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 \(x = 0.484848...\)
-
Since the repeating part, "48," has two digits, we multiply \(x\) by \(100\) to shift the decimal point two places to the right: \[ 100x = 48.484848... \]
-
Now, we can set up an equation: \[ 100x = 48.484848... \] \[ x = 0.484848... \]
-
Next, we subtract the second equation from the first: \[ 100x - x = 48.484848... - 0.484848... \] \[ 99x = 48 \]
-
Now solve for \(x\): \[ x = \frac{48}{99} \]
-
To simplify the fraction, we find the greatest common divisor (GCD) of 48 and 99. The GCD is 3.
-
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 \(\frac{16}{33}\).