Rounding Estimation And Bounds

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Rounding to Decimal Places

Identify the digit at the required decimal place; look at the next digit to the right.
If the next digit is 5 or more, round up (increase the digit). If less than 5, round down.
Write the answer with exactly the required number of decimal places (e.g., 2.40 for 2 d.p.).
Do not add extra zeros after the decimal place unless needed to show accuracy.

Rounding to Significant Figures

The first significant figure is the leftmost non-zero digit.
Count digits from the first significant figure to find 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 (e.g., 0.00344 to 3 s.f.).

Estimation

Round each number to 1 significant figure (or a convenient value) before calculating.
Avoid rounding to zero, especially in denominators.
For addition/multiplication: rounding up gives an overestimate; rounding down gives an underestimate.
For subtraction/division: rounding numerator up and denominator down gives overestimate; opposite gives underestimate.

Upper and Lower Bounds

Bounds are the smallest (LB) and largest (UB) values a rounded number can take.
For a number rounded to a given accuracy, half the degree of accuracy is added/subtracted to find UB/LB.
Error interval: LB ≤ x < UB (LB included, UB excluded).
Example: 3.6 (1 d.p.) → LB = 3.55, UB = 3.65.

Bounds of 24 800 (nearest 100)

Calculations with Bounds

For addition: UB = UB + UB, LB = LB + LB.
For subtraction: UB = UB – LB, LB = LB – UB.
For multiplication: UB = UB × UB, LB = LB × LB.
For division: UB = UB ÷ LB, LB = LB ÷ UB.

Applying Bounds in Context

To find maximum possible value, use upper bounds for quantities that increase the result and lower bounds for those that decrease it.
To find minimum possible value, use opposite bounds.
When counting discrete items (e.g., number of panels), round up for minimum needed, down for maximum possible.
Always check whether the context requires rounding up or down (e.g., people, crates).

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1.Write 0.046875 correct to 2 significant figures.
Easy
- A0.047
- B0.046
- C0.05
- D0.0469
2.Write the number 0.076499 correct to 2 significant figures.
Easy
- A0.076
- B0.077
- C0.0765
- D0.08
3.Calculate 2.38 + 6.42, writing down your full calculator display. Write your answer correct to 4 decimal places.
Easy
- A8.8000
- B8.8
- C8.80
- D8.800
4.A bag of crisps weighs 150 g to the nearest gram. Complete the statement about the weight, w g, in the bag. ... ≤ w < ...
Easy
- A149.5 ≤ w < 150.5
- B149 ≤ w < 151
- C149.5 ≤ w ≤ 150.5
- D150 ≤ w < 151
5.Calculate \frac{16.379 - 0.879}{4.2 \times 1.241} correct to 2 significant figures.
Medium
- A3.0
- B2.9
- C3.1
- D2.95
6.An equilateral triangle has side length 12 cm, correct to the nearest centimetre. Find the lower bound of the perimeter of the triangle.
Medium
- A34.5 cm
- B35 cm
- C34 cm
- D35.5 cm
7.The area of a square is 42.5 cm², correct to the nearest 0.5 cm². Calculate the lower bound of the length of the side of the square.
Medium
- A6.5 cm
- B6.48 cm
- C6.52 cm
8.Serge walks 7.9 km, correct to the nearest 100 metres. The walk takes 133 minutes, correct to the nearest minute. Calculate the maximum possible average speed of Serge’s walk in km/h.
Hard
- A3.6 km/h
- B3.58 km/h
- C3.59 km/h
- D3.57 km/h

