what's .38 (the 8 repeating) as a fraction in simplest form

Answer 1

To convert the repeating decimal \(0.38\overline{8}\) (which means \(0.388888...\)) into a fraction, we can use the following method:

Let \(x = 0.38888...\).

To eliminate the repeating part, we can multiply by 10 and then again by 10 to move the decimal point two places to the right (for the \(0.38\)) and then to the right once more to account for the repeat:

1. Multiply \(x\) by 10: \[ 10x = 3.8888... \]

2. Multiply \(x\) by 100 (to shift the decimal point two places): \[ 100x = 38.8888... \]

Now we have two equations:

1. \(10x = 3.8888...\)
2. \(100x = 38.8888...\)

Next, we can subtract the first equation from the second equation:

\[ 100x - 10x = 38.8888... - 3.8888... \]

This simplifies to:

\[ 90x = 35 \]

Now, divide both sides by 90:

\[ x = \frac{35}{90} \]

The fraction \(\frac{35}{90}\) can be simplified by finding the greatest common divisor (GCD) of 35 and 90. The GCD is 5.

Now, divide both the numerator and the denominator by 5:

\[ x = \frac{35 \div 5}{90 \div 5} = \frac{7}{18} \]

Thus, \(0.38\overline{8}\) as a fraction in simplest form is \(\frac{7}{18}\).