Bounds questions go wrong at two separate points. The first is finding the interval of each rounded value. The second is choosing which endpoint to use for each operation.
Treat them as two separate jobs and most of the confusion clears.
Step one: find the interval correctly
The interval depends on the rounding unit, not on the number of digits.
- Identify the rounding unit: nearest 10, nearest 1, 1 decimal place (0.1) and so on.
- Halve it. Half of 10 is 5; half of 0.1 is 0.05.
- Lower bound: value minus the half unit. Upper bound: value plus the half unit.
For a time of 15 s to the nearest second: the half unit is 0.5, so the time is from 14.5 s to 15.5 s.
The lesson on recovering an interval from a rounded measurement practises this part on its own, and the bounds and rounding explainer lets you test values.
Step two: choose endpoints by asking “largest or smallest?”
Do not memorise “upper times upper”. Ask what you need:
| Result wanted | Choose |
|---|---|
| Largest sum a + b | upper a, upper b |
| Smallest sum a + b | lower a, lower b |
| Largest difference a − b | upper a, lower b |
| Smallest difference a − b | lower a, upper b |
| Largest product (positive values) | upper, upper |
| Smallest product (positive values) | lower, lower |
| Largest quotient a ÷ b | upper a, lower b |
| Smallest quotient a ÷ b | lower a, upper b |
Worked example 1: area of a rectangle
A rectangular plot has length 8.4 m and width 3.2 m, each measured to 1 decimal place. Find the largest and smallest possible area.
Intervals: length 8.35 to 8.45. Width 3.15 to 3.25.
Largest area: upper × upper = 8.45 × 3.25 = 27.4625 m².
Smallest area: lower × lower = 8.35 × 3.15 = 26.3025 m².
Check: 8.4 × 3.2 = 26.88, which sits between 26.3025 and 27.4625. Good.
Worked example 2: a speed from rounded data
A runner covers 120 m (nearest 10 m) in 15 s (nearest second). Find the greatest and least possible average speed.
Intervals: distance 115 m to 125 m. Time 14.5 s to 15.5 s.
Greatest speed: largest distance over smallest time = 125 ÷ 14.5 = 8.62 m/s (3 s.f.).
Least speed: smallest distance over largest time = 115 ÷ 15.5 = 7.42 m/s (3 s.f.).
Check: 120 ÷ 15 = 8, which sits between 7.42 and 8.62.
The mistake to watch for
Mistaken answer for the least speed: 115 ÷ 14.5 = 7.93
What went wrong: the student used both lower values, copying the rule for products. For a quotient the bottom value must be as large as possible to make the result small.
Correction: least speed = lower distance ÷ upper time = 115 ÷ 15.5 = 7.42.
A second common slip is to take half the wrong unit, for example using 0.5 as the half unit for a length given to 1 decimal place. Half of 0.1 is 0.05.
Self-check
1. A mass is 62 kg to the nearest kg. Give its lower and upper bounds.
Show answer
Half unit = 0.5. Lower bound 61.5 kg, upper bound 62.5 kg.
2. Two lengths are 12 cm and 7 cm, each to the nearest cm. Find the greatest possible difference.
Show answer
Upper of first = 12.5, lower of second = 6.5. Greatest difference = 12.5 − 6.5 = 6 cm.
3. A side is 5.0 cm to 1 d.p. Find the greatest possible area of a square with that side.
Show answer
Upper bound of the side = 5.05. Area = 5.05² = 25.5025 cm².
Where this leads next
Try bounding a sum and a difference and bounding a product with positive measurements, then test yourself across the topic in precision, bounds and measurement. The non-calculator working trainer helps with the arithmetic.
If a wrong endpoint keeps creeping in, a teacher can watch your choice as you make it. Online one-to-one Mathematics tuition lets someone catch the slip on the spot.