Edexcel GCSE Maths Foundation (1MA1), Foundation tier · Number
Practise Rounding, estimation and error intervals. 1 exam-style questions plus unlimited generated ones on this subtopic, at up to four difficulty levels, with full mark schemes and a progress tracker. Free, no account needed.
Rounding to decimal places and significant figures, estimating answers by rounding to 1 significant figure, and writing error intervals for rounded or truncated values using inequalities. These come up on both papers, often as quick 1 to 3 mark questions.
Grade by grade
What you need to be able to do, from the first marks up to the top grade.
1
Round to the nearest 10, 100 or 1000Look at the next digit: 5 or more rounds up, so 3472 is 3500 to the nearest 100.
2
Round to a number of decimal placesCount digits after the decimal point: 6.4851 is 6.49 to 2 decimal places.
3
Round to significant figuresStart counting at the first non-zero digit: 0.03064 is 0.0306 to 3 significant figures.
3
Estimate by rounding to 1 significant figureRound every number to 1 s.f., then calculate: 38.2 × 5.9 ≈ 40 × 6 = 240.
4
Decide if an estimate is over or underUse how each number was rounded, e.g. rounding up the numbers you multiply gives an overestimate.
5
Write error intervals for rounded values7.3 to 1 decimal place means \(7.25 \le x \lt 7.35\).
5
Write error intervals for truncated values7.3 truncated to 1 decimal place means \(7.3 \le x \lt 7.4\).
Notes
Rounding
Find the last digit you are keeping and look at the next digit. 5 or more: round up. 4 or less: leave the digit as it is.
Decimal places (d.p.) count the digits after the decimal point: 12.3649 = 12.36 (2 d.p.).
Significant figures (s.f.) start at the first non-zero digit: 0.004718 = 0.0047 (2 s.f.) and 28,461 = 28,000 (2 s.f.).
Keep the place value: 28,461 to 2 s.f. is 28,000, not 28. The zeros hold the place.
Rounding can carry: 4.97 to 1 d.p. is 5.0, and you must write the 0.
If a question doesn't say how to round, write down the full answer, then give it to 3 s.f. (money to 2 d.p., e.g. £4.50).
Estimating
Round every number to 1 significant figure, then calculate: \(\frac{61.3 \times 4.82}{0.198} \approx \frac{60 \times 5}{0.2} = \frac{300}{0.2} = 1500\).
Dividing by 0.2 is the same as multiplying by 5; dividing by 0.5 is the same as multiplying by 2.
For a square root, round to a nearby square number: \(\sqrt{24.6} \approx \sqrt{25} = 5\).
Use ≈ ('is approximately equal to') for an estimate.
Overestimate or underestimate? Making the numbers you add or multiply bigger, or the number you divide by smaller, makes the answer bigger (an overestimate).
Error intervals
A rounded value could have come from a range of values: this is its error interval.
Rounding: go half a unit down and half a unit up. '\(x\) = 7.3 to 1 d.p.' gives \(7.25 \le x \lt 7.35\). The upper value is not included, because 7.35 would round to 7.4.
To the nearest 10: '\(n\) = 80' gives \(75 \le n \lt 85\).
Truncating means chopping off digits without rounding. '\(x\) = 7.3, truncated to 1 d.p.' gives \(7.3 \le x \lt 7.4\).
Write error intervals with \(\le\) on the left and \(\lt\) on the right.
Cheatsheet
Next digit 5 or more → round up; 4 or less → round down
Significant figures start at the first non-zero digit
Estimate: round every number to 1 s.f.
Rounded to 1 d.p.: 7.3 gives \(7.25 \le x \lt 7.35\)
Truncated to 1 d.p.: 7.3 gives \(7.3 \le x \lt 7.4\)
≈ means 'is approximately equal to'
÷ 0.1 is the same as × 10; ÷ 0.2 is × 5; ÷ 0.5 is × 2
How to answer each type of question
Round to decimal places or significant figures
1 mark each3
Count to the last digit you keep (for s.f., start at the first non-zero digit).
Look at the next digit to decide whether to round up.
Put back zeros to keep the place value of large numbers.
Example. (a) Round 0.06058 to 2 significant figures. (b) Round 1396.7 to 3 significant figures.
Show the model answer
(a) 0.061 (B1) (b) 1400 (B1)
Work out an estimate, and say if it is too big or too small
3 to 4 marks4
Round every number to 1 significant figure and write the rounded numbers down.
Work out the calculation with the rounded numbers.
For over or under: look at which way each number was rounded and what that does to the answer, and give that as your reason.
