What is the fractional form of 0.8 repeating

What is the fractional form of 0.8 repeating

The Correct Answer and Explanation is:

To convert the repeating decimal ( 0.\overline{8} ) (where the 8 repeats infinitely) into a fraction, we follow a systematic algebraic approach. Let’s go through the steps in detail:

Step 1: Represent the Repeating Decimal as a Variable

Let ( x = 0.\overline{8} ), which means:

[
x = 0.88888\ldots
]

Step 2: Eliminate the Repeating Part

Now, to eliminate the repeating decimal, we multiply both sides of the equation by 10, because the decimal repeats after one digit (the 8). This gives:

[
10x = 8.88888\ldots
]

Step 3: Set Up a System of Equations

We now have two equations:

1. ( x = 0.88888\ldots )
2. ( 10x = 8.88888\ldots )

Step 4: Subtract the First Equation from the Second

Subtract equation 1 from equation 2 to eliminate the repeating decimals:

[
(10x – x) = (8.88888\ldots – 0.88888\ldots)
]

This simplifies to:

[
9x = 8
]

Step 5: Solve for ( x )

Now, solve for ( x ) by dividing both sides of the equation by 9:

[
x = \frac{8}{9}
]

Final Answer:

Thus, the fractional form of ( 0.\overline{8} ) is ( \frac{8}{9} ).

Explanation:

The reason why ( 0.\overline{8} ) equals ( \frac{8}{9} ) is that repeating decimals are often equivalent to fractions, which can be determined by the algebraic method shown above. This approach works for any repeating decimal, not just for 0.8 repeating. It’s important to note that some people might assume that ( 0.\overline{8} ) is simply 1, because ( 0.\overline{9} = 1 ). However, through the above algebraic steps, we see that ( 0.\overline{8} ) is strictly equal to ( \frac{8}{9} ), a value that is slightly less than 1 but very close to it.