An expert question on cycles for P5–P6, with a step-by-step solution, diagrams and the answer. Cycle questions appear in the Hua Xia Cup from the P1 first round to the P5 papers: which comes in a given place, which day of the week, which last digit.
The question
Ching works out 23 ÷ 370 on a calculator, but the screen shows only the first few decimal places.
Her teacher says the answer is a recurring decimal: a few digits after the point that don't repeat, then a group of digits repeating for ever.
Among the 1st to the 1201st decimal places, how many times does the digit 2 appear?
Show the hints (3)
💡Hint 1: First do 23 ÷ 370 by long division. From which decimal place do the digits start repeating?
💡Hint 2: The 1st digit doesn't repeat; after that every 3 repeat. Any 2 there? How many 2s in each cycle?
💡Hint 3: Leaving out the 1st digit, there are 1200 places in groups of 3. Count the whole cycles, then see which places of a cycle the leftover ones are.
Worked solution
Have a go first, then read the solution step by step.
1.Long division of 23 ÷ 370: after the point come 0, then 6, 2, 1 over and over (from the 2nd place, every 3 digits repeat).
2.In the first 1 (0), 2 appears 0 times; in each cycle (6, 2, 1) it is in place 2: 1.
3.Leaving out the first 1: 1201 − 1 = 1200 places. 1200 ÷ 3 = 400 exactly: 400 whole cycles give 400 × 1 = 400.
4.In all 0 + 400 = 400.
⚠️Common mistake: treating the first 1 digit as part of the cycle and grouping from the 1st place.
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