Rounding, error intervals and bounds
A rounded number hides a range of possible values. Error intervals write that range down, and bounds tell you the biggest and smallest an answer could be when you calculate with rounded numbers.
Part 1 of 3: Learn it
In short
- Half the rounding unit either side gives the bounds: 7 to the nearest whole number lies from 6.5 up to (not including) 7.5.
- Write an error interval as lower bound ≤ x < upper bound.
- To make a result as big as possible, add or multiply upper bounds, but subtract or divide by a lower bound.
Where this is in your specification
Spec points: DfE N15 and N16 (bounds in calculations are Higher tier)
| Board | Topic: Place value, rounding and bounds |
|---|---|
| DfE content | N1, N2, N3, N4, N5, N14, N15, N16 |
| AQA 8300 | Number |
| Edexcel 1MA1 | Number |
| OCR J560 | 1 Number operations and integers; 4 Approximation and estimation |
Rounding
- Decimal places count digits after the decimal point: 3.14159 to 2 d.p. is 3.14.
- Significant figures count from the first non-zero digit: 0.004 562 to 2 s.f. is 0.0046, and 48 720 to 2 s.f. is 49 000.
- To estimate, round every number to 1 significant figure first, then calculate.
Error intervals
If x = 7 to the nearest whole number, the rounding unit is 1, so the true value is within 0.5 either side:
The lower bound is included; the upper bound is not, because 7.5 would round up to 8. For a value rounded to 1 decimal place (unit 0.1) go 0.05 either side, and for the nearest 10 go 5 either side.
Truncation just chops off digits, so the true value can only be bigger: if y = 4.3 truncated to 1 decimal place, 4.3 ≤ y < 4.4.
Bounds in calculations (Higher)
| Calculation | Upper bound of the answer | Lower bound of the answer |
|---|---|---|
| a + b | UB of a + UB of b | LB of a + LB of b |
| a − b | UB of a − LB of b | LB of a − UB of b |
| a × b | UB of a × UB of b | LB of a × LB of b |
| a ÷ b | UB of a ÷ LB of b | LB of a ÷ UB of b |
m = 3.6 to 1 decimal place. Write the error interval for m.
Show the answer
3.55 ≤ m < 3.65.
Part 2 of 3: See it worked
Worked examples
Example 1
A rectangle measures 8.4 cm by 5.2 cm, each to 1 decimal place. Find the upper bound of its area.
- Upper bounds: 8.45 cm and 5.25 cm
- Upper bound of area = 8.45 × 5.25
- = 44.3625 cm²
Answer: 44.3625 cm².
Example 2
A car travels 200 m, to the nearest 10 m, in 12 s, to the nearest second. Find the lower bound of its average speed.
- Distance bounds: 195 m and 205 m. Time bounds: 11.5 s and 12.5 s
- Smallest speed = smallest distance ÷ largest time
- = 195 ÷ 12.5 = 15.6 m/s
Answer: 15.6 m/s.
Common mistakes
- Writing the upper bound with ≤, as if 7.5 could round to 7.
- Using 0.5 either side for a value given to 1 decimal place. It is 0.05.
- Dividing by the upper bound when you want the biggest possible answer.
- Treating a truncated number like a rounded one.
Round 0.07385 to 2 significant figures.
Show the answer
0.074.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
m = 3.6 to 1 decimal place. Write the error interval for m.
3.55 ≤ m < 3.65.
Round 0.07385 to 2 significant figures.
0.074.
a = 20 and b = 4, both to the nearest whole number. Find the upper bound of a ÷ b.
20.5 ÷ 3.5 = 5.857... (5.86 to 3 s.f.).
Jobs that use this
- Civil engineer (opens a new tab)
- Actuary (opens a new tab)
- Data analyst-statistician (opens a new tab)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
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