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Rounding and Estimation - Edexcel GCSE Higher Maths

Typically 3-4 marks per paper for the estimation question itself, plus an accuracy mark wherever a paper names a rounding · Spec 1MA1

Two Edexcel 1MA1 spec points sit behind this page. N14 asks you to estimate answers and to check a calculation by approximation, including a calculation you carried out on a calculator. N15 asks you to round numbers and measures to a stated accuracy, either to a number of decimal places or to a number of significant figures. Neither point is Higher tier only, so the same skills are assessed at Foundation; what changes on Higher is that they turn up wrapped inside something longer, and that the numbers are chosen to be awkward.

The set-piece question opens 'Work out an estimate for the value of' and is followed by a fraction. It is normally worth 3 marks: one for writing each number correct to 1 significant figure, one for processing the numerator, one for the value at the end. Edexcel likes to put a small decimal underneath, something like 0.48 or 0.21, because dividing by a number below 1 makes the answer bigger and a lot of candidates refuse to believe that. Paper 1 gets the bare arithmetic version; Papers 2 and 3 dress it up as the cost of an order, the area of a circular lawn or the output of a machine over a year. Having a calculator does not rescue you, because an exact answer with no rounded values anywhere in the working scores nothing.

Part (b) carries the extra mark: is the estimate an underestimate or an overestimate, and why. A reason only earns the mark if it names the direction each number moved, and the denominator reverses the effect, so rounding 0.48 up to 0.5 pushes the estimate down rather than up. The rounding half of the topic bleeds marks across the rest of the paper instead of in one place, mostly through leading zeros (0.00478 to 2 significant figures is 0.0048, not 0.00) and through lost place holders (4863 to 2 significant figures is 4900, not 49). Round straight from the original number every time, never in stages.

Worked example

(a) Work out an estimate for the value of 3.86×19.40.213\frac{3.86 \times 19.4}{0.213} (b) Is your answer to part (a) an underestimate or an overestimate? Give a reason for your answer.
[4 marks]
  1. Write each number correct to 1 significant figure: 3.86≈43.86 \approx 4, 19.4≈2019.4 \approx 20 and 0.213≈0.20.213 \approx 0.2.
    M1: all three numbers rounded to 1 significant figure and written down; this mark is lost if the rounded values never appear on the page
  2. Work out the numerator first: 4×20=804 \times 20 = 80.
    M1: correct processing of the top of the fraction
  3. Divide by 0.2. Multiplying top and bottom by 10 turns it into 800÷2800 \div 2, so the estimate is 400.
    A1: 400. Dividing by 0.2 doubles then multiplies by 10; halving 80 to get 40 is the classic wrong turn here
  4. Both numbers on the top were rounded up and the number on the bottom was rounded down, so the estimate is larger than the true value. It is an overestimate.
    B1: overestimate with a reason that names the direction of the roundings, top and bottom

Practice questions

These are original questions in Edexcel 1MA1 Higher style. Do all of them without a calculator, including the ones set in a context, since the point of rounding to 1 significant figure is that the arithmetic then fits on paper. The underestimate and overestimate questions are multiple choice because the reason is what carries the mark: in the exam the right conclusion with the wrong reason scores nothing.

No account needed: the questions below are always free to check, plus 8 extra questions today. Your score so far updates as you go.

  1. Question 1 Warm-up [2 marks]
    By writing each number correct to 1 significant figure, work out an estimate for 18.9×4.218.9 \times 4.2
  2. Question 2 Exam pace [2 marks]
    Write 12.996 correct to 2 decimal places.
  3. Question 3 Exam pace [2 marks]
    Work out an estimate for 0.0398\sqrt{0.0398}
  4. Question 4 Exam pace [2 marks]
    Which of these is the best estimate for 150\sqrt{150}?
    Answer options for question 4
  5. Question 5 Exam pace [3 marks]
    A block of wood has a volume of 412 cm3^3 and a density of 0.78 g/cm3^3. By writing each number correct to 1 significant figure, work out an estimate for the mass of the block, in grams.
  6. Question 6 Stretch [3 marks]
    A rectangle has one side of length 41.3 cm. Kai estimates the area of the rectangle by rounding both side lengths to 1 significant figure. His estimate is 3200 cm2^2. Work out the value that the other side was rounded to, in cm.

