Convert the following repeating decimal to a fraction in simplest form.
., start overline, 7, 8, end overline
Let x = ., start overline, 7, 8, end overline
10x = 7.87878...
100x = 78.787878...
Subtracting the equation (10x = 7.87878...) from the equation (100x = 78.787878...), we get:
90x = 78.787878... - 7.87878...
90x = 70.90909...
Now we solve for x by dividing both sides of the equation by 90:
x = (70.90909...) / 90
To convert the repeating decimal to a fraction, we set it equal to another variable, let's say y:
y = (70.90909...) / 90
To eliminate the repeating decimal, we multiply both sides of the equation by 100, since there are two digits that repeat:
100y = (70.90909...) / 90 * 100
100y = 7090.90909...
Next, we subtract the original equation from the equation after multiplying by 100:
100y - y = 7090.90909... - (70.90909...) / 90 * 100
99y = 7090
Dividing both sides of the equation by 99 gives us:
y = 7090 / 99
To simplify the fraction, we find the greatest common divisor of 7090 and 99, which is 1. Therefore, the fraction in simplest form is:
y = 7090 / 99