Units, estimation and rounding

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Notes de leçon Diapos Questions d'entraînement

Notes de leçon

Rounding

Decimal places (d.p.) count digits after the decimal point; significant figures (s.f.) count from the first non-zero digit.
Look at the next digit: 5 or more rounds up. 3.4567 to 2 d.p. = 3.46; 0.004567 to 2 s.f. = 0.0046.
Zeros matter: 0.0500 to 3 s.f. keeps the trailing zeros to show precision.

Estimation

Round each number to 1 significant figure and calculate: \frac{39.2 \times 4.8}{0.52}\approx \frac{40 \times 5}{0.5} = 400.
For a quotient with positive inputs, increasing the numerator and decreasing a positive denominator gives an overestimate. If some changes increase and others decrease the result, inspect the effect rather than guessing.

Standard form

Write numbers as A \times 10ⁿ where 1 \leq A < 10. 45\,000 = 4.5 \times 10⁴; 0.0032 = 3.2 \times 10^-3.
Multiply: multiply the As and add the powers. Divide: divide the As and subtract powers. Adjust A if it leaves the range.

Units and approximations

Always give units. Round at the end, not during a calculation.
Use \approx for approximate equality, and the correct number of significant figures for the data given.

Length, area and volume units

Error intervals

Rounded values give an error interval: 4.6 to 1 d.p. means 4.55 \leq x < 4.65.

Values rounding to 4.6

Diapos

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Questions d'entraînement

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1.Round 7.8 to the nearest whole number.
Easy
- A7
- B8
- C9
- D7.5
2.Round 4.567 to 1 decimal place.
Easy
- A4.5
- B4.6
- C4.57
- D4.0
3.Round 3482 to 2 significant figures.
Easy
- A3500
- B3400
- C3480
- D3490
4.Round 0.004567 to 2 significant figures.
Easy
- A0.0045
- B0.0046
- C0.005
- D0.00457
5.Round 12.95 to 1 decimal place.
Easy
- A12.9
- B13.0
- C12.95
- D12.0
6.Estimate 19.8 \times 5.1 by rounding to 1 s.f.
Easy
- A100
- B90
- C120
- D80
7.Write 45 000 in standard form.
Easy
- A4.5 \times 10⁴
- B45 \times 10³
- C4.5 \times 10³
- D0.45 \times 10⁵
8.Write 0.0032 in standard form.
Easy
- A3.2 \times 10^-3
- B3.2 \times 10³
- C32 \times 10^-4
- D3.2 \times 10^-2

Plus de questions

9.What is 6.5 \times 10³ as an ordinary number?
Easy
- A650
- B6500
- C65 000
- D0.0065
10.Estimate (39.2 × 4.8)/0.52 by rounding each input to 1 significant figure.
Medium
- A400
- B40
- C4000
- D360
11.Calculate (3 \times 10⁴) \times (2 \times 10³).
Medium
- A6 \times 10⁷
- B6 \times 10¹²
- C5 \times 10⁷
- D6 \times 10¹
12.Calculate (8 \times 10⁶) \div (2 \times 10²).
Medium
- A4 \times 10⁴
- B4 \times 10⁸
- C16 \times 10⁴
- D4 \times 10³
13.Calculate (6 \times 10⁵) \times (5 \times 10⁴) in standard form.
Medium
- A3 \times 10¹⁰
- B30 \times 10⁹
- C3 \times 10⁹
- D11 \times 10⁹
14.Which is the largest?
Medium
- A3.2 \times 10⁴
- B5.1 \times 10³
- C9.9 \times 10³
- D1.2 \times 10⁴
15.A number rounded to 1 d.p. is 4.6. What is its error interval?
Medium
- A4.55 \leq x < 4.65
- B4.5 \leq x < 4.7
- C4.6 \leq x < 4.7
- D4.55 \leq x \leq 4.65
16.How many significant figures does 0.00402 have?
Medium
- A3
- B5
- C6
- D2
17.\frac{4.8 \times 10^-3}{1.2 \times 10²} equals:
Medium
- A4 \times 10^-5
- B4 \times 10^-1
- C5.76 \times 10^-1
- D4 \times 10⁵
18.Calculate (2.5 \times 10⁴) + (3 \times 10³) in standard form.
Medium
- A2.8 \times 10⁴
- B5.5 \times 10⁷
- C2.8 \times 10⁷
- D5.5 \times 10⁴
19.Estimate \frac{58 \times 102}{0.48} by rounding each number to 1 significant figure.
Medium
- A12 000
- B1200
- C120 000
- D6000
20.Which has more significant figures: 0.0450 or 4.5?
Easy
- A0.0450
- B4.5
- Cthe same
- Dcannot tell
21.When an accurate final result is required in a multi-step calculation, when should you round to the requested precision?
Easy
- Aat the end
- Bat every step
- Cat the beginning
- Dnever
22.Which number rounds to 5.2 to 1 decimal place?
Medium
- A5.14
- B5.17
- C5.25
- D5.3
23.In 0.0046 the leading zeros are significant.
Easy
True or false?
24.4.5 \times 10⁴ equals 45 000.
Easy
True or false?
25.42 \times 10³ is written in standard form.
Easy
True or false?
26.Rounding inputs to 1 significant figure is one useful way to make an estimate.
Easy
True or false?
27.A value rounded to 1 d.p. as 4.6 could be 4.64.
Medium
True or false?
28.Which numbers are in standard form?
Medium
- A3.5 \times 10⁶
- B35 \times 10⁵
- C9.99 \times 10^-2
- D0.5 \times 10³
- E1 \times 10⁰
29.Which round to 3.5 to 1 d.p.?
Medium
- A3.46
- B3.54
- C3.55
- D3.449
- E3.51
30.Round 6.049 to 2 decimal places.
Easy
- A6.04
- B6.05
- C6.0
- D6.1
31.Round 0.0830 to 1 significant figure.
Easy
- A0.08
- B0.09
- C0.1
- D0.083
32.Write (3.6 \times 10⁵) \div (1.2 \times 10²) in standard form and show your working.
Medium

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