Rounding Estimation And Bounds

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Rounding to a Given Place Value

Identify the digit in the required place value, then count in that unit to find the two possible rounded values (one below, one above).
Circle the digit immediately to the right of the required place; if it is 5 or more, round up; if less than 5, round down.
For decimal places, ensure the answer has exactly the specified number of decimal places (e.g., 2.40 for 2 d.p., not 2.4).
When rounding to a place value (e.g., nearest 100), fill any following places before the decimal point with zeros (e.g., 1567.45 → 1600).

Rounding to Significant Figures

The first significant figure is the leftmost non-zero digit (e.g., 0.006207 → first s.f. is 6).
Count digits from the first s.f. to locate the required s.f.; zeros between non-zero digits are significant.
Round using the same rules: circle the next digit; if ≥5 round up, else round down.
For large numbers, fill with zeros up to the decimal point (e.g., 34 568 to 2 s.f. → 35 000).
For decimals, insert zeros between decimal point and first s.f. as needed (e.g., 0.003 435 to 3 s.f. → 0.003 44).
Unless stated otherwise, give final answers to 3 significant figures; in money use 2 d.p.; in angles use 1 d.p.

Estimation

Estimate by rounding each number to 1 significant figure (or a convenient value) before calculating.
Avoid rounding small numbers to zero, especially in denominators (division by zero is undefined).
For addition/multiplication: rounding both numbers up gives an overestimate; rounding both down gives an underestimate.
For subtraction/division: rounding the first number up and the second down gives an overestimate; the opposite gives an underestimate.
Estimation is useful for checking if an exact answer is reasonable.

Upper & Lower Bounds

Bounds describe the range a rounded number can lie between: lower bound (LB) ≤ x < upper bound (UB).
To find bounds: divide the degree of accuracy by 2, then add to get UB and subtract to get LB.
Example: 3.6 km (1 d.p.) → degree = 0.1, half = 0.05 → LB = 3.55, UB = 3.65.
The error interval is written as LB ≤ x < UB (e.g., 3.55 ≤ l < 3.65).
Always identify the degree of accuracy correctly (e.g., nearest metre, 2 s.f., etc.).

Bounds on a number line

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1.Write 0.3728 correct to 1 decimal place.
Easy
- A0.4
- B0.37
- C0.3
- D0.38
2.The height, h metres, of a tower is 128 m, correct to the nearest metre. Complete the statement about the value of h: ___ ≤ h < ___
Medium
- A127.5 ≤ h < 128.5
- B127 ≤ h < 129
- C127.5 ≤ h ≤ 128.5
- D128 ≤ h < 129
3.Write 867 correct to the nearest ten.
Easy
- A870
- B860
- C900
- D800
4.The length, l cm, of a pencil is 18 cm, correct to the nearest centimetre. Complete the statement about the value of l: ___ ≤ l < ___
Medium
- A17.5 ≤ l < 18.5
- B17 ≤ l < 19
- C17.5 ≤ l ≤ 18.5
- D18 ≤ l < 19
5.Write 849.481 correct to 1 decimal place.
Easy
- A849.5
- B849.4
- C849.48
- D850.0
6.Write 97.4236 correct to 3 decimal places.
Easy
- A97.424
- B97.423
- C97.420
- D97.400
7.Write 3.72194 correct to 3 decimal places.
Easy
- A3.722
- B3.721
- C3.720
- D3.700
8.Write 8379 correct to the nearest hundred.
Easy
- A8400
- B8300
- C8000
- D9000

