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Chapter 5: Problem 213

Convert each fraction to a decimal. \(\frac{15}{11}\)

Short Answer

Expert verified
The decimal representation of \( \frac{15}{11} \) is approximately 1.36 with 36 repeating.

Step by step solution

01

Set Up the Division

To convert the fraction \( \frac{15}{11} \) to a decimal, interpret the fraction as a division problem: 15 divided by 11.
02

Perform Long Division

Divide 15 by 11 using long division. 11 goes into 15 once, so write down 1 and subtract 11 from 15, giving a remainder of 4. Bring down a zero to make it 40.
03

Continue the Division

11 goes into 40 three times (since 11 * 3 = 33). Write down 3 and subtract 33 from 40, resulting in a remainder of 7. Bring down another zero to make it 70.
04

Repeat the Process

Continue the division process. 11 goes into 70 six times (since 11 * 6 = 66). Write down 6 and subtract 66 from 70, leaving a remainder of 4. Bring down another zero to make it 40 and repeat.
05

Identify the Repeating Decimal

Notice the pattern that repeats every two digits, '36'. Thus, the result of the division indicates that \( \frac{15}{11} \) equals 1.363636... which can be written as 1.36 with a repeating bar over 36.

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Key Concepts

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

long division
To convert a fraction like \( \frac{15}{11} \) into a decimal, we use a method called long division. This is a process that helps break down the division step by step.
We start by setting up the division: the numerator (top number) 15 is divided by the denominator (bottom number) 11.
We then follow these steps:
  • Find how many times 11 can fit into the first digit of 15. In this case, 11 fits into 15 once. Write down 1.
  • Subtract 11 from 15 to get the remainder, 4.
  • Bring down a zero to make it 40 and repeat the process.

This step-by-step breakdown helps make the division easier to handle.
repeating decimals
When converting fractions to decimals, you may come across repeating decimals. A repeating decimal is a decimal that has one or more repeating numbers after the decimal point.
In our example, \( \frac{15}{11} \), the result is 1.363636..., where '36' repeats continuously.
To indicate that a number repeats, we place a bar over the repeating digits, like this: 1.36虆.
Repeating decimals are common when dividing certain fractions, and recognizing them helps in simplifying the final answer.
division process
The division process involves multiple steps and requires careful calculation. Here鈥檚 how it works using the fraction \( \frac{15}{11} \):
  • First, divide 15 by 11. Since 11 goes into 15 once, write 1 above the division line.
  • Subtract 11 from 15 to get 4.
  • Next, bring a zero down beside the 4 to make 40.
  • Repeat this process: 11 goes into 40 three times (33), write 3, subtract 33 from 40 to get 7.
  • Bring down another zero to get 70, and divide again. Continue until you notice a repeating pattern.

By following these repeating steps, the result is found to be a repeating decimal 1.363636... or 1.36虆.

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