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if a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator? 100 1...Question
If a repeating decimal has a repeating cycle of three digits, it will convert to a rational number with what denominator? (1
100
99
1,000
999
100
99
1,000
999
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Answered by
GPT-4o mini
A repeating decimal with a repeating cycle of three digits will convert to a rational number with a denominator of 999.
To explain why, consider a repeating decimal such as \(0.abcabcabc...\) where "abc" represents the three digits that repeat. When converting this repeating decimal to a fraction, the formula generally used is:
\[
\frac{N}{999}
\]
where \(N\) is the integer formed by the repeating digits "abc". The reason for using 999 is derived from the fact that the denominator is \(10^n - 1\), where \(n\) is the length of the repeating part—in this case, 3. Since \(10^3\) is 1000, we subtract 1 to find the denominator:
\[
10^3 - 1 = 1000 - 1 = 999
\]
Therefore, the correct answer is **999**.
To explain why, consider a repeating decimal such as \(0.abcabcabc...\) where "abc" represents the three digits that repeat. When converting this repeating decimal to a fraction, the formula generally used is:
\[
\frac{N}{999}
\]
where \(N\) is the integer formed by the repeating digits "abc". The reason for using 999 is derived from the fact that the denominator is \(10^n - 1\), where \(n\) is the length of the repeating part—in this case, 3. Since \(10^3\) is 1000, we subtract 1 to find the denominator:
\[
10^3 - 1 = 1000 - 1 = 999
\]
Therefore, the correct answer is **999**.
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