← IGCSE / O Level Mathematics

IGCSE / O Level Mathematics · Topic 1 Number · Chapter 2 · pp. 13–23

Accuracy

Real measurements are never exact, and a calculator gives far more digits than anyone needs. This chapter is about how accurate a number should be: rounding it sensibly, estimating an answer before you calculate, and working out the range a rounded number could really have come from.

Learning objectives

By the end of this chapter you can:

  1. Round to the nearest 10, 100, 1000, and to a number of decimal places (d.p.).
  2. Round to a number of significant figures (s.f.), including small decimals like 0.0043.
  3. Give a calculator answer to an appropriate accuracy, and estimate answers by rounding first.
  4. Find the upper and lower bounds of a rounded number, and write them as an inequality. (Extended)
  5. Find the largest and smallest possible results of a calculation with rounded numbers. (Extended)

Book exercises: 2.1 to 2.7 and Student assessments 1 to 3 (pp. 13–23).

Round10s, d.p., s.f. Estimateround, then work Boundswhat was it really? Calculatemax and min results Core Extended
The chapter in four ideas. The first two are on both tiers; bounds are Extended only.

Next: rounding and decimal places →

Rounding and decimal places

🔑 Key ideas

  • Find the last digit you are keeping, then look at the next digit along: the “decider”.
  • Decider 5 or more → round up. Decider 4 or less → leave it.
  • Whole numbers keep their size: fill the dropped places with zeros. Decimals just stop.
Mark the last digit you keep Look at the next digit (decider) 5 or more? yes no Add 1 to the kept digit7.86 → 7.9 Keep the digit as it is5.574 → 5.57
Every kind of rounding in this chapter follows this one flowchart. Only the “last digit you keep” changes.

Rounding to the nearest 10, 100 or 1000

28 617 people to the nearest…Keep up toDeciderResult
10 0002 (ten thousands)8 → up30 000
10008 (thousands)6 → up29 000
1006 (hundreds)1 → stay28 600

Decimal places (d.p.)

“To 2 d.p.” means keep two digits after the decimal point. Count from the point, not from the first digit.

📘 Example 1 · Write 7.864 to 1 d.p.

  1. Keep one digit after the point: 7.8|64.
  2. The decider is the next digit, 6.
  3. 6 is 5 or more, so the 8 rounds up to 9.
  4. Answer: 7.9 (1 d.p.).

📘 Example 2 · Write 99.996 to 2 d.p. (a carry)

  1. Keep two digits after the point: 99.99|6. The decider is 6, so round up.
  2. The last 9 becomes 10, which carries 1 into each place to its left: 99.99 → 100.00.
  3. Answer: 100.00. Keep both zeros: they show the answer is accurate to 2 d.p.

🧪 Try it: rounding machine

✏️ Try it

  1. Round 99 500 to the nearest 1000.
  2. Give 15.045 to 1 d.p.
  3. Give 0.0048 to 2 d.p.
Answers

1. 100 000 (decider 5) · 2. 15.0 (decider 4) · 3. 0.00 (decider 4, so 0.00 to 2 d.p.)

Next: significant figures →

Significant figures

🔑 Key ideas

  • The first significant figure is the first digit that is not zero, reading from the left.
  • Count from there, whether or not you pass the decimal point.
  • Zeros at the start of a decimal (0.00…) are never significant. Zeros in the middle (305) always are.
43.25 4 3 . 2 5 1st2nd3rd4th worth 40 → 3 → 0.2 → 0.05 0.0043 0 . 0 0 4 3 not significant1st2nd
Significant figures start at the first non-zero digit. In 0.0043 the leading zeros only hold the place, so the 4 is the first significant figure.

📘 Example 3 · Write 43.25 to 3 s.f.

  1. Find the 1st significant figure: 4. Count three: 4, 3, 2. Keep 43.2|5.
  2. The decider is 5, so round the 2 up to 3.
  3. Answer: 43.3 (3 s.f.).

📘 Example 4 · Write 48 599 to 3 s.f.

  1. Count three significant figures: 4, 8, 5. Keep 485|99.
  2. The decider is 9, so the 5 rounds up to 6: 486.
  3. This is a whole number, so keep its size: fill the two dropped places with zeros.
  4. Answer: 48 600 (3 s.f.). Not 486!

📘 Example 5 · Write 0.003 05 to 2 s.f.

