Question

if a repeating decimal has a repeating cycle of three digits it will convert to a rational number with what denominator?
a. 100
b. 999
c. 1,000
d. 99

Answer 1

If a repeating decimal has a repeating cycle of three digits, it means that the decimal repeats every three digits after the decimal point. To convert this repeating decimal to a rational number, we can write it as a fraction.

Let's assume the repeating decimal is represented by "0.abcabcabc...".

To convert it to a fraction, we define a variable x as "0.abc" and subtract it from the repeating decimal:

x = 0.abc
1000x = abc.abcabc...

Now, subtracting x from 1000x, we get:

1000x - x = abc.abcabc... - 0.abc
999x = abc

We can see that the resulting fraction is abc/999, which means the numerator is abc and the denominator is 999.

Therefore, the repeating decimal with a repeating cycle of three digits will convert to a rational number with the denominator 999.

The correct answer is b. 999.