Bounds and limits of accuracy

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Notes de leçon Diapos Questions d'entraînement

Notes de leçon

Rounded values and bounds

A measurement rounded to a given accuracy lies in a range. The lower bound is the smallest value that rounds to it; the upper bound is the limit it cannot reach.
Rounded to the nearest 10 cm, 240 cm lies in 235 \leq x < 245. To 1 d.p., 6.3 lies in 6.25 \leq x < 6.35.
Take half the degree of accuracy either side: if accurate to the nearest 5, add and subtract 2.5.

Numbers rounding to 5

Bounds in calculations

Addition: lower + lower, upper + upper.
Subtraction: a - b has upper bound a_\text{upper} - b_\text{lower} and lower bound a_\text{lower} - b_\text{upper}.
For positive bounds only: multiplication uses lower × lower and upper × upper. Division uses upper ÷ lower for the upper bound, and lower ÷ upper for the lower bound; the denominator must remain positive.
For signed intervals, compare endpoint products rather than assuming lower × lower works. If a divisor interval contains zero, these simple finite quotient rules do not apply.

Using bounds

Area of a rectangle 8.5 cm by 4.2 cm (1 d.p.): upper = 8.55 \times 4.25, lower = 8.45 \times 4.15.
To give an answer to a stated accuracy, work out both bounds and check they round to the same value; that value is the answer to that accuracy.

Error intervals

Written as a \leq x < b. Use \leq for the lower bound and < for the upper bound.
The ≤ lower / < upper form here describes positive measurements rounded using the standard GCSE convention. Bounds of a derived calculation can have different endpoint inclusion; a limiting bound need not be attainable.

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1.A number is 5 to the nearest whole number. What is the lower bound?
Easy
- A4
- B4.5
- C4.9
- D5
2.A length is 7.3 cm to 1 d.p. What is its upper bound?
Easy
- A7.35
- B7.4
- C7.31
- D7.30
3.A mass is 40 kg to the nearest 10 kg. What is its lower bound?
Easy
- A30 kg
- B35 kg
- C39 kg
- D36 kg
4.What is the error interval for 12.5 to 1 d.p.?
Easy
- A12.45 \leq x < 12.55
- B12.4 \leq x < 12.6
- C12.45 \leq x \leq 12.55
- D12.5 \leq x < 12.6
5.What is the error interval for 3400 to the nearest 100?
Easy
- A3350 \leq x < 3450
- B3300 \leq x < 3500
- C3399 \leq x < 3401
- D3395 \leq x < 3405
6.Two lengths are 4 cm and 6 cm, both to the nearest cm. What is the upper bound of their sum?
Easy
- A10 cm
- B11 cm
- C10.5 cm
- D9 cm
7.What is the lower bound of a length of 2.4 m measured to the nearest 0.1 m?
Easy
- A2.35
- B2.3
- C2.39
- D2.4
8.a = 8 and b = 3, both to the nearest whole number. What is the upper bound of a - b?
Medium
- A5
- B6
- C5.5
- D4

Plus de questions

9.a = 8 and b = 3, both to the nearest whole number. What is the lower bound of a - b?
Medium
- A5
- B4
- C3.5
- D4.5
10.x = 20 and y = 4 both to the nearest whole number. What is the upper bound of \frac{x}{y}? Give the numerical bound rounded to 2 decimal places.
Medium
- A5
- B5.86
- C5.57
- D6
11.x = 20 and y = 4 both to the nearest whole number. What is the lower bound of \frac{x}{y}? Give the numerical bound rounded to 2 decimal places.
Medium
- A4.33
- B5
- C4.4
- D4.0
12.A rectangle is 8 cm by 5 cm, both to the nearest cm. What is the upper bound of its area?
Medium
- A46.75 cm²
- B40 cm²
- C44 cm²
- D48 cm²
13.A rectangle is 8 cm by 5 cm, both to the nearest cm. What is the lower bound of its area?
Medium
- A33.75 cm²
- B40 cm²
- C36 cm²
- D35 cm²
14.A car travels 100 m (to the nearest m) in 8 s (to the nearest second). What is the upper bound of its speed (1 d.p.)?
Medium
- A13.4 m/s
- B12.5 m/s
- C14.4 m/s
- D15.0 m/s
15.p = 3.6 to 1 decimal place. What is the upper bound of p² (2 d.p.)?
Medium
- A13.32
- B12.96
- C13.00
- D13.69
16.Which of these is a possible value of a length that rounds to 15 cm (nearest cm)?
Medium
- A15.4
- B15.5
- C14.4
- D15.6
17.Which of these could NOT round to 8 (nearest whole number)?
Medium
- A7.5
- B8.49
- C8.5
- D7.99
18.If both bounds of a calculation round to 3.4, what can you say?
Easy
- Athe answer is 3.4 to 1 d.p.
- Bthe answer is exactly 3.4
- Cthe answer cannot be found
- Dthe answer is 3.5
19.When is the lower bound used with ≤ and the upper bound with <?
Easy
- Ain error intervals
- Bnever
- Cin all equations
- Din sequences
20.\frac{12}{3} with both values to the nearest whole number. What is the lower bound of the quotient (2 d.p.)?
Medium
- A3.29
- B4.00
- C3.55
- D3.00
21.The upper bound of 3.7 (1 d.p.) is:
Easy
- A3.75
- B3.8
- C3.71
- D3.7
22.The lower bound of 0.5 (1 d.p.) is:
Easy
- A0.45
- B0.4
- C0.55
- D0.49
23.The upper bound of a rounded value cannot actually be reached.
Easy
True or false?
24.5 (nearest whole number) has a lower bound of 4.5.
Easy
True or false?
25.The lower bound of a - b uses the lower bound of b.
Easy
True or false?
26.For two quantities whose lower bounds are positive, multiplying their lower bounds gives the lower bound of their product.
Easy
True or false?
27.An error interval is written a < x \leq b.
Medium
True or false?
28.Which values round to 3.6 to 1 d.p.?
Medium
- A3.55
- B3.649
- C3.65
- D3.56
- E3.549
29.Which are true?
Medium
- Aupper bound of 8 (whole number) is 8.5
- Blower bound of 8 (whole number) is 7.5
- Cupper bound of 20 (nearest 10) is 25
- Dlower bound of 20 (nearest 10) is 15
- Eupper bound of 8 (whole number) is 9
30.A distance is 50 m to the nearest 10 m. What is its lower bound?
Easy
- A45 m
- B40 m
- C49 m
- D55 m
31.A mass is 2.5 kg to 1 decimal place. What is its upper bound?
Easy
- A2.55 kg
- B2.6 kg
- C2.51 kg
- D2.45 kg
32.a = 6 and b = 2, both to the nearest whole number. What is the upper bound of a + b?
Easy
- A9
- B8
- C8.5
- D7
33.A rectangle measures 8 cm by 5 cm, both to the nearest cm. Calculate the upper and lower bounds of its area.
Medium

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