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You know that (1)/(7)=0. bar(142857). Can you predict what the decimal expansions of (2)/(7),(3)/(7) (4)/(7),(5)/(7),(6)/(7) are, without actually doing the long division? If so, how? [Hint: Study the remainders while finding the value of (1)/(7) carefully.]

You know that 17=0.142857‾ \frac{1}{7}=0 . \overline{142857} . Can you predict what the decimal expansions of 27,37 \frac{2}{7}, \frac{3}{7} 47,57,67 \frac{4}{7}, \frac{5}{7}, \frac{6}{7} are, without actually doing the long division? If so, how?\newline[Hint: Study the remainders while finding the value of 17 \frac{1}{7} carefully.]

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Full solution

Q. You know that 17=0.142857‾ \frac{1}{7}=0 . \overline{142857} . Can you predict what the decimal expansions of 27,37 \frac{2}{7}, \frac{3}{7} 47,57,67 \frac{4}{7}, \frac{5}{7}, \frac{6}{7} are, without actually doing the long division? If so, how?\newline[Hint: Study the remainders while finding the value of 17 \frac{1}{7} carefully.]
  1. Identify Repeating Pattern: Notice the repeating pattern in (1)/(7)=0.142857(1)/(7) = 0.142857. The decimal repeats every 66 digits.
  2. Shift Pattern for (2)/(7)(2)/(7): Realize that multiplying the repeating decimal by 22 shifts the pattern: (2)/(7)(2)/(7) should start with the second digit of the pattern and cycle through.
  3. Calculate (2)/(7)(2)/(7): Calculate (2)/(7)(2)/(7) as 0.2857140.285714 by shifting the pattern of (1)/(7)(1)/(7) to start from the second digit.
  4. Shift Pattern for (3)/(7)(3)/(7): Apply the same logic to (3)/(7)(3)/(7), which should start with the third digit of the pattern.
  5. Calculate (3)/(7)(3)/(7): Calculate (3)/(7)(3)/(7) as 0.4285710.428571 by shifting the pattern of (1)/(7)(1)/(7) to start from the third digit.
  6. Continue for (4)/(7)(4)/(7): Continue the pattern for (4)/(7)(4)/(7), starting with the fourth digit of the pattern.
  7. Calculate (4)/(7)(4)/(7): Calculate (4)/(7)(4)/(7) as 0.5714280.571428 by shifting the pattern of (1)/(7)(1)/(7) to start from the fourth digit.
  8. Move to (5)/(7)(5)/(7): Move to (5)/(7)(5)/(7), starting with the fifth digit of the pattern.
  9. Calculate (5)/(7)(5)/(7): Calculate (5)/(7)(5)/(7) as 0.7142850.714285 by shifting the pattern of (1)/(7)(1)/(7) to start from the fifth digit.
  10. Predict (6)/(7)(6)/(7): Finally, predict (6)/(7)(6)/(7) by starting with the sixth digit of the pattern.
  11. Calculate (6)/(7)(6)/(7): Calculate (6)/(7)(6)/(7) as 0.8571420.857142 by shifting the pattern of (1)/(7)(1)/(7) to start from the sixth digit.

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