Locate the following rational numbers on the number line.
- (i) 0.532
- (ii) 1.15
Step-by-step solution
Idea: Each decimal digit tells which tenth of the current interval the number is in. Enlarge that piece, divide it into ten again, and repeat. A repeating decimal never ends, so we stop when the point is located closely enough, or use its fraction form.
(i) 0.532
- 0.532 lies between 0.5 and 0.6. Divide 0.5 to 0.6 into 10 equal parts (0.51, 0.52, …).½ mark
- It lies between 0.53 and 0.54. Enlarge that piece and divide it into 10 equal parts (0.531, 0.532, …).½ mark
- Mark the 2nd point after 0.53: that is 0.532 (see the diagram).1 mark
(ii) 1.15
- 1.15 = 1.1555…. As a fraction: 10x = 11.5, 100x = 115.5, so 90x = 104 and x = 10490 = 5245.½ mark
- It lies between 1.1 and 1.2; enlarge and divide into hundredths: it lies between 1.15 and 1.16.½ mark
- Enlarge 1.15 to 1.16 into thousandths: 1.1555… lies between 1.155 and 1.156, a little past the 5th mark (exactly 59 of the way from 1.15 to 1.16, since 1.1555… − 1.15 = 0.00555… = 59 × 0.01). Mark it there (see the diagram).1 mark
Check: 5245 = 1.15555… ✓ (52 ÷ 45: 1 r 7; 70 ÷ 45 = 1 r 25; 250 ÷ 45 = 5 r 25; …).
Answer to write in the exam
(i)
0.5 < 0.532 < 0.6: divide 0.5–0.6 into 10 parts
0.53 < 0.532 < 0.54: divide 0.53–0.54 into 10 parts
∴ 0.532 is the 2nd mark after 0.53
(ii)
1.15 = 5245 = 1.1555…
1.1 < 1.1555… < 1.2; then 1.15 < 1.1555… < 1.16
1.155 < 1.1555… < 1.156
∴ Mark just past 1.155, 59 of the way from 1.15 to 1.16
Common mistakes that cost marks
- Placing 0.532 between 0.5 and 0.6 and stopping there; the question needs the exact point, so magnify until the third decimal.
- Reading 1.15 as 1.15 exactly. The 5s continue, so the point is slightly beyond 1.155.
- Dividing an interval into 9 or 11 parts instead of 10 when using decimals.
How this can come in the exam
When locating 2.374 by successive magnification, the second enlarged interval is
- 2 to 3
- 2.3 to 2.4
- 2.37 to 2.38
- 2.374 to 2.375
Show answer
(B) 2.3 to 2.4
First 2 to 3 is divided into tenths; the number lies in 2.3 to 2.4, which is enlarged next.
Describe how to locate 3.76 up to three decimal places on the number line.
Show answer
3.76 = 3.7666…: it lies between 3.7 and 3.8, then between 3.76 and 3.77 (1 mark). Divide 3.76 to 3.77 into 10 parts; the point lies between 3.766 and 3.767 (1 mark).Try one yourself
Between which two thousandths does 4.28 lie?
Show answer
4.2888…: between 4.288 and 4.289.
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