Question

what's .48 (its all repeating) as a fraction in simplest form

Answer 1

To convert the repeating decimal \(0.48\overline{48}\) to a fraction, we will use the method of multiplying by a power of 10.

Let \(x = 0.484848...\)

1. Since the repeating part, "48," has two digits, we multiply \(x\) by \(100\) to shift the decimal point two places to the right:
\[
100x = 48.484848...
\]

2. Now, we can set up an equation:
\[
100x = 48.484848...
\]
\[
x = 0.484848...
\]

3. Next, we subtract the second equation from the first:
\[
100x - x = 48.484848... - 0.484848...
\]
\[
99x = 48
\]

4. Now solve for \(x\):
\[
x = \frac{48}{99}
\]

5. To simplify the fraction, we find the greatest common divisor (GCD) of 48 and 99. The GCD is 3.

6. Now we simplify the fraction:
\[
x = \frac{48 \div 3}{99 \div 3} = \frac{16}{33}
\]

Thus, \(0.48\overline{48}\) as a fraction in simplest form is \(\frac{16}{33}\).