Common mistakes examiners see

  • Rounding a small denominator to the nearest whole number, so 0.51 becomes 1 and an estimate that should be 3000÷0.5=60003000 \div 0.5 = 6000 is written as 3000.

    One significant figure of a number below 1 is its first non-zero digit, wherever that digit sits: 0.51 rounds to 0.5, 0.048 rounds to 0.05, 0.0072 rounds to 0.007. A denominator must never be rounded to 0, and it should never gain a whole unit either.

  • Treating division by a decimal as if it shrinks the answer, so 3000÷0.53000 \div 0.5 is worked out as 1500 and 80÷0.280 \div 0.2 as 40.

    Scale the fraction until the bottom is a whole number: multiply top and bottom by 10 for 0.5, by 100 for 0.05. 3000÷0.53000 \div 0.5 becomes 30000÷5=600030000 \div 5 = 6000. Check the size before you write it down, because dividing by something less than 1 always gives an answer larger than the number you started with.

  • Dropping the place holders when rounding a large number, so 4863 to 2 significant figures is written as 49 and 27 418 to 3 significant figures as 274.

    The rounded value has to stay the same size as the original. Fill the remaining places up to the decimal point with zeros: 4900 and 27 400. Say the answer out loud as a size check, four thousand nine hundred against four thousand eight hundred and sixty three.

  • Counting the zeros after the decimal point, so 0.00478 to 2 significant figures is given as 0.00 or as 0.0047.

    Start counting at the first non-zero digit. In 0.00478 that is the 4, the second significant figure is the 7, and the next digit 8 rounds it up, giving 0.0048. Decimal places count from the point, significant figures count from the first digit that carries any size, and for numbers below 1 the two never agree.

  • Justifying the answer from the numerator alone. Estimating 6.83×39.20.46\frac{6.83 \times 39.2}{0.46} as 7×400.5=560\frac{7 \times 40}{0.5} = 560 and calling it an overestimate because 6.83 and 39.2 both went up, when the true value is about 582 and the estimate is too small.

    Check all of the numbers, and remember that the denominator works backwards: rounding it up makes the estimate smaller, rounding it down makes the estimate larger. Write the reason as a sentence that names both halves, such as 'top rounded up and bottom rounded down, so the estimate is too big'. When the roundings pull in opposite directions, say so rather than guessing.

Frequently asked questions

Is estimation on the Higher paper or only on Foundation?
Both. N14 and N15 are common content in Edexcel 1MA1, so a Foundation and a Higher candidate can meet the same estimation question. Higher versions tend to carry a decimal on the bottom of the fraction, a square root, or a context such as an area or a rate, and they are more likely to ask for the underestimate or overestimate reason as well.
Do I always round to 1 significant figure when estimating?
Yes, unless the question tells you otherwise, and the first mark is awarded for doing it. The one exception is a square root: round the number under the root to the nearest square number instead, so an estimate for 35.7\sqrt{35.7} comes from 36=6\sqrt{36} = 6. Rounding 35.7 to 40 would leave you with a root you cannot work out.
How many marks is an estimation question worth on Edexcel Higher?
The standard 'work out an estimate for the value of' question is 3 marks. Where a part (b) asks whether the answer is an underestimate or an overestimate with a reason, that is usually 1 more, so 3-4 marks in total. Rounding instructions attached to other questions, such as 'give your answer correct to 3 significant figures', are worth an accuracy mark each and add up across a paper.
Can I just use my calculator on Paper 2 or Paper 3?
You can hold it, but it will not earn the marks. The method marks on an estimation question are for the rounded numbers and for the calculation done with them, so a page showing only the exact value from the display scores zero even when that value is right. Write the 1 significant figure versions down first, then work with those.

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