If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator? Please explain your answer, or guide me through it step by step. (=

Answer 1

To convert a repeating decimal with a repeating cycle of three digits into a rational number, we need to determine the denominator of the final fraction.

Let's take a general example of a repeating decimal with a repeating cycle of three digits. We can represent this repeating decimal as:

0.xyzxyzxyz...

Where x, y, and z represent the three repeating digits.

To convert this repeating decimal into a rational number, we need to express it as a fraction. Let's call this fraction "a/b", where "a" and "b" are integers.

To begin, we can multiply the repeating decimal by a power of 10 that shifts the repeating cycle to the left of the decimal point. In this case, multiplying by 10^3 (= 1000) works because the repeating cycle has three digits. So, we have:

1000 * 0.xyzxyzxyz... = xyz.xyzxyzxyz...

Notice that the repeating cycle now starts immediately after the decimal point.

Next, we subtract the original repeating decimal from this new number to eliminate the repeating cycle. Subtracting these two numbers, we get:

(1000 * 0.xyzxyzxyz...) - (0.xyzxyzxyz...) = xyz.xyzxyzxyz... - 0.xyzxyzxyz... = xyz

The repeating cycle is eliminated, leaving us with the three-digit number "xyz".

Now, we equate this number "xyz" to the fraction a/b:

xyz = a/b

To find the value of "b", we recognize that the numerator "a" is just the number "xyz", while the denominator "b" is equal to the power of 10 that corresponds to the number of digits in the repeating cycle. In this case, since the repeating cycle has three digits, the denominator "b" will be 10^3 (= 1000).

So, the repeating decimal with a repeating cycle of three digits will convert to a rational number with a denominator of 1000.

Note that in this explanation, we assumed that the repeating decimals are purely periodic, meaning the cycle repeats infinitely and does not have any non-repeating digits before or after the cycle. If a non-repeating part or additional digits are present, they would need to be included in the calculations.