Thêm câu hỏi

9.Write 6789 correct to the nearest 100.
Easy
- A6800
- B6700
- C7000
- D6000
10.Write 6789 correct to 3 significant figures.
Medium
- A6790
- B6780
- C6800
- D6700
11.The mass, m kilograms, of a horse is 429 kg, correct to the nearest kilogram. Complete this statement about the value of m: ___ ≤ m < ___
Medium
- A428.5 ≤ m < 429.5
- B428 ≤ m < 430
- C428.5 ≤ m ≤ 429.5
- D429 ≤ m < 430
12.Write 8473 correct to the nearest ten.
Easy
- A8470
- B8480
- C8500
- D8400
13.Write 16.086 correct to 2 decimal places.
Easy
- A16.09
- B16.08
- C16.10
- D16.00
14.Write 3.058 correct to 3 significant figures.
Medium
- A3.06
- B3.05
- C3.058
- D3.07
15.Genaro measures the length, l cm, of his desk as 120 cm, correct to the nearest centimetre. Complete the statement about the value of l: ___ ≤ l < ___
Medium
- A119.5 ≤ l < 120.5
- B119 ≤ l < 121
- C119.5 ≤ l ≤ 120.5
- D120 ≤ l < 121
16.Round 1850 to 2 significant figures.
Medium
- A1900
- B1800
- C1850
- D2000
17.Round 5.789 to 2 decimal places.
Easy
- A5.79
- B5.78
- C5.80
18.The height, h metres, of a tower is 76.3 m, correct to 1 decimal place. Complete the statement about the value of h.
Medium
- A76.25 ≤ h < 76.35
- B76.2 ≤ h < 76.4
- C76.3 ≤ h < 76.4
19.By rounding each number correct to 1 significant figure, estimate the value of (71 × 32.4) / 4.82.
Medium
- A420
- B400
- C350
- D450
20.Calculate √40 correct to 4 decimal places.
Easy
- A6.3249
- B6.3248
- C6.3246
- D6.3250
21.Write 0.047883 correct to 2 significant figures.
Medium
- A0.048
- B0.047
- C0.0479
22.Write 6.8167 correct to 3 significant figures.
Medium
- A6.82
- B6.81
- C6.817
23.T = (49.2 - 9.59) / (4.085 × 2.35). By writing each number correct to 1 significant figure, work out an estimate for T.
Hard
- A4
- B5
- C3
- D6
24.The length, l cm, of a line is 18.3 cm, correct to the nearest millimetre. Complete this statement about the value of l.
Easy
- A18.25 ≤ l < 18.35
- B18.2 ≤ l < 18.4
- C18.3 ≤ l < 18.4
25.By rounding each number correct to 1 significant figure, estimate the value of 90006 / 10.012.
Medium
- A9000
- B900
- C90000
26.The amount of water, w litres, in a jug is 1.5 litres, correct to the nearest 0.1 litre. Complete this statement about the value of w.
Easy
- A1.45 ≤ w < 1.55
- B1.4 ≤ w < 1.6
- C1.5 ≤ w < 1.6
27.The length, p cm, of a pencil is 9.8 cm, correct to 2 significant figures. Complete the statement about the value of p.
Easy
- A9.75 ≤ p < 9.85
- B9.7 ≤ p < 9.9
- C9.8 ≤ p < 9.9
28.The length of a truck, L metres, is 8.2 m, correct to 1 decimal place. Complete this statement about the value of L.
Easy
- A8.15 ≤ L < 8.25
- B8.1 ≤ L < 8.3
- C8.2 ≤ L < 8.3
29.When rounding 5.678 to 2 decimal places, the answer is 5.68.
Easy
True or false?
30.What is 0.00456 correct to 2 significant figures?
Easy
- A0.0045
- B0.0046
- C0.005
- D0.00456
31.The length, l cm, of a sheet of paper is 29.7 cm, correct to the nearest millimetre. Which error interval is correct?
Medium
- A29.65 ≤ l < 29.75
- B29.7 ≤ l < 29.8
- C29.6 ≤ l < 29.8
- D29.65 ≤ l ≤ 29.75
32.The number 0.047883 rounded to 2 significant figures is 0.048.
