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Chapter 5: Problem 215

Convert each fraction to a decimal. \(\frac{15}{111}\)

Short Answer

Expert verified
0.135 with a repeating bar over 135

Step by step solution

01

Understand the Fraction

To convert the fraction \(\frac{15}{111}\) to a decimal, recognize that this is a division problem where 15 is divided by 111.
02

Set Up the Division

Write the division as \(15 \div 111\). This means you will determine how many times 111 can fit into 15, but since 111 is bigger than 15, you are essentially finding the fraction in decimal form.
03

Perform the Division

Using a calculator, divide 15 by 111 to get a decimal value. The result is approximately 0.135135135.
04

Notate the Answer

Recognize that the decimal 0.135135135 repeats. So, \(\frac{15}{111}\) equals 0.135135135... which can be simplified to 0.135 with a repeating bar over 135.

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Key Concepts

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

Division
When converting a fraction to a decimal, you essentially perform a division. For the fraction \(\frac{15}{111}\), think of it as 15 divided by 111.

Here's how you can do the division step-by-step:
  • Place 15 (the numerator) under the division bar.
  • Place 111 (the denominator) outside the division bar.
  • Since 111 is larger than 15, it won't go into 15 any whole number of times.

You add a decimal point immediately after 15 and introduce zeros to proceed with the division. Keep dividing until you recognize a repeating pattern or until you reach sufficient decimal places.
Repeating Decimal
When you performed the division 15 divided by 111, the result was approximately 0.135135135.

This pattern, where the group of digits repeats forever, is called a repeating decimal.

For \(\frac{15}{111}\), the repeating sequence is '135'.

Hence, you write the answer as 0.135 with a bar over the digits 135, indicating they repeat indefinitely.
  • Fraction: \(\frac{15}{111}\)
  • Decimal: 0.135135135...
  • Repeating Decimal: 0.135 with a bar over 135

    To express a repeating decimal, draw a horizontal line (called a vinculum) over the repeating digits.
Fraction Simplification
Fraction simplification makes calculations simpler and results more understandable.

Although \(\frac{15}{111}\) itself cannot be further simplified, recognizing its decimal form helps in various applications.

In scenarios where exact values are needed, fractions can be more useful. Conversely, for approximate values and practical use, decimals are often easier to work with.
  • Simplifying fractions involves dividing the numerator and the denominator by their greatest common divisor (GCD).
  • \frac{15}{111} simplified is easiest handled through conversion to decimal form as previously seen: 0.135135135...

Remember, while most fractions simplify easily, in some cases like repeating decimals, understanding their decimal form can be very helpful.

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