Thêm câu hỏi

9.Saafia has a barrel containing 6000 millilitres of oil, correct to the nearest 100 ml. She uses the oil to fill bottles which each hold exactly 50 ml. Calculate the upper bound for the number of bottles she can fill.
Hard
- A120
- B121
- C119
- D122
10.A rectangle measures 8.5 cm by 10.7 cm, both correct to 1 decimal place. Calculate the upper bound of the perimeter of the rectangle.
Medium
- A38.5 cm
- B38.4 cm
- C38.6 cm
- D38.7 cm
11.The formula s = 1/2 a t² is used to calculate the distance, s, travelled by a bicycle. When a = 3 and t = 10, each correct to the nearest integer, calculate the lower bound of the distance, s.
Medium
- A112.8
- B125
- C100.5
- D118.6
12.A road is 4530 m long, correct to the nearest 10 metres. Kirsty drove along the road in 205 seconds, correct to the nearest 5 seconds. The average speed limit for the road is 80 km/h. Could Kirsty's average speed have been greater than 80 km/h?
Challenge
- AYes, maximum speed is 80.6 km/h
- BNo, maximum speed is 79.8 km/h
- CYes, maximum speed is 81.2 km/h
- DNo, maximum speed is 78.9 km/h
13.Write 0.047883 correct to 2 significant figures.
Easy
- A0.048
- B0.047
- C0.0479
- D0.05
14.Write 1.8972 correct to 2 decimal places.
Easy
- A1.90
- B1.89
- C1.9
- D1.897
15.Write the number 10 100 correct to 2 significant figures.
Easy
- A10000
- B11000
- C10100
- D1000
16.Write 209 802 correct to the nearest thousand.
Easy
- A210000
- B209000
- C200000
- D209800
17.Write 4123 correct to 3 significant figures.
Easy
- A4120
- B4130
- C4123
- D4100
18.The mass of a cat is 4.3 kg correct to 2 significant figures. Write down the upper bound of the weight of the cat.
Easy
- A4.35 kg
- B4.4 kg
- C4.34 kg
- D4.45 kg
19.The length of a book is 33.8 cm, correct to one decimal place. Write down the lower bound of the length.
Easy
- A33.75 cm
- B33.7 cm
- C33.85 cm
- D33.8 cm
20.Calculate an estimate for \frac{17.3 \times 3.81}{11.5}.
Medium
- A6
- B8
- C7
- D9
21.The length of a rectangle is 12.3 cm, correct to 1 decimal place. The width is 8.7 cm, correct to 1 decimal place. Calculate the lower bound of the perimeter of the rectangle.
Medium
- A41.6 cm
- B41.8 cm
- C42.0 cm
- D42.2 cm
22.A car travels a distance of 250 km, correct to the nearest 10 km. The journey takes 4.5 hours, correct to the nearest 0.1 hour. Calculate the upper bound of the average speed in km/h.
Medium
- A56.8 km/h
- B57.0 km/h
- C57.3 km/h
- D57.5 km/h
23.The mass of a bag of flour is 1.5 kg, correct to the nearest 0.1 kg. What is the upper bound of the mass?
Easy
- A1.45 kg
- B1.50 kg
- C1.55 kg
- D1.6 kg
24.Write the number 0.05037 correct to 3 significant figures.
Easy
- A0.0503
- B0.0504
- C0.05037
- D0.050
25.A rectangle measures 15.2 cm by 8.6 cm, both correct to 1 decimal place. Calculate the lower bound of the area of the rectangle.
Medium
- A129.2 cm²
- B129.5 cm²
- C129.8 cm²
- D130.1 cm²
26.The length of a side of a square is 6.3 cm, correct to 1 decimal place. Find the upper bound of the perimeter of the square.
Medium
- A25.0 cm
- B25.2 cm
- C25.4 cm
- D25.6 cm
27.A cylindrical can has a radius of 4.2 cm, correct to 1 decimal place, and a height of 10.5 cm, correct to 1 decimal place. Calculate the upper bound of the volume of the can. (Volume = \π r² h, use \π = 3.14)
Hard
- A580 cm³
- B590 cm³
- C600 cm³
- D610 cm³
28.The number of students in a school is 850, correct to the nearest 10. What is the lower bound of the number of students?
Easy
- A845
- B849
- C850
- D855
29.A baker uses 2.5 kg of flour, correct to the nearest 0.1 kg, to make 12 cakes, each of mass 200 g, correct to the nearest 10 g. Does he definitely have enough flour to make 12 cakes?
Hard
- AYes, because the lower bound of flour is greater than the upper bound of total cake mass.
- BNo, because the upper bound of flour is less than the lower bound of total cake mass.
- CYes, because the upper bound of flour is greater than the lower bound of total cake mass.
- DNo, because the lower bound of flour is less than the upper bound of total cake mass.
30.Round 0.006 408 to 2 significant figures.
Easy
- A0.0064
- B0.00641
- C0.00640
31.The length of a rectangle is 8.4 cm, correct to 1 decimal place, and the width is 5.2 cm, correct to 1 decimal place. Calculate the lower bound of the area.
Medium
- A43.0 cm²
- B42.9 cm²
- C43.2 cm²
- D43.5 cm²
32.What is the first significant figure of 0.00308?
Easy
- A0
- B3
- C8
- D1
33.When rounding to 2 decimal places, the number 3.456 becomes 3.46.
Easy
True or false?
34.A rectangle has length 8.4 cm and width 5.2 cm, each correct to 1 decimal place. Which calculation gives the lower bound of the area?
Medium
- A8.35 × 5.15
- B8.45 × 5.25
- C8.35 × 5.25
- D8.45 × 5.15
35.A number is rounded to 2 significant figures and becomes 4700. Which of the following could be the original number? (Select all that apply)
Medium
- A4650
- B4749
- C4649
- D4750
- E4699
36.Arrange these numbers in increasing order: 0.345, 34%, 17/6, 0.345 (repeating).
Medium
- 17/6
- 34%
- 0.345
- 0.345 (repeating)
37.Match each rounded number to its correct description. (Round each to the stated degree of accuracy.)
Easy
- 0.00308 (3 s.f.)
- 209802 (3 s.f.)
- 4123 (3 s.f.)
- 1.8972 (2 d.p.)
- 0.00308
- 210000
- 4120
- 1.90
38.Round 0.05037 to 3 significant figures.
Easy
39.Complete the error interval for a length L = 6.3 cm correct to 1 decimal place.
Easy
____ ≤ L < ____
40.Write the number 0.003 059 correct to 2 significant figures.
Easy
0.003 059 correct to 2 significant figures is ____.
41.A length is measured as 8.2 cm, correct to the nearest 0.1 cm. Find the lower bound of the length.
Easy
42.Estimate the value of (9.82 × 4.13) / 2.97 by rounding each number to 1 significant figure. State, with a reason, whether your estimate is an overestimate or an underestimate.
Medium
43.A cylindrical can has a radius of 4.2 cm, correct to 1 decimal place, and a height of 10.5 cm, correct to 1 decimal place. Calculate the upper bound of the volume of the can. Use π = 3.142.
Hard
44.Estimate the value of (9.82 × 4.13) / 2.97 by rounding each number to 1 significant figure.
Medium
45.A rectangle has length 15.2 cm, correct to the nearest 0.1 cm, and width 8.6 cm, correct to the nearest 0.1 cm. Calculate the lower bound of the area of the rectangle. Show your working.
Medium
46.A length is measured as 8.2 cm, correct to the nearest 0.1 cm. Find the lower bound of the length.
Easy
- A7.15 cm
- B10.15 cm
- C9.15 cm
- D8.15 cm
47.Estimate the value of (9.82 × 4.13) / 2.97 by rounding each number to 1 significant figure.
Medium
- A14.3
- B15.3
- C12.3
- D13.3

