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Chapter 4: Problem 93

Find decimal notation for \(\frac{1}{7}, \frac{2}{7}, \frac{3}{7}, \frac{4}{7},\) and \(\frac{5}{7} .\) Observe the pattern and guess the decimal notation for \(\frac{6}{7}\).

Short Answer

Expert verified
The decimal notations are: \ \( \frac{1}{7} = 0.142857... \), \ \( \frac{2}{7} = 0.285714... \), \ \( \frac{3}{7} = 0.428571... \), \ \( \frac{4}{7} = 0.571428... \), \ \( \frac{5}{7} = 0.714285... \, and \ \( \frac{6}{7} = 0.857142... \.

Step by step solution

01

Convert \ \( \frac{1}{7} \) to Decimal

Perform the division \ \( 1 \div 7 \) to find the decimal notation for \ \( \frac{1}{7} \). This gives \ \( 0.142857... \). The decimal repeats every 6 digits.
02

Convert \ \( \frac{2}{7} \) to Decimal

Perform the division \ \( 2 \div 7 \) to find the decimal notation for \ \( \frac{2}{7} \). This gives \ \( 0.285714... \). The decimal repeats every 6 digits.
03

Convert \ \( \frac{3}{7} \) to Decimal

Perform the division \ \( 3 \div 7 \) to find the decimal notation for \ \( \frac{3}{7} \). This gives \ \( 0.428571... \). The decimal repeats every 6 digits.
04

Convert \ \( \frac{4}{7} \) to Decimal

Perform the division \ \( 4 \div 7 \) to find the decimal notation for \ \( \frac{4}{7} \). This gives \ \( 0.571428... \). The decimal repeats every 6 digits.
05

Convert \ \( \frac{5}{7} \) to Decimal

Perform the division \ \( 5 \div 7 \) to find the decimal notation for \ \( \frac{5}{7} \). This gives \ \( 0.714285... \). The decimal repeats every 6 digits.
06

Observe the Pattern

Notice the repeating patterns of the decimals: \ \( \frac{1}{7} = 0.142857... \), \ \( \frac{2}{7} = 0.285714... \), \ \( \frac{3}{7} = 0.428571... \), \ \( \frac{4}{7} = 0.571428... \), and \ \( \frac{5}{7} = 0.714285... \). Each fraction's decimal notation is a cyclic permutation of \ \( 142857 \).
07

Guess \ \( \frac{6}{7} \)鈥檚 Decimal Notation

By observing \ \( 142857 \), deduce that \ \( \frac{6}{7} \)鈥檚 decimal notation will follow the cycle, making it \ \( 0.857142... \).

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Key Concepts

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

fractions
Fractions represent parts of a whole. When you see a fraction like \(\frac{1}{7}\), it's telling you that 1 is being divided by 7. The top number (numerator) shows how many parts we have, and the bottom number (denominator) tells us how many parts make up a whole.
For example, if you cut a pizza into 7 slices and take 1, you've taken \(\frac{1}{7}\) of the pizza.

Understanding fractions is key to moving on to their decimal equivalents. So, in this exercise, we'll convert fractions like \(\frac{1}{7}\) into their decimal forms by using division.
division
Division is a way of sharing or grouping numbers. It's the process used to convert fractions to decimals. When we say perform the division 1 梅 7, it means we're splitting 1 into 7 equal parts.
Here are the steps to convert \(\frac{1}{7}\) to its decimal form using division:
  • Divide 1 by 7. This equals approximately 0.142857.
  • This decimal repeats every 6 digits because 1 梅 7 doesn鈥檛 result in a neat, non-repeating number.
For other fractions like \(\frac{2}{7}\) and \(\frac{3}{7}\), you鈥檒l follow a similar method:
  • 2 梅 7 equals approximately 0.285714.
  • 3 梅 7 equals approximately 0.428571.
Each result also repeats every 6 digits.
repeating decimals
Repeating decimals are decimals in which one or more digits repeat infinitely. These often result from dividing numbers that don't divide evenly.
In our exercise, dividing by 7 leads to repeating decimals:
  • \(\frac{1}{7} = 0.142857 \ldots\)

    here, 142857 repeats.
  • \(\frac{2}{7} = 0.285714 \ldots\)
  • \(\frac{3}{7} = 0.428571 \ldots\)
  • \(\frac{4}{7} = 0.571428 \ldots\)
  • \(\frac{5}{7} = 0.714285 \ldots\)
To represent repeating parts, we use a line (called vinculum) over digits: 0.142857 becomes 0.\overline{142857\overline}.
This notation tells us that 142857 repeats forever.
pattern recognition
Pattern recognition involves finding regularities in data. When studying decimals of fractions like \(\frac{1}{7}\), you can see a repeating pattern.
If you look closely at the results:
  • \(\frac{1}{7} = 0.142857 \ldots\)
  • \(\frac{2}{7} = 0.285714 \ldots\)
  • \(\frac{3}{7} = 0.428571 \ldots\)
  • \(\frac{4}{7} = 0.571428 \ldots\)
  • \(\frac{5}{7} = 0.714285 \ldots\)
Each decimal sequence is a cyclic permutation of the numbers 142857.
To find the decimal notation of \(\frac{6}{7}\), follow this pattern. Since 142857 cyclically shifts, \(\frac{6}{7}\) will be 0.857142...
Recognizing patterns makes it easier to anticipate results and understand underlying mathematical relationships.

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Most popular questions from this chapter

The Colavitos own a house in Louisiana with an assessed value of \(\$ 187,300\). For every \(\$ 1000\) of assessed value, they pay \(\$ 4.90\) in property taxes each year. How much do they pay in property taxes each year?

Estimate by rounding as directed. $$ \begin{aligned} &6.03+0.007+0.214 ;\\\ &\text { nearest one } \end{aligned} $$

Rita's gross pay (before deductions) is \$495.72. Her deductions are \(\$ 59.60\) for federal income tax, \(\$ 29.00\) for FICA, and \(\$ 29.00\) for medical insurance. What is her take-home pay?

Estimate. Indicate which of the choices is an appropriate estimate. \(98.4 \times 0.083\) a) 80 b) 12 c) 8 d) 0.8

Add. $$ \begin{array}{r} 81.008 \\ +\quad 3.409 \\ \hline \end{array} $$
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