Convert the following decimal numbers in the form of pq.
- (i) 12.6
- (ii) 0.0120
- (iii) 3.052
- (iv) 1.235
- (v) 0.23
- (vi) 2.05
- (vii) 2.125
- (viii) 3.125
- (ix) 2.1625
Step-by-step solution
Idea: Terminating decimals: write over a power of 10 and simplify. Repeating decimals: let x be the number; multiply by 10m (m = non-repeating digits after the point) and then by 10n more (n = digits in the repeating block); subtract the two equations so the endless tail cancels.
(i) 12.6
- One decimal place: 12.6 = 12610 = 635.½ mark
(ii) 0.0120
- 0.0120 = 0.012 = 121000 = 3250 (divide by 4).½ mark
(iii) 3.052
- x = 3.052. 10x = 30.52 and 1000x = 3052.52. Subtract: 990x = 3022, x = 3022990 = 1511495.½ mark
(iv) 1.235
- x = 1.235. 10x = 12.35 and 1000x = 1235.35. Subtract: 990x = 1223, x = 1223990.½ mark
(v) 0.23
- x = 0.23. 100x = 23.23. Subtract: 99x = 23, x = 2399.½ mark
(vi) 2.05
- x = 2.05. 10x = 20.5 and 100x = 205.5. Subtract: 90x = 185, x = 18590 = 3718.½ mark
(vii) 2.125
- x = 2.125. 100x = 212.5 and 1000x = 2125.5. Subtract: 900x = 1913, x = 1913900.½ mark
(viii) 3.125
- x = 3.125. 100x = 312.5 and 1000x = 3125.5. Subtract: 900x = 2813, x = 2813900.½ mark
(ix) 2.1625
- x = 2.1625: four digits repeat, so multiply by 104: 10000x = 21625.1625.½ mark
- Subtract: 9999x = 21623, x = 216239999 (no common factor: 9999 = 32 × 11 × 101, and 21623 is divisible by none of 3, 11, 101).½ mark
Check: Divide back: 1511 ÷ 495 = 3.05252…, 1223 ÷ 990 = 1.23535…, 37 ÷ 18 = 2.0555…, 1913 ÷ 900 = 2.12555…, 21623 ÷ 9999 = 2.16251625… ✓.
Answer to write in the exam
(i)
12.6 = 12610
∴ = 635
(ii)
0.0120 = 121000
∴ = 3250
(iii)
x = 3.052
10x = 30.52, 1000x = 3052.52
990x = 3022
∴ x = 3022990 = 1511495
(iv)
x = 1.235
10x = 12.35, 1000x = 1235.35
990x = 1223
∴ x = 1223990
(v)
x = 0.23
100x = 23.23
99x = 23
∴ x = 2399
(vi)
x = 2.05
10x = 20.5, 100x = 205.5
90x = 185
∴ x = 18590 = 3718
(vii)
x = 2.125
100x = 212.5, 1000x = 2125.5
900x = 1913
∴ x = 1913900
(viii)
x = 3.125
100x = 312.5, 1000x = 3125.5
900x = 2813
∴ x = 2813900
(ix)
x = 2.1625
10000x = 21625.1625
9999x = 21623
∴ x = 216239999
Common mistakes that cost marks
- Putting the bar over the wrong digits: in 3.052 the 0 does not repeat, so it is not 3.052.
- Multiplying by the wrong power of 10: in (iv) the block 35 has two digits, so the second multiplier is 100, not 10.
- Treating 0.0120 as 0.12: the zero after the point matters (0.012 = 121000).
How this can come in the exam
1.2 in the form pq is
- 65
- 119
- 129
- 1210
Show answer
(B) 119
10x − x = 12.2 − 1.2 = 11, so x = 119.
Express 0.318 in the form pq.
Show answer
x = 0.318; 10x = 3.18 (1 mark); 1000x = 318.18 (1 mark); 990x = 315, x = 315990 = 722 (1 mark).Try one yourself
Convert 4.06 and 0.132 into the form pq.
Show answer
4.06: 90x = 406 − 40 = 366, x = 6115. 0.132: 999x = 132, x = 44333.