Bounds (AQA GCSE Maths): Revision Notes
Bounds
Understanding bounds is essential when working with measurements and calculations. When we measure something, the actual value might be slightly different from what we record, and bounds help us understand this uncertainty.
What are bounds?
Bounds tell us the range of possible actual values when we have a measurement that has been rounded or truncated. Think of bounds as the "wiggle room" around a measured value - they show us the smallest and largest values that the actual measurement could be.
Understanding Measurement Uncertainty
Every measurement has some degree of uncertainty. When we see a measurement like "2.4 kg", this doesn't mean the actual mass is exactly 2.4 kg - it could be anywhere within a small range around this value. Bounds help us define exactly what this range is.
Upper and lower bounds
When we have a measurement, we need to consider how it was obtained to find the correct bounds. The method used to obtain the measurement (rounding or truncation) determines how we calculate the bounds.
Rounded measurements
When a measurement has been rounded to a given unit, the actual measurement could be anything up to half a unit bigger or smaller than the recorded value.
Worked Example: Finding Bounds for Rounded Measurements
If a cake's mass is given as 2.4 kg to the nearest 0.1 kg:
Step 1: Identify the rounding unit = 0.1 kg Step 2: Calculate half the unit = 0.1 ÷ 2 = 0.05 kg Step 3: Find the bounds:
- Lower bound = kg
- Upper bound = kg Step 4: Write the interval:
Critical Point About Bounds
The actual value is greater than or equal to the lower bound but strictly less than the upper bound. This is because if the actual mass was exactly 2.45 kg, it would round up to 2.5 kg instead.
Truncated measurements
When a measurement has been truncated to a given unit, the actual measurement can be up to a whole unit bigger but no smaller than the given value.
Truncation means chopping off decimal places rather than rounding. If the mass of 2.4 kg was truncated to 1 decimal place, the interval would be .
Truncation vs Rounding
- Rounding: Can be ±0.5 units from the given value
- Truncation: Can be +1 unit bigger but no smaller than the given value
Truncation is less common in everyday measurements but important to understand for complete coverage of bounds.
Maximum and minimum values for calculations
When we perform calculations using rounded values, there will be a discrepancy between the calculated value and the actual value. This is where understanding bounds becomes crucial for determining the range of possible answers.
Finding bounds in area calculations
Let's say we have a pinboard measured as 0.89 m wide and 1.23 m long, both to the nearest centimetre.
Worked Example: Area Bounds Calculation
Given: Width = 0.89 m, Length = 1.23 m (both to nearest cm)
Step 1: Find bounds for each measurement
- Width:
- Length:
Step 2: Calculate area bounds
- Minimum area =
- Maximum area =
Step 3: Express the result Area bounds:
Addition and subtraction with bounds
When adding or subtracting values with bounds, remember this important rule:
- Maximum value of = maximum value of + maximum value of
- Minimum value of = minimum value of + minimum value of
Worked Example: Addition with Bounds
If and (both to 1 decimal place):
Step 1: Find bounds for each value
- Bounds for :
- Bounds for :
Step 2: Apply the addition rule
- Maximum value of
- Minimum value of
Division Bounds - A Common Mistake
When dividing, remember that dividing by a larger number gives a smaller answer. For division:
- Maximum value: Use upper bound ÷ lower bound
- Minimum value: Use lower bound ÷ upper bound
This is the opposite of what many students expect!
Important considerations
Whole Number Constraints
Be careful with bounds when the quantity must be a whole number. For example, if you're finding bounds for the number of people, the maximum value of 154.99999... would still be 154 people, not 155.
Always consider the real-world context of your problem.
When working with bounds in exam questions, always consider:
- Whether the measurement was rounded or truncated
- What unit it was rounded to
- Whether your final answer needs to be a whole number
- The appropriate degree of accuracy for your final answer
Key Points to Remember:
- Rounded measurements: Can be up to half a unit bigger or smaller ( units)
- Truncated measurements: Can be up to a whole unit bigger but no smaller ( unit only)
- Lower bound: Use (greater than or equal to)
- Upper bound: Use (strictly less than)
- Maximum calculations: Use upper bounds for multiplication, lower bounds for division
- Minimum calculations: Use lower bounds for multiplication, upper bounds for division
- Real-world context: Always consider whether your answer makes sense in the given situation