1. Overview
Limits of accuracy deal with the fact that measurements are never perfectly precise. When a value is rounded, it could have been slightly higher or lower than the stated value. The upper bound is the largest possible value before rounding down, and the lower bound is the smallest possible value before rounding up. This topic is essential for understanding the range of possible values in practical situations and calculations.
Key Definitions
- Upper Bound (UB): The maximum possible value that a measurement could have been before it was rounded up.
- Lower Bound (LB): The minimum possible value that a measurement could have been before it was rounded down.
- Degree of Accuracy: The unit to which a number has been rounded (e.g., "nearest 10", "1 decimal place", "3 significant figures").
- Error Interval: The range of values written as an inequality: $LB \le x < UB$.
Core Content: Finding Bounds for Data
To find the limits of accuracy for a single measurement, follow the "Half-Unit" Rule:
- Identify the unit of accuracy (the degree to which the number was rounded).
- Divide that unit by 2.
- Subtract this value from the measurement to find the Lower Bound.
- Add this value to the measurement to find the Upper Bound.
Worked example 1 — Nearest Whole Number
A length $L$ is given as $12 \text{ cm}$, correct to the nearest centimeter. Find the upper and lower bounds.
- Step 1: The unit of accuracy is $1 \text{ cm}$.
- Step 2: Half the unit is $1 \div 2 = 0.5 \text{ cm}$.
- Step 3 (LB): $12 - 0.5 = 11.5 \text{ cm}$.
- Step 4 (UB): $12 + 0.5 = 12.5 \text{ cm}$.
- Error Interval: $11.5 \le L < 12.5$.
Worked example 2 — Significant Figures
A mass $M$ is $4.8 \text{ kg}$, correct to 2 significant figures. Find the bounds.
- Step 1: The last significant figure is in the first decimal place ($0.1$).
- Step 2: Half the unit is $0.1 \div 2 = 0.05$.
- Step 3 (LB): $4.8 - 0.05 = 4.75 \text{ kg}$.
- Step 4 (UB): $4.8 + 0.05 = 4.85 \text{ kg}$.
- Error Interval: $4.75 \le M < 4.85$.
Worked example 3 — Decimal Places
The height of a plant is recorded as $15.6 \text{ cm}$, measured to the nearest tenth of a centimeter. Find the lower and upper bounds for the actual height.
- Step 1: The unit of accuracy is $0.1 \text{ cm}$ (one decimal place).
- Step 2: Half of the unit of accuracy is $0.1 \div 2 = 0.05 \text{ cm}$.
- Step 3 (LB): To find the lower bound, subtract half the unit of accuracy from the measurement: $15.6 - 0.05 = 15.55 \text{ cm}$.
- Step 4 (UB): To find the upper bound, add half the unit of accuracy to the measurement: $15.6 + 0.05 = 15.65 \text{ cm}$.
- Error Interval: $15.55 \le \text{height} < 15.65$.
Worked example 4 — Nearest 10
The attendance at a concert is recorded as 350 people, to the nearest 10 people. What are the upper and lower bounds for the number of attendees?
- Step 1: The unit of accuracy is 10.
- Step 2: Half of the unit of accuracy is $10 \div 2 = 5$.
- Step 3 (LB): To find the lower bound, subtract half the unit of accuracy from the measurement: $350 - 5 = 345$.
- Step 4 (UB): To find the upper bound, add half the unit of accuracy to the measurement: $350 + 5 = 355$.
- Error Interval: $345 \le \text{attendance} < 355$.
4. Supplement Content (Extended Curriculum Only)
When performing calculations (addition, subtraction, multiplication, division) using rounded numbers, you must combine the bounds carefully to find the maximum or minimum possible result. The goal is to determine the largest and smallest possible outcomes given the uncertainty in the initial values.
Rules for Combined Bounds
| Operation | To get the Upper Bound (UB) | To get the Lower Bound (LB) |
|---|---|---|
| Addition | $UB_1 + UB_2$ | $LB_1 + LB_2$ |
| Multiplication | $UB_1 \times UB_2$ | $LB_1 \times LB_2$ |
| Subtraction | $UB_{first} - LB_{second}$ | $LB_{first} - UB_{second}$ |
| Division | $UB_{top} \div LB_{bottom}$ | $LB_{top} \div UB_{bottom}$ |
Worked Example: Speed (Division)
A car travels a distance $d = 100 \text{ m}$ (to the nearest metre) in a time $t = 20 \text{ s}$ (to the nearest second). Calculate the upper bound for the average speed.
1. Find individual bounds first:
- $d$: Unit is $1$. Half-unit is $0.5$. $LB = 99.5$, $UB = 100.5$.
- $t$: Unit is $1$. Half-unit is $0.5$. $LB = 19.5$, $UB = 20.5$.
2. Choose the correct formula for UB of Speed: To make a fraction as large as possible (Upper Bound), you need the largest numerator and the smallest denominator. $$\text{Speed}{UB} = \frac{\text{Distance}{UB}}{\text{Time}_{LB}}$$
3. Calculate: $$\text{Speed}{UB} = \frac{100.5}{19.5}$$ $$\text{Speed}{UB} = 5.1538...$$ Final Answer: $5.15 \text{ m/s}$ (3 s.f.)
Worked example 2 — Area of a Rectangle
The length of a rectangle is measured as $8.5 \text{ cm}$ to the nearest $0.1 \text{ cm}$, and the width is measured as $6.2 \text{ cm}$ to the nearest $0.1 \text{ cm}$. Calculate the lower bound for the area of the rectangle.