Ghi chú ôn tập

## Rounding, Estimation & Bounds ### Rounding - **Significant figures (s.f.)**: First non-zero digit is first s.f. Count zeros between non-zero digits. Round using the digit to the right (≥5 round up, <5 round down). Fill with zeros for large numbers. - **Decimal places (d.p.)**: Identify required d.p., check next digit, round accordingly. Keep trailing zeros to show accuracy. - **General rule**: If not specified, round final answer to 3 s.f. For money, round to 2 d.p. For angles, round to 1 d.p. ### Estimation - Round each number to 1 s.f. (avoid rounding to 0). Perform calculation. - **Over/underestimate**: For +/×, rounding up both gives overestimate; rounding down both gives underestimate. For −/÷, rounding up numerator/denominator down gives overestimate; rounding down numerator/denominator up gives underestimate. ### Upper & Lower Bounds - **Finding bounds**: Given number rounded to nearest unit, divide accuracy by 2. LB = number – half, UB = number + half. Error interval: LB ≤ x < UB. - **Calculations with bounds**: - Addition: UB = UB+UB, LB = LB+LB - Subtraction: UB = UB–LB, LB = LB–UB - Multiplication: UB = UB×UB, LB = LB×LB - Division: UB = UB÷LB, LB = LB÷UB - **Context**: For maximum/minimum, use appropriate bounds (e.g., max speed = UB distance ÷ LB time). Round up/down based on real-life constraints (e.g., number of items).

Significant figures: The digits in a number that carry meaning, starting from the first non-zero digit. Used to round numbers to a specified accuracy.
Decimal places: The number of digits after the decimal point to which a number is rounded.
Estimation: Finding an approximate value by rounding numbers (usually to 1 significant figure) and performing the calculation.
Upper bound: The largest value that a rounded number can be, but not including. Given by number + half the degree of accuracy.
Lower bound: The smallest value that a rounded number can be, including. Given by number – half the degree of accuracy.
Error interval: The range between the lower and upper bounds, written as LB ≤ x < UB.

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