Question

0.78 repeating decimal into a simplified fraction

Answer 1

To convert the repeating decimal \( 0.78\overline{78} \) into a fraction, we can follow these steps:

Let \( x = 0.787878...\)

Now, to eliminate the repeating part, we can multiply \( x \) by 100 (since the repeating part consists of two digits):

\[
100x = 78.787878...
\]

Next, we can set up the equation:

\[
100x = 78.78\overline{78}
\]
\[
x = 0.78\overline{78}
\]

Now, we subtract the first equation from the second:

\[
100x - x = 78.787878... - 0.787878...
\]

This simplifies to:

\[
99x = 78
\]

Now, solve for \( x \):

\[
x = \frac{78}{99}
\]

Next, we need to simplify the fraction \( \frac{78}{99} \). To do this, we can divide both the numerator and the denominator by their greatest common divisor (GCD).

The GCD of 78 and 99 is 3. Dividing both by 3 gives us:

\[
x = \frac{78 \div 3}{99 \div 3} = \frac{26}{33}
\]

Thus, the repeating decimal \( 0.78\overline{78} \) as a simplified fraction is:

\[
\frac{26}{33}
\]