  1. Skip the leading zeros. The 1st significant figure is 3.
  2. Count two: 3, 0. The middle zero counts, because it is after the 3. Keep 0.0030|5.
  3. The decider is 5: round the 0 up to 1.
  4. Answer: 0.0031 (2 s.f.).
Decimal placesSignificant figures
Count fromthe decimal pointthe first non-zero digit
Good formoney, measurements on one scalenumbers of very different sizes
3.0472 to 23.053.0

✏️ Try it (use the rounding machine on the last tab to check)

  1. 990 to 1 s.f.
  2. 2.5728 to 3 s.f.
  3. 0.954 to 1 s.f.
  4. 0.009 73 to 2 s.f.
Answers

1. 1000 · 2. 2.57 · 3. 1 (0.954 rounds up past the point) · 4. 0.0097

Next: estimating →

Appropriate accuracy and estimating

🔑 Key ideas

  • Don't copy every calculator digit. Unless told otherwise, give 1 or 2 d.p. (or 3 s.f.).
  • To estimate, round every number to 1 s.f. (or a friendly number), then work it out in your head.
  • An estimate catches typing mistakes: if the calculator says 2000 and you estimated 200, check again.

📘 Example 6 · 4.64 ÷ 2.3 to an appropriate accuracy

  1. Calculator: 4.64 ÷ 2.3 = 2.017 391 3…
  2. The question's numbers have 1 or 2 d.p., so 1 d.p. is plenty.
  3. Answer: 2.0 (1 d.p.).
6386 ÷ 27the real question round 6000 ÷ 30friendly numbers work ≈ 200exact: 236.5… ≈ means “is approximately equal to”
Estimating in three moves: round each number, do the easy sum, compare with the calculator.

📘 Example 7 · Estimate 57 × 246

  1. Round each number to 1 s.f.: 57 ≈ 60 and 246 ≈ 200.
  2. Multiply: 60 × 200 = 12 000.
  3. Estimate: ≈ 12 000. (50 × 250 = 12 500 is just as good. Exact: 14 022.)

📘 Example 8 · Estimate ∛120 × 48

  1. Pick a nearby cube number: 125 = 5³, so ∛120 ≈ ∛125 = 5.
  2. Round 48 to 50.
  3. Multiply: 5 × 50 = ≈ 250.

📘 Example 9 · Is 95 × 212 = 20 140 definitely wrong?

  1. Estimate: 100 × 200 = 20 000, so the size looks right.
  2. Check the last digit: 5 × 2 = 10, so the answer must end in 0. It does.
  3. So it may be correct (it is: 95 × 212 = 20 140). An estimate can only prove an answer wrong, never right.

✏️ Try it, without a calculator

  1. 4950 × 28
  2. 610 ÷ 0.22
  3. Is 689 × 413 = 28 457 definitely wrong?
Answers

1. 5000 × 30 = 150 000 · 2. 600 ÷ 0.2 = 3000 · 3. Yes: 700 × 400 = 280 000, ten times bigger.

Next: upper and lower bounds →

Upper and lower bounds Extended

🔑 Key ideas

  • A rounded number stands for a whole range of real values.
  • The bounds are half a unit either side, where the unit is what it was rounded to.
  • The lower bound is included (≤). The upper bound is not (<), because it would round up.
4.4 4.6 4.5 4.45lower bound ● 4.55upper bound ○
4.5 (to 1 d.p.) could be anything from 4.45 up to, but not including, 4.55: 4.45 ≤ x < 4.55. A filled dot is included; an open dot is not.
Rounded valueAccuracyHalf a unitRange
162 cmnearest 1 cm0.5161.5 ≤ h < 162.5
4.502 d.p. (0.01)0.0054.495 ≤ x < 4.505
620 m³2 s.f. (nearest 10)5615 ≤ x < 625
400 mnearest 10 m5395 ≤ W < 405

📘 Example 10 · A girl's height is 162 cm to the nearest cm

  1. The unit is 1 cm, so half a unit is 0.5 cm.
  2. Lower bound = 162 − 0.5 = 161.5 cm. Upper bound = 162 + 0.5 = 162.5 cm.
  3. Write the inequality: 161.5 ≤ h < 162.5.

📘 Example 11 · 0.045 correct to 2 s.f.

  1. The 2nd significant figure (5) is in the thousandths place, so the unit is 0.001.
  2. Half a unit = 0.0005.
  3. Bounds: 0.045 − 0.0005 = 0.0445 and 0.045 + 0.0005 = 0.0455, so 0.0445 ≤ x < 0.0455.

