take the result from the division, 0.777777..., and round it to 0.77778.

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Key Concepts

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

Rounding Decimals
Rounding decimals is an important skill used to simplify numbers while maintaining their approximate value. It is especially useful when you need a number that is easier to work with or when you need to fit a decimal into a limited space. For our exercise, we rounded the decimal obtained from dividing \( \frac{7}{9} \) to five decimal places. Here鈥檚 a quick guide on how to round a decimal number:
  • Identify the decimal place to which you need to round. For example, rounding to five decimal places means looking at the sixth decimal place.
  • If the decimal digit in the next place (the one you are looking at to decide) is 5 or greater, increase your chosen rounding place by one. If it's less than 5, keep the digit as it is.
  • Adjust the number while keeping the rest to the right unchanged.
In our case, for the number 0.777777..., the fifth decimal is 7 and the sixth is also 7, so we round up the fifth decimal place, resulting in 0.77778.
Repeating Decimals
Repeating decimals occur when the digits in a decimal repeat infinitely. These fun numerical patterns typically appear when dividing certain fractions. In the exercise, when we convert the fraction \( \frac{7}{9} \) to a decimal, we get 0.777777..., with 7 endlessly repeating.
There are a few ways to recognize and represent repeating decimals:
  • You can see repeating sequences by examining the output of long division or from a calculator.
  • These decimals are often denoted by placing a bar (vinculum) over the repeating part. For example, 0.777777... is written as \( 0.\overline{7} \).
It's crucial to understand repeating decimals when rounding and performing calculations since they affect the accuracy of your results.
Long Division
Long division is a method used to divide numbers and is particularly handy when turning fractions into decimals. Even though calculators make the process faster, understanding long division is vital for grasping how fractions and decimals relate. Here's how long division works with our example \( \frac{7}{9} \):
  • Set up the division by writing 7 inside the division bracket and 9 outside.
  • Begin the division process by seeing how many times 9 fits into the initial digits of the number you want to divide. Since 9 doesn鈥檛 fit into 7, you start by extending to decimals.
  • Continue the division by adding decimal places, noting how often 9 fits into each new number combination. This is where you'll start to see the repeating pattern, producing the repeating decimal 0.777...
Mastering long division allows you to observe and analyze the process behind the conversion of fractions to decimals, reinforcing your understanding of numbers.

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Most popular questions from this chapter

Find the following products. \(10,000 \cdot 16.52187\)

(11.1106)(12.08) $$\begin{array}{|l|l|l|l|}\hline \text { Actual product } & \text { Tenths } & \text { Hundreds } & \text { Thousandths } \\ \hline & & & \\\\\hline\end{array}$$

Write each decimal fraction. Five hundred eleven and four thousandths.

Convert the following decimals to fractions or mixed numbers. Be sure to reduce. 6,646.0107

Write each decimal fraction. Nine hundred forty-seven thousandths.
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