Rounding Estimation And Bounds

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Notes

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 $\le x <$ UB** (LB included, UB excluded).
Example: 3.6 (1 d.p.) → $LB = 3.55, UB = 3.65$ .

Calculations with Bounds

For addition: $UB = UB + UB, LB = LB + LB$ .
For subtraction: $UB = UB$ – $LB, LB = LB$ – UB.
For multiplication: $UB = UB \times UB, LB = LB \times LB$ .
For division: $UB = UB \div LB, LB = LB \div UB$ .

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).

Rounding to Decimal Places

Significant Figures

Upper and Lower Bounds

Bounds in Calculations

Practice questions

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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. ... $\le w <$ ...
Easy
- A $149.5 \le w < 150.5$
- B $149 \le w < 151$
- C $149.5 \le w \le 150.5$
- D $150 \le w < 151$
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 $42.5 cm^{2}$ , correct to the nearest $0.5 cm^{2}$ . 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

Summary

## 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).

Key terms

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.

Slides

Rounding to Significant Figures

The first significant figure is the first non-zero digit from the left.
To round to a given number of significant figures, count that many digits from the first significant figure.
Look at the next digit: if it's 5 or more, round up; otherwise, round down.
Fill with zeros if needed to preserve place value.

Round 0.047883 to 2 significant figures.

0.048
0.047
0.05
0.0479

Rounding to Decimal Places

Identify the digit at the required decimal place.
Look at the next digit to the right: if it's 5 or more, round up; otherwise, round down.
Always show the required number of decimal places, even if the last digit is zero (e.g., 2.40 not 2.4).

Round 1.8972 to 2 decimal places: ____

Estimation by Rounding to 1 Significant Figure

Estimate calculations by rounding each number to 1 significant figure.
Avoid rounding small numbers to zero, especially in denominators.
Use the estimate to check if your exact answer is reasonable.

Estimate $17.3 \times 3.81 \div 11.5$ by rounding each number to 1 significant figure.

Overestimates and Underestimates

For addition and multiplication: rounding up gives an overestimate; rounding down gives an underestimate.
For subtraction and division: rounding up the first number and down the second gives an overestimate; the opposite gives an underestimate.
If both are rounded the same way, the effect is not straightforward.

Which of these are true?

Rounding both numbers up in a multiplication gives an overestimate.
Rounding the numerator up and denominator down in a division gives an overestimate.
Rounding both numbers down in a subtraction gives an overestimate.

Upper and Lower Bounds

Bounds are the range of possible values for a rounded number.
Lower bound (LB) is the smallest possible value; upper bound (UB) is the largest value less than the bound.
Error interval: $LB \le x < UB$ .
To find bounds: divide the degree of accuracy by 2, then add/subtract from the rounded value.

A length is 3.6 km correct to 1 decimal place. Lower bound $= ____ km$ , Upper bound $= ____ km$ .

Bounds in Calculations

For addition: $UB = UB + UB, LB = LB + LB$ .
For subtraction: $UB = UB - LB, LB = LB - UB$ .
For multiplication: $UB = UB \times UB, LB = LB \times LB$ .
For division: $UB = UB \div LB, LB = LB \div UB$ .

A rectangle measures 4 m by 7 m, each to the nearest metre. What is the upper bound of the area?

$33.75 m^{2}$
$28 m^{2}$
$30.25 m^{2}$
$35 m^{2}$

Using Bounds in Context

To find the maximum possible result, use upper bounds for quantities that increase the result and lower bounds for those that decrease it.
To find the minimum possible result, do the opposite.
Often need to round up or down to whole numbers in real-life problems.

Fencing panels are 50 cm long (nearest 10 cm). Path is 6 m (nearest 10 cm). What is the minimum number of panels needed?

Determining Accuracy from Bounds

If the lower and upper bounds of a calculation round to the same number at a given degree of accuracy, that is the true value to that accuracy.
Compare the bounds to find the highest precision where they agree.

A value has $LB = 8.33217$ and $UB = 8.33198$ . To how many significant figures do both round to the same number? $____ s.f$ .

Error Intervals

An error interval shows the range of possible values for a rounded number.
Written as: lower bound $\le x <$ upper bound.
The lower bound is included, the upper bound is not.

A mass is 429 kg correct to the nearest kilogram. Which error interval is correct?

$428.5 \le m < 429.5$
$428 \le m < 430$
$428.5 \le m \le 429.5$
$429 \le m < 430$

Summary

Rounding: to significant figures, decimal places, or nearest unit.
Estimation: round to 1 s.f. for quick checks.
Bounds: find the range of possible values and use them in calculations to find maximum/minimum results.

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