M5 · Rounding, estimation and accuracy | Edexcel GCSE Maths

Revise the key ideas

Decimal places count digits after the decimal point; significant figures start at the first non-zero digit. 0.004762 to 2 significant figures is 0.0048; to 3 decimal places it is 0.005.
To round, inspect the next digit: 5 or more rounds up, less than 5 stays. Include trailing zeros when they communicate precision: 3.5 to 2 decimal places is 3.50.
Truncation removes digits without rounding. Truncating 4.789 to 2 decimal places gives 4.78, while rounding gives 4.79. For positive measurements truncated to 4.78, 4.78 ≤ x < 4.79.
For a negative value truncated towards zero, the interval has the opposite included end: a value truncated to −4.78 at 2 decimal places satisfies −4.79 < x ≤ −4.78. Distinguish truncation from rounding.
Estimate by rounding inputs to convenient values, often one significant figure. 19.8 × 5.120.49 ≈ 20 × 50.5 = 200. An estimate checks the size, not the exact answer.
Use standard units and convert before combining measurements. 2.3 m + 45 cm = 2.75 m, not 47.3 m.

A measured length rounded to 8.4 cm to the nearest 0.1 cm has interval 8.35 ≤ l < 8.45 cm. Half the rounding step lies on each side of the reported value.
Error interval for 8.4 centimetres
Enlarge diagram: Error interval for 8.4 centimetres
The lower bound is included and the upper bound is excluded under the usual GCSE convention. Rounding 8.45 cm would give 8.5 cm, so it cannot be included in the interval for 8.4 cm.
Worked example: A mass of 350 g to the nearest 10 g means 345 ≤ m < 355 g. A whole-number-looking measurement is not necessarily accurate to the nearest 1 g.
Limits of accuracy describe possible actual measurements; they are not probabilities and do not imply the reported measurement is always exactly in the middle.
Give final answers to the requested precision and retain full calculator values during intermediate steps. Write ≈ for rounded values, rather than joining unequal exact values with =.

For positive a and b, the largest possible a + b uses both upper bounds and the smallest uses both lower bounds. For a − b, the upper limit uses upper a minus lower b.
For positive quantities, a product's upper limit uses both upper bounds. A quotient's upper limit uses the largest numerator and smallest denominator; its lower limit reverses those choices.
Worked example: A rectangle is 8.4 cm by 5.2 cm, each to the nearest 0.1 cm. Its area lies between 8.35 × 5.15 = 43.0025 cm² and 8.45 × 5.25 = 44.3625 cm², with the upper limit excluded.
Worked example: Distance is 100 m to the nearest metre and time is 20 s to the nearest second. Speed's lower limit is 99.520.5 ≈ 4.854 m/s; upper limit is 100.519.5 ≈ 5.154 m/s. Do not divide the two upper bounds to find maximum speed.
Only report a precision guaranteed by the entire possible interval. If the bounds round differently to a given precision, that precision is not justified for a single certain rounded result.

Practice questions with explained answers will be added here.

For now, cover the worked answers, try the calculations yourself, then compare each step. Include units and reasons where needed.