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Chapter 9: Problem 3

Without working out the decimal expansions, determine the period lengths of the decimal expansions for the fractions \(3 / 21,12 / 43,53 / 96,7 / 23\).

Short Answer

Expert verified
Period lengths: \( \frac{3}{21} = 6 \), \( \frac{12}{43} = 42 \), \( \frac{53}{96} = 0 \), \( \frac{7}{23} = 22 \).

Step by step solution

01

- Simplify the fractions

Simplify each fraction to its lowest terms:- \( \frac{3}{21} \) simplifies to \( \frac{1}{7} \)- \( \frac{12}{43} \) is already in its simplest form- \( \frac{53}{96} \) is already in its simplest form- \( \frac{7}{23} \) is already in its simplest form
02

- Determine the period length conditions

The period length of the decimal expansion of \( \frac{a}{b} \) depends on the form of \( b \):- If \( b \) has any prime factors other than 2 and 5, \( \frac{a}{b} \) will have a repeating decimal.
03

- Apply the period length condition

Check the prime factors of the denominators of the simplified fractions:- For \( \frac{1}{7} \), the prime factor of 7 is not 2 or 5, so \( \frac{1}{7} \) has a repeating decimal. The length of the period is 6 (as 7 is a prime number).- For \( \frac{12}{43} \), 43 is a prime number not equal to 2 or 5, so \( \frac{12}{43} \) has a repeating decimal. The length of the period is 42 (as 43 is a prime number).- For \( \frac{53}{96} \), since the prime factors of 96 are 2 and 3, \( \frac{53}{96} \) has a finite decimal expansion.- For \( \frac{7}{23} \), 23 is a prime number not equal to 2 or 5, so \( \frac{7}{23} \) has a repeating decimal. The length of the period is 22 (as 23 is a prime number).
04

- List results for each fraction

Determine the period lengths for each simplified fraction:- \( \frac{3}{21} \) (i.e., \( \frac{1}{7} \)): period length is 6.- \( \frac{12}{43} \): period length is 42.- \( \frac{53}{96} \): no repeating decimal (finite).- \( \frac{7}{23} \): period length is 22.

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Key Concepts

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

repeating decimals
When a fraction is converted into a decimal, it can result in either a terminating decimal or a repeating decimal.
Repeating decimals occur in cases where the decimal expansion continues indefinitely with a recurring sequence of digits.
For example, the fraction \(\frac{1}{3}\) converts to 0.333..., where '3' repeats endlessly.
In the exercise, fractions like \(\frac{1}{7}\) and \(\frac{7}{23}\) have repeating decimals.
The repeating part in the decimal expansion is known as the 'period'.
The length of this repeating sequence varies depending on the fraction's denominator.
Understanding repeating decimals helps in identifying whether a fraction's decimal form will terminate or repeat.
prime factors
Prime factors play a crucial role in determining whether a fraction's decimal expansion will terminate or repeat.
Prime factors are the building blocks of any number and are crucial in understanding the properties of that number.
For a fraction \(\frac{a}{b}\) to have a finite decimal expansion, the simplified denominator 'b' must only have the prime factors 2 and 5.
If 'b' contains any other prime factors, the decimal expansion will be a repeating pattern.
For instance, in the exercise, the fractions \(\frac{12}{43}\) and \(\frac{7}{23}\) have denominators which are prime numbers other than 2 and 5.
This results in repeating decimals with period lengths depending on those prime numbers.
Recognizing and understanding prime factors in the denominator is essential to solving problems related to the period of repeating decimals.
fraction simplification
Simplifying fractions is the first crucial step in determining their decimal properties.
To simplify a fraction, divide both its numerator and denominator by their greatest common divisor (GCD).
This reduces the fraction to its lowest terms, making it easier to analyze further.
For example, \(\frac{3}{21}\) simplifies to \(\frac{1}{7}\) since both 3 and 21 are divisible by 3.
It is essential to simplify fractions before assessing their decimal expansion because the denominator鈥檚 prime factors in its simplest form can be easily identified.
Understanding the concept of fraction simplification helps in accurately determining whether the decimal is repeating and calculating the period length if it does.
Simplification is foundational for clear and correct mathematical analysis.

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Most popular questions from this chapter

Investigate the cyclic patterns occurring within the decimals for fractions with denominator \(13 .\)
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