Question

what is the fraction form of 0.8412? (12 repeats.)

Answer 1

To convert the repeating decimal \(0.84121212...\) (where "12" repeats) into a fraction, we can follow these steps:

1. Let \(x = 0.84121212...\)

2. Recognize that the decimal part has a repeating section of "12", which is 2 digits long. To eliminate the repeating part, we can multiply \(x\) by \(100\) (to shift the decimal point two places to the right):

\[
100x = 84.12121212...
\]

3. Also, note that we can multiply \(x\) by \(10\) to move just past the non-repeating part:

\[
10x = 8.4121212...
\]

4. Next, we subtract the second equation from the first:

\[
100x - 10x = 84.12121212... - 8.4121212...
\]

This simplifies to:

\[
90x = 75.7090909...
\]

5. To deal with the repeating part, we can also express \(75.7090909...\) in a similar way. Let's first isolate the fraction part. Let \(y = 0.090909...\), which is \(0.\overline{09}\).

\[
y = 0.090909...
\]

To convert \(y\) to a fraction:

\[
100y = 9.090909...
\]

Subtracting gives:

\[
100y - y = 9.090909... - 0.090909...
\]
\[
99y = 9 \implies y = \frac{9}{99} = \frac{1}{11}
\]

6. Now, we can express \(75.7090909...\) as:

\[
75 + y = 75 + \frac{1}{11} = \frac{75 \times 11 + 1}{11} = \frac{825 + 1}{11} = \frac{826}{11}
\]

7. Now we can substitute this back into our equation:

\[
90x = \frac{826}{11}
\]

8. Solving for \(x\):

\[
x = \frac{826}{11 \times 90} = \frac{826}{990}
\]

9. Finally, we simplify \(\frac{826}{990}\) by finding the greatest common divisor (GCD) of \(826\) and \(990\).

The GCD of \(826\) and \(990\) is \(2\):

\[
\frac{826 \div 2}{990 \div 2} = \frac{413}{495}
\]

So the fraction form of \(0.8412\) (with "12" repeating) is:

\[
\boxed{\frac{413}{495}}
\]