Rounding, Estimation & BoundsEdexcel GCSE Maths: Revision notes
Section 1
How do we round to decimal places and significant figures?
To round to a number of decimal places (d.p.), look at the digit after the required place; if it is 5 or more, round up.
- 3.4678 to 2 d.p. = 3.47
To round to a number of significant figures (s.f.), count from the first non-zero digit; the same 'round up if 5 or more' rule applies to the digit after the required precision.
- 0.02864 to 2 s.f. = 0.029
- 34,650 to 2 s.f. = 35,000
When rounding to significant figures, place-holding zeros must still be included to keep the number's size correct, e.g. 34,650 to 2 s.f. is 35,000, not 35.
Section 2
How do we estimate and check calculations?
To estimate an answer, round every number in the calculation to 1 significant figure first, then calculate.
- Estimate 38.7 × 5.2: round to 40 × 5 = 200
Estimation is used to check that a calculator answer is sensible — including answers obtained using technology, which can be mistyped. Always sanity-check a final answer against the estimate.
When a question says 'estimate', you must show rounding to 1 s.f. as your method — an exact calculation without rounding will not gain the estimation marks.
Section 3
How do we use inequality notation for error intervals?
When a number is rounded or truncated, the true value lies within an error interval, written using inequality notation.
For a number rounded to a given degree of accuracy, the true value x satisfies:
(rounded value − half the unit) ≤ x < (rounded value + half the unit)
- A length is 12 cm to the nearest cm: 11.5 ≤ x < 12.5
- A mass is 3.4 kg to 1 d.p.: 3.35 ≤ x < 3.45
Section 4
What are limits of accuracy and upper/lower bounds?
Limits of accuracy describe the range within which a measurement could truly lie, given its precision. The upper bound is the largest possible true value; the lower bound is the smallest possible true value.
For a measurement rounded to the nearest unit u: lower bound = measurement − u/2, upper bound = measurement + u/2.
- A length of 8 cm to the nearest cm: lower bound 7.5 cm, upper bound 8.5 cm
Section 5
How do we use bounds in calculations, including compound measures?
When combining measurements in a calculation, choose bounds carefully to get the correct overall upper or lower bound:
- For a maximum result of addition/multiplication, use the upper bounds of all values
- For a minimum result of addition/multiplication, use the lower bounds of all values
- For subtraction or division, the maximum result uses (upper bound of first) with (lower bound of second); the minimum uses (lower bound of first) with (upper bound of second)
This is especially important for compound measures like speed = distance ÷ time or density = mass ÷ volume.
Distance = 100 m (to nearest m), time = 20 s (to nearest s). Maximum speed uses max distance (100.5) ÷ min time (19.5) = 5.15 m/s
Must Know
- Round using the digit after the required place/figure: 5 or more rounds up
- To estimate, round every value to 1 significant figure before calculating
- Error interval for a rounded value: (value − half unit) ≤ x < (value + half unit)
- Upper bound = measurement + half the rounding unit; lower bound = measurement − half the rounding unit
- For maximum of a sum/product, use upper bounds; for minimum, use lower bounds
- For division/subtraction, mix bounds correctly: max = (upper ÷ lower), min = (lower ÷ upper)
That's the notes covered.
Carry on to the next subtopic.