Easy
True or false?
33.Which of the following are correct to 3 significant figures? (Select all that apply)
Medium
- A0.00296
- B0.0030
- C0.003
- D0.0029
34.Arrange these numbers in order, starting with the smallest: 17/6, 34%, 25/9, 0.345
Medium
- 17/6
- 34%
- 25/9
- 0.345
35.The mass, m grams, of a banana is 115 g, correct to the nearest 5 g. Which error interval is correct?
Medium
- A112.5 ≤ m < 117.5
- B110 ≤ m < 120
- C115 ≤ m < 120
- D112.5 ≤ m ≤ 117.5
36.Match each number to its correct rounding to 1 significant figure.
Medium
- 37.8
- 13.2
- 28.5
- 22.1
- 40
- 10
- 30
- 20
37.The mass of a bag of flour is 2.5 kg, correct to the nearest 0.1 kg. What is the upper bound of the mass?
Medium
- A2.55 kg
- B2.6 kg
- C2.45 kg
- D2.5 kg
38.Complete the statement about the value of p when p = 9.8 cm, correct to 2 significant figures.
Medium
____ ≤ p < ____
39.The length, l cm, of a line is 18.3 cm, correct to the nearest millimetre. Write down the upper bound of l.
Easy
40.By rounding each number to 1 significant figure, estimate the value of (37.8 × 13.2) / (28.5 + 22.1).
Hard
41.Complete the error interval for the length l = 29.7 cm, correct to the nearest millimetre.
Medium
____ ≤ l < ____
42.A number is 3.6 correct to 1 decimal place. Calculate the lower bound of this number.
Easy
43.The length of a rope is 15.8 m correct to the nearest 0.1 m. Find the upper bound of its length in metres.
Easy
44.Complete the error interval for a number x = 2400 correct to the nearest 100.
Medium
____ ≤ x < ____
45.By rounding each number to 1 significant figure, estimate the value of (19.6 × 5.2) / (0.48).
Easy
46.A rectangular field has length 45.6 m and width 28.3 m, each measured correct to the nearest 0.1 m. Calculate the maximum possible area of the field.
Medium
47.A car travels 48.7 miles on 4.2 gallons of petrol. By rounding each number to 1 significant figure, estimate the number of miles per gallon.
Easy
48.A rectangle has length 8.2 cm and width 3.6 cm, both correct to 1 decimal place. Calculate the maximum possible area of the rectangle. Show your working.
Medium
49.The length, l cm, of a line is 18.3 cm, correct to the nearest millimetre. Write down the upper bound of l.
Easy
- A17.35 cm
- B19.35 cm
- C20.35 cm
- D18.35 cm
50.By rounding each number to 1 significant figure, estimate the value of (37.8 × 13.2) / (28.5 + 22.1).
Medium
- A9
- B7
- C8
- D10
51.A number is 3.6 correct to 1 decimal place. Calculate the lower bound of this number.
Easy
- A5.55
- B3.55
- C4.55
- D2.55
52.The length of a rope is 15.8 m correct to the nearest 0.1 m. Find the upper bound of its length in metres.
Easy
- A14.85 m
- B15.85 m
- C16.85 m
- D17.85 m
53.A rectangular field has length 45.6 m and width 28.3 m, each measured correct to the nearest 0.1 m. Calculate the maximum possible area of the field.
Medium
- A1422.4 m²
- B1293.1 m²
- C1551.7 m²
- D1163.8 m²
54.A car travels 48.7 miles on 4.2 gallons of petrol. By rounding each number to 1 significant figure, estimate the number of miles per gallon.
Easy
- A10 mpg
- B11.5 mpg
- C12.5 mpg
- D15 mpg