🧪 Try it: bounds finder

    ✏️ Try it

    1. A sack is 5.4 kg (1 d.p.). Write the range of its mass M.
    2. 25 000 to 2 s.f. Write the range of x.
    3. A 100 m race was won in 11.8 s. Give the bounds of the time.
    Answers

    1. 5.35 ≤ M < 5.45 · 2. 24 500 ≤ x < 25 500 · 3. 11.75 s and 11.85 s

    Next: calculating with bounds →

    Calculating with upper and lower bounds Extended

    🔑 Key ideas

    • To get the biggest answer, choose whichever bound of each number pushes the answer up.
    • Adding and multiplying: big with big. Subtracting and dividing: big with small.
    • Write down every bound before you calculate, then pick.
    OperationMaximum answerMinimum answer a + bUB + UBLB + LB a × bUB × UBLB × LB a − bUB − LBLB − UB a ÷ bUB ÷ LBLB ÷ UB
    UB = upper bound, LB = lower bound. Below the line the second number flips: taking away or dividing by less leaves you with more.

    📘 Example 12 · 34 × 65, each to the nearest whole number

    1. Bounds: 33.5 ≤ a < 34.5 and 64.5 ≤ b < 65.5.
    2. Minimum product = LB × LB = 33.5 × 64.5 = 2160.75.
    3. Maximum product = UB × UB = 34.5 × 65.5 = 2259.75.

    📘 Example 13 · 33.5 ÷ 22.0, each to 1 d.p.

    1. Bounds: 33.45 ≤ a < 33.55 and 21.95 ≤ b < 22.05.
    2. Minimum = LB ÷ UB = 33.45 ÷ 22.05 = 1.52 (2 d.p.).
    3. Maximum = UB ÷ LB = 33.55 ÷ 21.95 = 1.53 (2 d.p.).

    📘 Example 14 · How long could the rocket take? (Exercise 2.7, Q11)

    Earth to Moon: 400 000 km to the nearest 100 000 km. Rocket speed: 3500 km/h to the nearest 500 km/h.

    1. Bounds: 350 000 ≤ d < 450 000 and 3250 ≤ s < 3750.
    2. time = distance ÷ speed, so this is a division: big ÷ small for the greatest time.
    3. Greatest time = 450 000 ÷ 3250 ≈ 138.5 hours.
    4. Least time = 350 000 ÷ 3750 ≈ 93.3 hours.

    ✏️ Try it

    1. Parcels of 1.5 kg and 2.5 kg are each weighed to the nearest 0.5 kg. Give the bounds of their total.
    2. A rectangle is 10.0 cm by 7.5 cm (1 d.p.). Give the bounds of its area.
    3. 17.6 − 4.2, each to 1 d.p. Give the bounds of the answer.
    Answers

    1. 3.5 kg and 4.5 kg · 2. 9.95 × 7.45 = 74.1275 cm² and 10.05 × 7.55 = 75.8775 cm² · 3. 17.55 − 4.25 = 13.3 and 17.65 − 4.15 = 13.5

    Next: summary and quiz →

    Chapter summary

    • To round, mark the last digit you keep and look at the decider after it: 5 or more rounds up.
    • d.p. counts from the decimal point; s.f. counts from the first non-zero digit. Whole numbers keep their size with zeros.
    • Give calculator answers to a sensible accuracy, usually 1 or 2 d.p. or 3 s.f. Estimate by rounding to 1 s.f. first.
    • A rounded number lies within half a unit either side: lower bound ≤ x < upper bound.
    • For the biggest result: UB + UB, UB × UB, UB − LB, UB ÷ LB. For the smallest, swap every bound.

    📄 Practice questions

    The book's Exercises 2.1 to 2.7 and Student assessments 1 to 3 are in the printable assignment for this chapter.

    Mini-quiz

    1. What is 28 617 rounded to the nearest 1000?
    2. What is 5.574 written to 2 decimal places?
    3. What is 0.0043 written to 1 significant figure?
    4. What is 48 599 written to 3 significant figures?
    5. Which is the best estimate of 6386 ÷ 27?
    6. A height is 162 cm to the nearest cm. What is its lower bound?
    7. Which inequality shows 4.5 correct to 1 decimal place?
    8. a and b are rounded. Which gives the largest possible value of a − b?

    Source: Cambridge IGCSE Mathematics Core and Extended (4th edition, Pimentel and Wall), Chapter 2 (Accuracy). Explanations, the Try it answers, the tools and the quiz are Guru Jazzy's own. This page is not made or endorsed by Cambridge or Hodder Education.