Limits of accuracy.
A rounded measurement hides a range. Use that range to bound a calculation.
Write error intervals · identify rounding units · bound perimeter, area and speed
A rounded value represents an interval
A displayed measurement is not always the exact original value. A length reported as 8 cm to the nearest centimetre could have come from several measurements close to 8, rather than exactly 8.
The lower and upper bounds describe the edge of that possible interval. They sit half a rounding unit below and above the reported value. We are describing uncertainty caused by rounding, not claiming the measuring instrument definitely made a particular error.
A measurement of 7.5 cm to the nearest 0.1 cm does not mean the length is exactly 7.5 cm. It represents every value that rounds to 7.5.
Upper bound = reported value + 12 × rounding unit
The lower bound is included, and the upper bound is excluded under the usual rounding convention used here. The upper bound is the boundary 7.55, not 7.54999…
Half the rounding unit is 0.5. Subtract it for the lower bound; add it for the upper bound.
Pause and explainCould 8.4 cm round to 8 cm to the nearest centimetre?
Yes. It lies below the halfway point 8.5 cm and is nearer to 8 than to 9.
Identify the rounding unit first
The phrase “nearest” tells you which rounding unit to use. Nearest metre has unit 1 m; nearest tenth has unit 0.1; nearest hundred has unit 100. Half that unit is the distance to either boundary.
Keep the stated units while finding the interval. For conventional rounding of positive measurements, include the lower boundary but exclude the upper: the upper halfway point would round to the next displayed value. The inequality signs record this distinction.
| Reported value | Rounding unit | Error interval |
|---|---|---|
| 12 m, nearest metre | 1 m | 11.5 ≤ L < 12.5 |
| 3.40 kg, 2 d.p. | 0.01 kg | 3.395 ≤ m < 3.405 |
| 2500 people, nearest 100 | 100 people | 2450 ≤ n < 2550 |
| 0.037, 2 s.f. | 0.001 | 0.0365 ≤ x < 0.0375 |
Pause and explainWhat is half a rounding unit for a value rounded to the nearest 0.1?
0.05. The interval extends 0.05 below and 0.05 above the reported value.
Choose bounds to make the result extreme
When uncertain measurements enter a calculation, the answer is uncertain too. Think about how each input affects the result before choosing a bound: does making that input larger increase or decrease the answer?
For a positive rectangle, larger side lengths give a larger perimeter and area. For speed, a larger distance increases speed but a larger time decreases it. This is why an upper speed bound needs an upper distance bound and a lower time bound, not two upper bounds.
For positive measurements, use these choices. Ask whether increasing each input increases or decreases the answer.
| Expression | Lower bound | Upper bound |
|---|---|---|
| a + b | a lower + b lower | a upper + b upper |
| a − b | a lower − b upper | a upper − b lower |
| ab | a lower × b lower | a upper × b upper |
| ab, b > 0 | a lower ÷ b upper | a upper ÷ b lower |
A rectangle is 12 cm by 8 cm, each to the nearest cm. Bound its perimeter and area.
11.5 ≤ L < 12.5 and 7.5 ≤ W < 8.5.
Lower perimeter: 2(11.5 + 7.5) = 38 cm.
Upper perimeter: 2(12.5 + 8.5) = 42 cm.
Lower area: 11.5 × 7.5 = 86.25 cm².
Upper area: 12.5 × 8.5 = 106.25 cm².
Distance is 150 m to the nearest 10 m; time is 12 s to the nearest second. Find the lower bound of speed.
145 ≤ d < 155 and 11.5 ≤ t < 12.5.
= 14512.5 = 11.6 m/s
A smaller distance and a longer time make the speed smaller. The upper bound is 15511.5 ≈ 13.4783 m/s.
Pause and explainFor positive distance and time, which choices give the lowest possible speed bound?
Use the lower distance bound and upper time bound, because speed is distance divided by time.
Your turn. Track the precision.
Write an interval for every rounded input before combining them. Label each bound clearly so that it is not mistaken for the reported measurement.
Next explain whether each input should increase or decrease for the required extreme result. Only then substitute. Keep enough digits in the calculation; rounding a bound prematurely can weaken or invalidate the stated limit.
A mass is 6.2 kg to 1 decimal place. Write its error interval.
Reveal worked answer
A square has side 9 cm to the nearest cm. Find the upper bound of its area.
Reveal worked answer
A distance is 80 km to the nearest km and time is 2.0 h to the nearest 0.1 h. Find the upper speed bound.
Reveal worked answer
a = 20 and b = 7, both rounded to the nearest whole number. Give bounds for a − b.