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Chapter 19: Problem 28

Find the fractions equal to the given decimals. $$0.184561845618456 \ldots$$

Short Answer

Expert verified
Decimal 0.184561845618456... equals \( \frac{6152}{33333} \).

Step by step solution

01

Identify the Repeating Sequence

Observe the given decimal: 0.184561845618456... Now, identify the repeating sequence. You can see that '18456' repeats indefinitely.
02

Express as a Fraction with a Repeating Part

A repeating decimal can be expressed as a fraction where the repeating part is over a series of 9s equal to the length of the repeating sequence. Here, the sequence '18456' has 5 digits, so we can write it as \( \frac{18456}{99999} \).
03

Simplify the Fraction

Now, simplify the fraction \( \frac{18456}{99999} \). Start finding the greatest common divisor (GCD) for the numerator and the denominator, which is 3 in this case. Divide both by 3 to simplify the fraction: \( \frac{18456 \div 3}{99999 \div 3} = \frac{6152}{33333} \).

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Key Concepts

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

Understanding Repeating Decimals
Repeating decimals are decimals where one or more digits repeat infinitely. This often happens when fractions are converted into decimals. The recurring portion is called the 'repeating sequence'. For the decimal 0.184561845618456..., the repeating sequence is '18456'. Knowing this repeating sequence is the first step towards converting a repeating decimal back into a fraction. When identifying the repeating sequence, look for the pattern in the decimal places. It helps to remember that sometimes the repetition is subtle, so examining for at least ten places can provide a clearer picture.
Expressing Decimals as Fractions
To convert a repeating decimal into a fraction, the repeating sequence becomes key. We use a method that constructs a fraction where the repeating parts are represented over a series of 9s, and any non-repeating parts are handled separately.
For instance, when you have a repeating sequence like '18456' with 5 digits, you could write this repeating decimal as a fraction of the repeating part over a denominator with the same number of nines, which would be \( \frac{18456}{99999} \) here.
  • Identify the repeating sequence.
  • Write the repeating sequence over the same number of 9s as there are digits in the sequence.
This straightforward approach ensures that every repeating decimal can be expressed precisely as a fraction.
Simplifying Fractions
Once you've expressed a repeating decimal as a fraction using its repeating sequence, the next task is to simplify this fraction.
Simplifying means reducing the fraction to its smallest whole number form. This involves calculating the greatest common divisor (GCD) of the numerator and the denominator.
  • Calculate the GCD of both numbers in the fraction.
  • Divide the numerator and the denominator by the GCD.
  • Write down the simplified fraction.
For example, with the fraction \( \frac{18456}{99999} \), find that its GCD is 3. Then, divide both parts by 3 to simplify it to \( \frac{6152}{33333} \). This version is simpler and easier to work with or further convert.

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