Round to decimal places, significant figures or the nearest 10, 100 or 1000; estimate a calculation by rounding every number to 1 significant figure; and give answers to a sensible accuracy for the situation.
Find the last digit you are keeping, then look at the next digit only: 5 or more rounds up, 4 or less leaves it. Keep the size of the number with zeros (38 652 is 38 700 to 3 s.f., not 387). To estimate, round every number to 1 significant figure first.
Fig. 1: 5764 lies between 5000 and 6000 and is beyond the halfway value 5500, so to the nearest thousand it is 6000
Section 01
Rounding to a place value.
In the lab, press Next number, choose decimal places, significant figures or a nearest power of ten, and decide which way it rounds before the lab shows you. Watch the deciding digit it highlights.
Worked example Nearest 10, 100, 1000
Write 5764 correct to (a) the nearest thousand (b) the nearest ten.
(a) Thousands digit is 5; the next digit is 7, which is 5 or more, so round up (Fig. 1).
Fill the places after it with zeros: 6000.
(b) Tens digit is 6; the next digit is 4, so the 6 stays: 5760.
Answer: (a) 6000 (b) 5760
Worked example Decimal places
Write (a) 3.14159 (b) 7.9963 correct to 2 decimal places.
(a) Keep 3.14; the next digit is 1, so 3.14.
(b) Keep 7.99; the next digit is 6, so round up: 7.99 becomes 8.00 (the 9s roll over).
Answer: (a) 3.14 (b) 8.00
Examiner trap
7.9963 to 2 d.p. is 8.00, not 8. The zeros show the accuracy asked for; writing 8 gives only 0 decimal places.
Section 02
Significant figures.
The first significant figure is the first non-zero digit, counting from the left. Every digit after it counts, including zeros in the middle.
Leading zeros are NOT significant: 0.004 637 has first significant figure 4.
Zeros between digits ARE significant: 2.0496 to 3 s.f. keeps 2, 0, 4 and looks at the 9.
In whole numbers, fill with zeros to keep the place value: 38 652 to 2 s.f. is 39 000.
Worked example Three cases
Write (a) 0.004 637 to 2 s.f. (b) 38 652 to 3 s.f. (c) 2.0496 to 3 s.f.
(a) Significant digits 4, 6; next digit 3, so 0.0046.
(b) Keep 3, 8, 6; next digit 5, so round up to 387 hundreds: 38 700.
(c) Keep 2, 0, 4; next digit 9, so round up: 2.05.
Answer: (a) 0.0046 (b) 38 700 (c) 2.05
Examiner trap
38 652 to 3 s.f. is 38 700, not 387. Rounding must not change the size of the number.
Stage 1 · Learn
Check your understanding
Four quick checks. Each correct answer earns XP and records this skill on your profile.
Quick check+10 XP
Write 5764 correct to the nearest thousand.
Quick check+10 XP
Write 0.046 82 correct to 3 significant figures.
Quick check+10 XP
Write 12.996 correct to 2 decimal places.
Quick check+10 XP
By rounding to 1 significant figure, estimate \(9.8 \times 31.2\).
Section 03
Estimating a calculation.
Round every number to 1 significant figure, then work it out in your head. Show the rounded numbers: that is where the marks are. Use a nearby square number for a square root.
Worked example Rounding to 1 s.f.
By rounding each number to 1 significant figure, estimate \(\dfrac{48.7 \times 0.312}{5.86}\).
\(= 2.5\). (The exact value is 2.59..., so the estimate is good.)
Answer: 2.5
Worked example Dividing by a decimal
Estimate \(\dfrac{21.4^{2}}{0.487}\).
\(21.4 \approx 20\) and \(0.487 \approx 0.5\).
\(20^{2} = 400\).
\(400 \div 0.5 = 800\) (dividing by a half doubles).
Answer: 800 (exact value 940.4...)
Worked example A square root
Estimate \(\sqrt{98.6} \times 4.92\).
98.6 is close to the square number 100, so \(\sqrt{98.6} \approx 10\).
\(4.92 \approx 5\), so \(10 \times 5 = 50\).
Answer: 50 (exact value 48.85...)
Examiner trap
Dividing by 0.5 gives a bigger answer, not a smaller one: \(400 \div 0.5 = 800\), not 200.
Section 04
A sensible degree of accuracy.
Answers in context must make sense. Money goes to the nearest cent (2 d.p.); people, buses and tins are whole numbers, and whether to round up or down depends on the question. Otherwise Cambridge expects 3 significant figures.
Worked example Up or down?
(a) 230 students go on a trip. A coach seats 52. How many coaches are needed? (b) Shelves 0.75 m long are cut from a 5 m plank. How many shelves can be cut?
(a) \(230 \div 52 = 4.42\dots\). Four coaches leave students behind, so round UP.
(b) \(5 \div 0.75 = 6.66\dots\). A seventh shelf would be too short, so round DOWN.
Answer: (a) 5 coaches (b) 6 shelves
Your turn
A 2-litre bottle of juice is poured into cups that each hold 0.3 litres. How many cups can be filled completely?
\(2 \div 0.3 = 6.66\dots\); only whole cups count, so round down: 6 cups.
Examiner trap
Do not round half-way through a calculation. Keep the full calculator value and round only the final answer, or your 3 s.f. answer may be wrong.
Stage 2 · Exam
Exam-style questions
Unlocks once the four checks above are done. Worth more XP, written to IGCSE Paper 2 and Paper 4 standard.
Finish the four checks above to unlock the exam questions
Exam style+20 XP
By rounding each number to 1 significant figure, estimate \(\dfrac{48.7 \times 0.312}{5.86}\).
Exam style+20 XP
Write 38 652 correct to 3 significant figures.
Exam style+20 XP
By rounding each number to 1 significant figure, estimate \(\dfrac{21.4^{2}}{0.487}\).
Exam style+20 XP
230 students go on a trip. Each coach carries 52 students. How many coaches are needed?
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