Convert the following repeating decimal to a fraction in simplest form.

Question

point, 7, start overline, 8, end overline
.7
8

Answer 1

Let's represent the repeating decimal as x:

x = 0.78̅

To convert this repeating decimal to a fraction, we can first multiply both sides of the equation by a power of 10 to eliminate the repeating part. Since there is 1 digit repeating (8), we will multiply by 10:

10x = 7.88̅

Next, we subtract the original equation from the new equation to eliminate the repeating part:

10x - x = 7.88̅ - 0.78̅

9x = 7.1

Now, we can solve for x by dividing both sides of the equation by 9:

x = 7.1/9

To simplify the fraction 7.1/9, we can divide both the numerator and denominator by 0.1:

x = (7.1/0.1)/(9/0.1)
x = 71/10 divided by 9/0.1
x = (71/10) * (10/9)
x = 71/9

Therefore, the repeating decimal 0.78̅ can be expressed as the fraction 71/9.