Convert 0.16 into the form pq.

Step-by-step solution

Idea: Here 1 does not repeat and 6 does. First multiply by 10 to move the non-repeating 1 in front of the point; then multiply by 10 again to move one full repeating block. The two results have the same tail, so subtracting them cancels it.

1. Let x = 0.16 = 0.1666…½ mark
2. One non-repeating digit: multiply by 10¹ = 10 → 10x = 1.6. Now the repeating part starts right after the point.½ mark
3. One repeating digit: multiply by 10 again → 100x = 16.6.½ mark
4. Subtract: 100x − 10x = 16.6 − 1.6 = 15, so 90x = 15.1 mark
5. x = 1590 = 16.½ mark

Check: 1 ÷ 6 = 0.1666… ✓.

Answer to write in the exam

Common mistakes that cost marks

• Multiplying only by 10 and subtracting x: 9x = 1.5 still has a decimal. Multiply by 10 and by 100 so that 100x − 10x gives the whole number 15.
• Reading 0.16 as 0.16 = 0.1616… and getting 1699.
• Subtracting x from 100x: 99x = 16.5 leaves a decimal and is easy to get wrong.

How this can come in the exam

Try one yourself

Convert 0.41 into the form pq.

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