How is 0.24\overline{6} written as a fraction in simplest form

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How is 0.24\overline{6} written as a fraction in simplest form?

The Correct Answer and Explanation is:

Step-by-Step Conversion:

Let
x=0.246666…x = 0.246666\ldotsx=0.246666…

Split it into non-repeating and repeating parts:

  1. Let’s isolate the repeating part:
    • Multiply by 10 to move past the non-repeating 2 digits: 10x=2.46666…10x = 2.46666\ldots10x=2.46666…
    • Multiply by 1000 to move past the repeating digit: 1000x=246.666…1000x = 246.666\ldots1000x=246.666…
  2. Subtract the two equations: 1000x−10x=246.666…−2.4666…1000x – 10x = 246.666\ldots – 2.4666\ldots1000x−10x=246.666…−2.4666… 990x=244.2990x = 244.2990x=244.2 Now convert 244.2 into a fraction: 244.2=244210244.2 = \frac{2442}{10}244.2=102442​ So, 990x=244210990x = \frac{2442}{10}990x=102442​ Multiply both sides by 10: 9900x=24429900x = 24429900x=2442 x=24429900x = \frac{2442}{9900}x=99002442​
  3. Simplify the fraction:
    Find the greatest common divisor (GCD) of 2442 and 9900.
    GCD of 2442 and 9900 is 6. 2442÷69900÷6=4071650\frac{2442 \div 6}{9900 \div 6} = \frac{407}{1650}9900÷62442÷6​=1650407​

✅ Final Answer:

4071650\boxed{\frac{407}{1650}}1650407​​

✏️ Explanation

To convert the repeating decimal 0.246‾0.24\overline{6}0.246 into a fraction, we need to understand the structure of repeating decimals. This number has two parts: a non-repeating part (“24”) and a repeating part (“6”).

We begin by setting the decimal equal to a variable, say x=0.246666…x = 0.246666\ldotsx=0.246666…. Then we use multiplication to shift the decimal so the repeating digits align. First, multiplying by 10 moves the decimal just past the non-repeating digits, giving us 10x=2.46666…10x = 2.46666\ldots10x=2.46666…. Next, we multiply both sides by 100 to get 1000x=246.666…1000x = 246.666\ldots1000x=246.666…. This allows us to subtract the equations and eliminate the repeating part, resulting in 990x=244.2990x = 244.2990x=244.2.

Now, we convert 244.2 to a fraction by recognizing it is 244210\frac{2442}{10}102442​. So we get: 990x=244210⇒x=24429900990x = \frac{2442}{10} \Rightarrow x = \frac{2442}{9900}990x=102442​⇒x=99002442​

Then we simplify this fraction. The GCD of 2442 and 9900 is 6, so dividing both numerator and denominator by 6 gives 4071650\frac{407}{1650}1650407​, which is the simplest form.

This method is a systematic way to convert any repeating decimal to a fraction using algebra, subtraction, and simplification techniques.

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