Bounds and Limits of Accuracy
The "Half Unit" Rule
When a number is rounded, the "true" value could be slightly higher or lower. We call these the Upper Bound (UB) and Lower Bound (LB).
Step 1: Identify what the number was rounded to (e.g., nearest 10, nearest 0.1, nearest 0.01). This is your "Unit".
Step 2: Halve that value.
Step 3: Add it for the UB, subtract it for the LB.
Notation: The interval is usually written as: \( \text{LB} \le x < \text{UB} \)
1. Finding Basic Bounds
Example: 60 (rounded to the nearest 10)
- Unit = 10. Half Unit = 5.
- Upper Bound: \(60 + 5 = 65\)
- Lower Bound: \(60 - 5 = 55\)
Note: Even though 65 would round up to 70, we treat 65 as the mathematical boundary (the Upper Bound).
Example: 67 (rounded to the nearest integer)
- Unit = 1 (nearest whole number). Half Unit = 0.5.
- Upper Bound: \(67 + 0.5 = 67.5\)
- Lower Bound: \(67 - 0.5 = 66.5\)
Example: 3.5 (rounded to 1 decimal place)
- Unit = 0.1. Half Unit = 0.05.
- Upper Bound: \(3.5 + 0.05 = 3.55\)
- Lower Bound: \(3.5 - 0.05 = 3.45\)
Example: 3.4 (rounded to 1 decimal place)
- Unit = 0.1. Half Unit = 0.05.
- Upper Bound: \(3.4 + 0.05 = 3.45\)
- Lower Bound: \(3.4 - 0.05 = 3.35\)
Example: 3.40 (rounded to 2 decimal places)
- Unit = 0.01. Half Unit = 0.005.
- Upper Bound: \(3.40 + 0.005 = 3.405\)
- Lower Bound: \(3.40 - 0.005 = 3.395\)
Tip: The trailing zero in 3.40 is important! It tells us the precision is to 2 decimal places.
2. Calculating with Bounds (Max & Min)
When you add, subtract, multiply, or divide numbers that have been rounded, you need to be careful which bound you use.
Addition (Max)
\( \text{UB} + \text{UB} \)
Addition (Min)
\( \text{LB} + \text{LB} \)
Subtraction (Max)
\( \text{UB} - \text{LB} \)
(Start big, take away small)
Subtraction (Min)
\( \text{LB} - \text{UB} \)
(Start small, take away big)
Division (Max)
\( \text{UB} \div \text{LB} \)
(Big numerator, small denominator)
Division (Min)
\( \text{LB} \div \text{UB} \)
(Small numerator, big denominator)
Practice Questions & Solutions
Click on the question to check your working.
Q1: A length is 50cm to the nearest cm. Write the error interval.
Rounded to nearest 1 cm.
Half unit = \(0.5\).
LB = \(50 - 0.5 = 49.5\)
UB = \(50 + 0.5 = 50.5\)
Interval: \(49.5 \le x < 50.5\)
Q2: Weight \(w = 4.2\)kg (1 d.p). Find the Upper Bound.
Rounded to nearest 0.1.
Half unit = \(0.05\).
UB = \(4.2 + 0.05\)
Upper Bound = 4.25
Q3: \(x = 20\) (nearest 10), \(y = 5\) (nearest whole number). Calculate the Maximum value of \(x + y\).
Step 1: Find UBs
\(x\) (nearest 10): UB = 25
\(y\) (nearest 1): UB = 5.5
Step 2: Calculate Max Sum
Max = \(UB_x + UB_y = 25 + 5.5\)
Answer = 30.5
Q4: \(a = 15\) (nearest integer), \(b = 10\) (nearest integer). Calculate the Maximum value of \(a - b\).
Step 1: Find Bounds
\(a\): UB = 15.5
\(b\): LB = 9.5
Step 2: Logic
To get the biggest difference, we want a BIG starting number minus a SMALL number.
Max Diff = \(UB_a - LB_b = 15.5 - 9.5\)
Answer = 6
Q5: Area of a rectangle. Length = 8.4cm, Width = 3.5cm (both 1 d.p). Calculate Minimum Area.
To find the Minimum Area, we need the smallest possible length and width.
LB Length: \(8.35\)
LB Width: \(3.45\)
Min Area = \(8.35 \times 3.45\)
Answer = 28.8075 cm²
Q6: Distance = 100m (nearest 10m), Time = 9.5s (nearest 0.1s). Calculate the Maximum Speed.
Formula: \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)
To get Max Speed, we need Max Distance divided by Min Time.
Max Dist (UB) = 105
Min Time (LB) = 9.45
Max Speed = \( \frac{105}{9.45} \)
Answer = 11.11... m/s
Q7: \(y = \frac{10}{x}\). If \(x = 2.5\) (1 d.p), what is the Lower Bound of \(y\)?
To make a fraction as small as possible (Lower Bound), you need to divide by the largest possible number.
We need the UB of \(x\).
UB of 2.5 is 2.55.
Calculation: \( \frac{10}{2.55} \)
Answer = 3.92 (approx)
Q8: \(x = 5.3\) truncated to 1 decimal place. What is the Upper Bound?
Watch out! "Truncated" means the digits were just cut off, not rounded.
If a number is 5.3... truncated, it could have been 5.3000... up to 5.3999...
LB = 5.3
UB = 5.4
Upper Bound = 5.4
Q9: Calculate the error interval for \(x\) if \(x = 300\) rounded to 1 s.f.
Rounding to 1 s.f means rounding to the nearest 100 in this case.
Unit = 100. Half Unit = 50.
LB = \(300 - 50 = 250\)
UB = \(300 + 50 = 350\)
Interval: \(250 \le x < 350\)
Q10: \(P = 2x + y\). \(x=4.5\), \(y=12\) (both to nearest integer). Calculate Max P.
This is a trick question on finding bounds first.
Let's assume the question meant: \(x\) and \(y\) were rounded to the nearest integer to get these values.
\(x = 4.5\) (rounded to nearest integer). Wait, 4.5 rounded to nearest integer is 5. The question is slightly confusing as written for a student.
Let's clarify the standard exam interpretation: The values given are the result of rounding.
If \(x\) was rounded to the nearest integer and the result is 4.5, that's impossible. 4.5 is the boundary between 4 and 5.
Alternative Interpretation: Perhaps \(x\) and \(y\) are just exact values? No, the topic is bounds.
Most likely Interpretation for this level: The inputs to the rounding process were rounded to the nearest integer. E.g., "x is 5 (to nearest int), y is 12 (to nearest int)". Let's solve that version as it's mathematically sound.
Revised Q10: \(P = 2x + y\). \(x=5\) (nearest integer), \(y=12\) (nearest integer). Calculate Max P.
UB \(x\) = 5.5
UB \(y\) = 12.5
Max P = \(2(\text{UB } x) + (\text{UB } y)\)
Max P = \(2(5.5) + 12.5 = 11 + 12.5\)
Answer = 23.5