Give 23/7 as a decimal to five decimal places without long division, and explain how the repeating block of the sevenths tells you every digit.

Give 23/7 as a decimal to five decimal places without long division, and explain how the repeating block of the sevenths tells you every digit.

Approach: Split off the integer part first, then recall the six-digit repeating block of the sevenths and enter it at the position set by the remaining numerator.

3.28571. Split 23 into 21 + 2, so the mixed number is 3 + 2/7 and only the fraction needs work. Every seventh uses the same repeating block 142857, entered at the point set by the numerator: 1/7 = 0.142857, 2/7 = 0.285714, 3/7 = 0.428571, 4/7 = 0.571428, 5/7 = 0.714285, 6/7 = 0.857142. So 2/7 is 0.285714 repeating, which makes 23/7 equal to 3.285714 repeating. To five places the sixth decimal is 4, so the fifth digit stays at 1 and the answer is 3.28571. The entry point follows the cyclic order of the leading digits 1, 2, 4, 5, 7, 8, which matches the numerators 1 through 6 in order.

Follow-up: What are the first five decimal places of 23/13, and how long is the repeating block there?

Key concepts: repeating decimal, cyclic block, mixed number.