08333 as a fraction

Full AI 2025

2 min read

·

19-02-2025

1

0

Share

The decimal 0.8333... (where the 3s repeat infinitely) represents a recurring decimal. Converting recurring decimals to fractions requires a bit of algebra. Let's break down the process step-by-step.

Understanding Repeating Decimals

The key to solving this is understanding that the repeating part of the decimal, in this case "3", continues infinitely. This is different from a terminating decimal (like 0.75) which ends. We'll use this property to our advantage.

Step-by-Step Conversion of 0.8333...

1. Set up an Equation:

Let's represent the decimal as 'x':

x = 0.8333...

2. Multiply to Shift the Decimal:

We need to manipulate the equation so the repeating part lines up. Multiply both sides by 10:

10x = 8.3333...

3. Subtract the Original Equation:

Now, subtract the original equation (x = 0.8333...) from the multiplied equation (10x = 8.3333...). Notice what happens to the repeating part:

10x - x = 8.3333... - 0.8333...

This simplifies to:

9x = 7.5

4. Solve for x:

Divide both sides by 9 to isolate x:

x = 7.5 / 9

5. Simplify the Fraction:

The fraction 7.5/9 isn't in its simplest form. To simplify, we can multiply both the numerator and denominator by 2 to get rid of the decimal:

x = (7.5 * 2) / (9 * 2) = 15/18

Now, we can simplify further by finding the greatest common divisor (GCD) of 15 and 18, which is 3. Divide both the numerator and the denominator by 3:

x = 15/3 / 18/3 = 5/6

Therefore, 0.8333... as a fraction is 5/6.

Verification

You can verify this result using a calculator: divide 5 by 6. The result will be 0.8333..., confirming our conversion.

Alternative Method (for decimals with only repeating digits after the decimal point)

If the decimal had only repeating digits after the decimal (like 0.333... or 0.121212...), there is a slightly faster method. For example, for 0.333...:

1. The repeating digit is 3.
2. The fraction is that digit over 9 (3/9 which simplifies to 1/3).

This shortcut doesn't work for 0.8333... because there's a non-repeating digit (8) before the repeating digits.

Conclusion

Converting recurring decimals like 0.8333... to fractions involves a systematic process of multiplying, subtracting, and simplifying. By following these steps, you can accurately represent any recurring decimal as a fraction. Remember the key is to align the repeating part of the decimal through multiplication before subtraction. Now you know how to tackle similar conversions with confidence!