Step 1: Find the lower bound for the length: The unit of accuracy is $0.1 \text{ cm}$, so half the unit is $0.05 \text{ cm}$. $L_{LB} = 8.5 - 0.05 = 8.45 \text{ cm}$.
Step 2: Find the lower bound for the width: The unit of accuracy is $0.1 \text{ cm}$, so half the unit is $0.05 \text{ cm}$. $W_{LB} = 6.2 - 0.05 = 6.15 \text{ cm}$.
Step 3: Calculate the lower bound for the area: To find the lower bound of the area, multiply the lower bounds of the length and width. $A_{LB} = L_{LB} \times W_{LB} = 8.45 \times 6.15 = 51.9675 \text{ cm}^2$.
Step 4: Round to a reasonable number of significant figures (e.g., 3 s.f.): $A_{LB} = 52.0 \text{ cm}^2$.
Final Answer: The lower bound for the area of the rectangle is $52.0 \text{ cm}^2$.
Worked example 3 — Calculating a Difference
A shelf is supposed to be $200 \text{ cm}$ long, to the nearest cm. Two books are placed on the shelf. The first book is $35 \text{ cm}$ long to the nearest cm, and the second book is $42 \text{ cm}$ long to the nearest cm. Calculate the upper bound for the remaining length of the shelf.
Step 1: Find the upper bound for the shelf length: The unit of accuracy is $1 \text{ cm}$, so half the unit is $0.5 \text{ cm}$. $Shelf_{UB} = 200 + 0.5 = 200.5 \text{ cm}$.
Step 2: Find the lower bound for the length of the first book: The unit of accuracy is $1 \text{ cm}$, so half the unit is $0.5 \text{ cm}$. $Book1_{LB} = 35 - 0.5 = 34.5 \text{ cm}$.
Step 3: Find the lower bound for the length of the second book: The unit of accuracy is $1 \text{ cm}$, so half the unit is $0.5 \text{ cm}$. $Book2_{LB} = 42 - 0.5 = 41.5 \text{ cm}$.
Step 4: Calculate the lower bound for the total length of the two books: $TotalBooks_{LB} = Book1_{LB} + Book2_{LB} = 34.5 + 41.5 = 76 \text{ cm}$.
Step 5: Calculate the upper bound for the remaining length of the shelf: To maximize the remaining length, subtract the minimum length of the books from the maximum length of the shelf. $RemainingLength_{UB} = Shelf_{UB} - TotalBooks_{LB} = 200.5 - 76 = 124.5 \text{ cm}$.
Final Answer: The upper bound for the remaining length of the shelf is $124.5 \text{ cm}$.
5. Key Equations & Notation
| Concept | Notation/Formula | Notes |
|---|---|---|
| Error Interval | $LB \le x < UB$ | Note: $\le$ for LB, but strictly $<$ for UB. |
| Range of Bounds | $\pm \frac{1}{2} \times \text{Degree of Accuracy}$ | The "Half-Unit" rule. |
| Area Bounds | $Area_{UB} = L_{UB} \times W_{UB}$ | For rectangles. |
| Speed Bounds | $S_{LB} = \frac{D_{LB}}{T_{UB}}$ | Divide small by big to get the smallest result. |
Half-Unit Rule: $\text{Adjustment} = \frac{1}{2} \times \text{Unit of Accuracy}$
Upper Bound: $UB = \text{Measurement} + \text{Adjustment}$
Lower Bound: $LB = \text{Measurement} - \text{Adjustment}$
Note: These formulas are NOT provided on the formula sheet.
6. Common Mistakes to Avoid
- ❌ Wrong: Rounding intermediate answers during calculations. This introduces errors.
- ✅ Right: Keep full calculator displays until the very final step to avoid "rounding errors." Only round your final answer to the required precision.
- ❌ Wrong: Thinking that $12.49$ or $12.499$ is the upper bound for $12$ (nearest whole number).
- ✅ Right: Always use the exact half-way point ($12.5$). The inequality sign ($<$) in the error interval ($11.5 \le x < 12.5$) indicates that the value can be arbitrarily close to $12.5$ but not equal to it.
- ❌ Wrong: Always adjusting by $0.5$, regardless of the degree of accuracy.
- ✅ Right: The adjustment depends on the unit of accuracy. If rounded to the nearest $0.1$, the adjustment is $0.05$. If rounded to the nearest $5$, the adjustment is $2.5$. Always find half of the unit of accuracy.
- ❌ Wrong: For subtraction, finding the upper bound by subtracting the upper bounds.
- ✅ Right: For subtraction, to find the upper bound of the difference, subtract the lower bound from the upper bound: $UB = UB_{first} - LB_{second}$.
- ❌ Wrong: Forgetting to consider units.
- ✅ Right: Always check that all measurements are in the same units before performing calculations. Convert if necessary.
7. Exam Tips
- Command Words: Look for "Calculate the upper/lower bound of..." or "Give the limit of accuracy for...".
- Real-World Contexts: Expect questions involving fences (perimeter), flooring (area), fuel consumption, or average speed.
- Calculator Tip: In the Extended paper, if you are asked to show your calculation, write down the full unrounded values for the bounds before plugging them into the formula. This demonstrates your understanding and avoids losing marks due to premature rounding.
- Memorisation: The rules for subtraction and division bounds are not on the formula sheet. You must memorize that for division, $UB = UB \div LB$ and $LB = LB \div UB$.
- Check Units: Ensure your bounds are in the same units as the question asks for in the final answer (e.g., converting grams to kilograms).
- Show Your Working: Even if the answer seems obvious, clearly show how you calculated the upper and lower bounds. This helps you get partial credit even if you make a small mistake.