This set has twelve original questions, ordered from easier to harder, covering all five lessons in precision bounds and measurement. Questions 1 to 3 are warm-ups on significant figures, 4 to 6 are intervals, 7 to 10 are bounds of calculations, and 11 and 12 test accuracy.
Attempt each question on paper and write your working as you would in an exam. Then open the answer, mark the ones you got wrong and use the routing list at the end.
Questions
1. Write 0.0060954 to 3 significant figures.
Show answer
The first significant figure is 6. The first three figures are 6, 0, 9 and the next digit is 5, so 609 rounds up to 610. Keep the final zero, because it is the third significant figure.
0.00610
2. Write 85 600 to 2 significant figures.
Show answer
The first two figures are 8 and 5. The next digit is 6, so 85 rounds up to 86. Keep the size with zeros.
86 000
3. A runner covers 345 m in 27 s. Find the average speed in m/s, giving a sensible number of significant figures.
Show answer
345 ÷ 27 = 12.777… The data 345 has 3 significant figures and 27 has 2, so the weaker value limits the result to 2 significant figures.
13 m/s
4. A time is 3.6 s to 1 decimal place. Write the interval for the true time, t.
Show answer
The unit is 0.1, so the half-unit is 0.05. Lower bound 3.6 − 0.05 = 3.55. Upper bound 3.6 + 0.05 = 3.65.
3.55 ≤ t < 3.65 s
5. The number of people at a match is 2400 to 2 significant figures. Write the interval for the true number, n.
Show answer
The second figure is the 4 in the hundreds place, so the unit is 100 and the half-unit is 50. Lower bound 2350. Upper bound 2450.
2350 ≤ n < 2450
6. A length is 0.85 m to 2 decimal places. Write the interval, and state the width of the interval.
Show answer
The unit is 0.01, so the half-unit is 0.005. Lower bound 0.845. Upper bound 0.855. Width 0.855 − 0.845 = 0.01.
0.845 ≤ L < 0.855 m, width 0.01 m
7. Two lengths are a = 15.2 cm and b = 8.7 cm, each to 1 decimal place. Find the bounds of a + b and of a − b.
Show answer
Intervals: 15.15 ≤ a < 15.25 and 8.65 ≤ b < 8.75.
Sum: lower 15.15 + 8.65 = 23.80, upper 15.25 + 8.75 = 24.00.
Difference: lower 15.15 − 8.75 = 6.40, upper 15.25 − 8.65 = 6.60.
23.80 ≤ a + b < 24.00 and 6.40 ≤ a − b < 6.60
8. A wall is 3.4 m long and a cabinet is 1.8 m wide, each to 1 decimal place. The space left beside the cabinet is s metres. Find the bounds of s, and say whether a 1.6 m shelf is certain to fit in the space.
Show answer
Intervals: 3.35 ≤ wall < 3.45 and 1.75 ≤ cabinet < 1.85. Space = wall − cabinet.
Lower bound 3.35 − 1.85 = 1.50. Upper bound 3.45 − 1.75 = 1.70.
1.50 ≤ s < 1.70 m. The space could be as small as 1.50 m, which is less than 1.6 m, so the shelf is not certain to fit.
9. A rectangle is 8 cm by 5 cm, each to the nearest centimetre. Find the bounds of its area.
Show answer
Intervals: 7.5 ≤ length < 8.5 and 4.5 ≤ width < 5.5. Lower bound 7.5 × 4.5 = 33.75. Upper bound 8.5 × 5.5 = 46.75.
33.75 ≤ area < 46.75 cm²
10. A metal block has mass 54 g (nearest gram) and volume 20 cm³ (nearest cm³). Density = mass ÷ volume. Find the bounds of the density, and the accuracy justified.
Show answer
Intervals: 53.5 ≤ mass < 54.5 and 19.5 ≤ volume < 20.5. For a quotient, the lower bound uses the smallest top and largest bottom.
Lower bound 53.5 ÷ 20.5 = 2.609… Upper bound 54.5 ÷ 19.5 = 2.794…
To 2 significant figures these are 2.6 and 2.8, which differ. To 1 significant figure both are 3.
2.609… ≤ density < 2.794… g/cm³, so the density is 3 g/cm³ to 1 significant figure.
11. A student writes a length as 7.50 m and another as 7.5 m (1 decimal place). Write both intervals and explain which length is known more precisely.
Show answer
7.50 to 2 decimal places: 7.495 ≤ x < 7.505, width 0.01. 7.5 to 1 decimal place: 7.45 ≤ x < 7.55, width 0.1.
7.50 m is known more precisely, because its interval is ten times narrower.
12. The lower bound of a quantity is 41.96 and its upper bound is 42.04. To how many significant figures can it be given? State the value.
Show answer
To 4 significant figures the bounds are 41.96 and 42.04, which differ. To 3 significant figures both round to 42.0. To 2 significant figures both round to 42, so 3 is the largest that agrees.
42.0 (3 significant figures)
If you got these wrong
Match the type of error to the lesson that repairs it.
- Questions 1, 2 and 3, rounding and choosing figures: go to choose sensible significant figures.
- Questions 4, 5 and 6, building the interval: go to recover an interval from a rounded measurement.
- Questions 7 and 8, sums and differences: go to bound a sum and a difference.
- Questions 9 and 10, products and quotients: go to bound a product with positive measurements.
- Questions 11 and 12, accuracy claims: go to distinguish accuracy from displayed decimal places.
Record each slip in the mistake log and retest queue with the error type, so a fresh question can be retested later. The bounds and rounding explainer draws each interval so you can check your own, and the non-calculator working trainer is useful for the exact steps.
When the same error returns after a lesson, a teacher in online one-to-one Mathematics tuition can trace it through your written working and set questions that test exactly that step.