It repeats: The division never reaches a remainder of 0, but the sequence of digits begins to loop infinitely. 511 = 0.454545 … = 0.45.

Step-by-step solution

Idea: Do the long division and watch the remainders. As soon as a remainder comes back, the digits from that point repeat for ever. The bar is written over the block that repeats.

1. 50 ÷ 11 = 4 remainder 6; 60 ÷ 11 = 5 remainder 5; 50 ÷ 11 = 4 remainder 6 again.1 mark
2. The remainder 5 has come back (we again divide 50), so the digits 4, 5 repeat for ever: 511 = 0.4545… = 0.45. The remainder is never 0, so the decimal does not terminate.1 mark

Check: 0.45 = 4599 = 511 ✓.

Answer to write in the exam

Common mistakes that cost marks

• Putting the bar over only the 5 (0.45 = 0.4555…). Both digits 4 and 5 repeat.
• Writing 0.45 and stopping. That is a terminating decimal and is not equal to 511.
• Bringing down a 0 and forgetting to write a digit in the quotient when the divisor does not go in.

How this can come in the exam

Try one yourself

Find the decimal expansion of 211.

More questions like this

• Try to find the decimal expansions of 103 and 1112. What do you observe about the repetition of the digits after the decimal point?
• Why do some rational numbers have repeating decimal representations? Imagine calculating 17 using long division. You are dividing by 7. What are the possible remainders at each step? They can only be 1, 2, 3, 4, 5, or 6 (why not 0?).
• The decimal expansion of pq will be terminating precisely when the prime factors of q are only 2, only 5 or both 2 and 5. Can you explain why?
• Convert 0.35 into the form pq.
• Convert 0.6 into the form pq.

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