Example. (a) Work out an estimate for \(\frac{3.87 \times 59.6}{0.203}\) (b) Is your answer to (a) an overestimate or an underestimate? Give a reason for your answer.
Show the model answer
(a) \(\frac{4 \times 60}{0.2}\) (M1 for at least two of 4, 60, 0.2) \(= \frac{240}{0.2} = 1200\) (A1) (b) Overestimate: 3.87 and 59.6 were rounded up, and 0.203 was rounded down, so you divide a bigger number by a smaller number (C1)
Write down the error interval
1 to 2 marks each5
Find the unit it was rounded to (e.g. 0.1 for 1 d.p.).
Rounding: subtract and add half of that unit. Truncation: the value itself up to one whole unit more.
Write it as an inequality with the letter in the middle: \(\le\) on the left, \(\lt\) on the right.
Example. (a) The length of a pencil, \(L\) cm, is 8.4 cm correct to 1 decimal place. Write down the error interval for \(L\). (b) Nia's time, \(t\) seconds, is 12 seconds when truncated to a whole number of seconds. Write down the error interval for \(t\).
Show the model answer
(a) \(8.35 \le L \lt 8.45\) (B2, or B1 for 8.35 and 8.45 with the wrong signs, or one end correct) (b) \(12 \le t \lt 13\) (B1)
Shortcuts and memory tricks
For s.f., the first significant figure is the first digit that isn't 0; after that, zeros count.
Error interval for rounding: half a unit down, half a unit up; \(\le\) on the left, \(\lt\) on the right.
Truncation just chops: the value can't be below the truncated number, and goes up to (but not including) the next one.
Dividing by a decimal in an estimate? Multiply top and bottom by 10: 240 ÷ 0.2 = 2400 ÷ 2 = 1200.
Where marks are lost
Losing the place value: 28,461 to 2 s.f. is 28,000, not 28.
Counting zeros at the front as significant: 0.00471 to 1 s.f. is 0.005.
Not writing down the rounded numbers in an estimate: the method mark is for seeing them.
Rounding to 1 d.p. instead of 1 s.f. in an estimate, or using the exact values.
Writing \(\le\) at both ends of a rounding error interval, e.g. \(8.35 \le L \le 8.45\).
Writing 3.96 to 1 d.p. as 3.10 or 4 instead of 4.0.
Exam technique
In 'work out an estimate' questions, show the rounded numbers. An exact answer from a calculator gets no marks.
When asked for 3 significant figures, write the full calculator value first, then round.
For error intervals, use the letter from the question and put it in the middle of the inequality.
If asked 'overestimate or underestimate?', give a reason linked to how each number was rounded.
Sample questions
Written for this site in the style of Edexcel exam questions. They are not taken from real past papers.
Question 1Easy3 marks
(a) Round \(32{,}202\) to the nearest 1000.[1]
(b) Round \(58.95\) to 1 decimal place.[1]
(c) Round \(83{,}099\) to 2 significant figures.[1]
Show the answer and mark scheme
(a)Answer: \(32{,}000\)
B1 for 32,000
Worked solution: \(32{,}202 \approx 32{,}000\)
(b)Answer: \(59.0\)
B1 for 59.0 cao
Worked solution: Look at the second decimal place: \(58.95 \approx 59.0\)
(c)Answer: \(83{,}000\)
B1 for 83,000
Worked solution: \(83{,}099 \approx 83{,}000\) (2 s.f.)
Question 2Medium4 marks
Nia wants an estimate for the value of \(\frac{38.7 \times 0.493}{0.0213}\).
(a) By rounding each number to 1 significant figure, work out an estimate for the value.[2]
(b) Without working out the exact value, explain whether your estimate is an overestimate or an underestimate.[2]
Show the answer and mark scheme
(a)Answer: 1000
M1 for \(\frac{40 \times 0.5}{0.02}\) (all three numbers correctly rounded)
A1 for 1000
Worked solution: \(\frac{40 \times 0.5}{0.02} = \frac{20}{0.02} = 1000\)
(b)Answer: Overestimate: 38.7 and 0.493 were both rounded up, so the numerator is too big, and 0.0213 was rounded down, and dividing by a smaller number gives a bigger answer.
C1 for stating that 38.7 and 0.493 were both rounded up, so the top of the fraction is too large
C1 for stating that 0.0213 was rounded down, so dividing by the smaller number makes the answer larger, with the conclusion 'overestimate'
Worked solution: 40 > 38.7 and 0.5 > 0.493, so the numerator is larger than it should be. 0.02 < 0.0213, and dividing by a smaller number gives a larger result. Every change makes the value larger, so 1000 is an overestimate (the actual value is about 896).