Convert the following repeating decimal to a fraction in simplest form.
point, 6, start overline, 3, end overline
.6
3
To convert the repeating decimal .6¯3 to a fraction, we will first define the variable x as the repeating decimal:
x = .6¯3
Since the decimal repeats indefinitely, we can multiply both sides of the equation by 1000 to eliminate the repeating decimal:
1000x = 633.333¯
Next, we subtract the original equation from the multiplied equation, which will eliminate the repeating decimal:
1000x - x = 633.333¯ - .6¯3
Simplifying both sides of the equation gives:
999x = 633.333 - .6
999x = 632.733
To isolate x, we divide both sides of the equation by 999:
x = 632.733/999
To simplify the fraction, we find the greatest common divisor (GCD) of the numerator and denominator:
GCD(632.733, 999) = 31
Dividing both the numerator and denominator by the GCD:
632.733/999 = 632.733/999 ÷ 31/31 = (632.733÷31)/(999÷31) = 20.415/32.29
The fraction 20.415/32.29 is already in simplest form, as there are no common factors greater than 1 that can divide both the numerator and denominator.