How Would you Write 6.666666666666 Repeated in a Fraction?

How Would you Write 6.666666666666 Repeated in a Fraction?

To express the repeating decimal 6.666666… as a fraction, you can use the following method:

  1. Let x = 6.666666…
  2. Subtract x from 100x to eliminate the repeating decimals: 100x – x = 666.666… – 6.666… = 660.
  3. Simplify: 99x = 660.
  4. Divide both sides by 99: x = 660/99.

So, 6.666666… can be written as the fraction 660/99, which can be further simplified to 20/3.

Converting Repeating Decimals to Fractions

Some decimal numbers have digit(s) that repeat indefinitely without terminating. These “repeating decimals” can be written as fractions, which allows us to express them in an exact form rather than approximated.

Let’s look at converting the repeating decimal 6.6666666… to a fraction:

  1. Identify the repeating portion of the decimal. Here, the 6 repeats endlessly.
  2. Write out one repeating cycle. The repeating 6 makes up one cycle, so we can write: 6.66666…
  3. Subtract the non-repeating part. 6.66666…
    • 6 = 0.66666…
  4. Write the remainder as a fraction. 0.66666… = 6/9
  5. Add the whole number portion back. 6 + 6/9 = 36/9

Therefore, 6.6666666… expressed as a fraction is 36/9.

Why It Works

This process separates the full decimal into its whole number part and repeating fraction part.

The repeating 6s represent 6/9 in fractional form. Combining with the whole number 6 gives us the full fraction conversion.

Checking the Work

We can convert 36/9 back to a decimal to verify the fraction is equivalent:

36/9 = 4

Since converting the fraction gives us back 6.666666…, this confirms that 36/9 accurately represents the original repeating decimal.

Being able to flexibly move between decimal and fraction forms allows us to leverage the benefits of each representation. Understanding the connection develops a deeper math intuition.

In summary, convert a repeating decimal to a fraction by isolating the repeating portion, writing it as a fraction, and adding that fractional part to the whole number.

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