Ghi chú ôn tập

## Rounding, Estimation & Bounds ### Rounding to a Given Place Value - Identify the digit in the required place value. - Count in the units you are rounding to (e.g., nearest 100 → count in 100s). - Circle the digit to the right: if ≥5 round up, else round down. - Put zeros in following place values before the decimal point. - For decimal places, give exactly the number of decimal places asked (e.g., 2.40 not 2.4). ### Rounding to Significant Figures - First significant figure: first non-zero digit from left. - Count to the required significant figure; use normal rounding rules. - For large numbers, fill with zeros up to the decimal point. - For decimals, fill zeros between decimal point and first significant figure. - Unless stated, round final answers to 3 significant figures. ### Estimation - Round each number to 1 significant figure (avoid rounding to 0). - Perform the calculation with rounded numbers. - Overestimate: numerator rounded up and/or denominator rounded down (for division). - Underestimate: numerator rounded down and/or denominator rounded up. ### Upper & Lower Bounds - Bounds: LB ≤ x < UB. - Degree of accuracy (e.g., nearest 100, 1 d.p., 2 s.f.). - Half the degree of accuracy: add to get UB, subtract to get LB. - Error interval: LB ≤ x < UB. - Examples: - 128 m (nearest m): 127.5 ≤ h < 128.5 - 18.3 cm (nearest mm): 18.25 ≤ l < 18.35 - 115 g (nearest 5 g): 112.5 ≤ m < 117.5 - 1.5 L (nearest 0.1 L): 1.45 ≤ w < 1.55

Rounding: Approximating a number to a given place value or significant figure by looking at the next digit (≥5 round up, else round down).
Significant figures: The digits in a number that carry meaning, starting from the first non-zero digit from the left.
Estimation: Finding an approximate value by rounding numbers (usually to 1 significant figure) and calculating.
Upper bound (UB): The largest value that a rounded number can be, but not including (x < UB).
Lower bound (LB): The smallest value that a rounded number can be, including (LB ≤ x).
Error interval: The range between the lower and upper bounds, written as LB ≤ x < UB.

Thẻ ghi nhớ

Round 0.3728 to 1 decimal place.

0.4

The height h of a tower is 128 m, correct to the nearest metre. Complete: ... ≤ h < ...

127.5 ≤ h < 128.5

Round 867 to the nearest ten.

870

The length l of a pencil is 18 cm, correct to the nearest centimetre. Complete: ... ≤ l < ...

17.5 ≤ l < 18.5

Round 849.481 to 1 decimal place.

849.5

Round 97.4236 to 3 decimal places.

97.424

Round 3.72194 to 3 decimal places.

3.722

Round 8379 to the nearest hundred.

8400

Round 6789 to the nearest 100.

6800

Round 6789 to 3 significant figures.

6790

The mass m of a horse is 429 kg, correct to the nearest kilogram. Complete: ... ≤ m < ...

428.5 ≤ m < 429.5

Round 8473 to the nearest ten.

8470

Round 16.086 to 2 decimal places.

16.09

Round 3.058 to 3 significant figures.

3.06

Genaro measures his desk as 120 cm, correct to the nearest centimetre. Complete: ... ≤ l < ...

119.5 ≤ l < 120.5

Round 1850 to 2 significant figures.

1900

By rounding each number to 1 significant figure, estimate (37.8 × 13.2) / (28.5 + 22.1).

(40 × 10) / (30 + 20) = 400 / 50 = 8

The height h of a tower is 76.3 m, correct to 1 decimal place. Complete: ... ≤ h < ...

76.25 ≤ h < 76.35

By rounding each number to 1 significant figure, estimate (71 × 32.4) / 4.82.

(70 × 30) / 5 = 2100 / 5 = 420

Calculate ∛40 (cube root of 40) correct to 4 decimal places.

∛40 ≈ 3.4200

Write 0.047883 correct to 2 significant figures.

0.048

Write 6.8167 correct to 3 significant figures.

6.82

The amount of fuel t is 135 litres, correct to the nearest litre. Complete: ... ≤ t < ...

134.5 ≤ t < 135.5

The length l of a line is 18.3 cm, correct to the nearest millimetre. Complete: ... ≤ l < ...

18.25 ≤ l < 18.35

The mass m of a banana is 115 g, correct to the nearest 5 g. Complete: ... ≤ m < ...

112.5 ≤ m < 117.5

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