Convert 2.357 into the form pq.
Answer: Let x = 2.357. 100x = 235.7, 1000x = 2357.7. Subtracting, 900x = 2122, so x = 2122900 = 1061450.
Step-by-step solution
Idea: ’35’ does not repeat (2 digits) and ‘7’ repeats (1 digit). Multiply by 102 to move the non-repeating digits in front of the point, then by 101 more to move one repeating block. Subtract to cancel the tail.
- Let x = 2.357 = 2.35777…½ mark
- Two non-repeating digits (3, 5): multiply by 102 = 100 → 100x = 235.7.½ mark
- One repeating digit (7): multiply by 10 more → 1000x = 2357.7.½ mark
- Subtract: 1000x − 100x = 2357.7 − 235.7 = 2122, so 900x = 2122.1 mark
- x = 2122900 = 1061450 (divide by 2).½ mark
2.357 = 1061/450
Check: 1061 ÷ 450 = 2.357777… ✓.
Answer to write in the exam
Let x = 2.357
100x = 235.7 … (1)
1000x = 2357.7 … (2)
(2) − (1): 900x = 2122
∴ x = 2122900 = 1061450
Common mistakes that cost marks
- Multiplying by 10 and 100 (not 100 and 1000), so the tails do not line up.
- Forgetting the whole-number part 2 and finding 0.357 instead.
- Subtraction slip: 2357 − 235 = 2122, not 2112 or 2132.
How this can come in the exam
MCQ (1 mark)
To convert 4.125 by this method, the two equations subtracted are
- 10x and x
- 100x and 10x
- 1000x and 100x
- 1000x and x
Show answer
(C) 1000x and 100x
Two non-repeating digits (×100) and one repeating digit (×10 more = ×1000).
Short answer (3 marks)
Express 1.243 in the form pq.
Show answer
100x = 124.3 (1 mark); 1000x = 1243.3 (1 mark); 900x = 1119, x = 1119900 = 373300 (1 mark).Try one yourself
Convert 0.583 into the form pq.
Show answer
900x = 583 − 58 = 525, x = 525900 = 712.
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