Upper and lower bounds

When a measurement is rounded, the real value could be a little more or a little less. The smallest it could be is the lower bound and the largest is the upper bound: half a unit either side.

Key facts

  • Bounds are half the rounding unit either side: 7 cm to the nearest cm is between 6.5 and 7.5 cm.
  • Adding or multiplying: use upper with upper for the maximum.
  • Subtracting or dividing: for the maximum use the upper bound of the first and the lower bound of the second.
  • Error interval: 6.5 ≤ x < 7.5.

Worked examples

Example 1

A pencil is 14 cm long to the nearest cm. Write the error interval.

  1. Half of 1 cm is 0.5 cm.
  2. Lower bound: 14 − 0.5 = 13.5. Upper bound: 14 + 0.5 = 14.5. 13.5 ≤ l < 14.5

Answer: 13.5 ≤ l < 14.5

Example 2

A car travels 120 km (nearest 10 km) in 1.5 h (nearest 0.1 h). Find the maximum average speed.

  1. Distance upper bound: 125 km. Time lower bound: 1.45 h.
  2. Speed = distance ÷ time. 125 ÷ 1.45 = 86.2…
  3. Maximum speed ≈ 86.2 km/h.

Answer: 86.2 km/h (1 d.p.)