What is 3 repeating expressed as a fraction?

by

To express the repeating decimal 3.333… (where the digit ‘3’ repeats indefinitely) as a fraction, we can follow a systematic approach:

  1. Let x equal the repeating decimal:

    2. Let x = 3.333…

  2. Multiply by 10:

    3. To shift the decimal point, multiply both sides by 10:
    10x = 33.333…

  3. Subtract the original equation from the new one:

    4. Now, subtract the first equation (x = 3.333…) from the second (10x = 33.333…):
    10x – x = 33.333… – 3.333…

    This simplifies to:

    9x = 30

  4. Solve for x:

    5. Now simplify by dividing both sides by 9:

    x = 30 / 9

    This can be further simplified:

    x = 10 / 3

  5. Final Result:

    6. Therefore, the repeating decimal 3.333… can be expressed as the fraction 10/3.

In conclusion, if you encounter 3 with repeating 3’s after the decimal, remember that it translates to the fraction 10/3. This method of converting repeating decimals to fractions can be applied to other repeating decimals as well, making it a handy technique in mathematics.

Leave a Comment