Upper and Lower Bounds
This topic deals with the uncertainty in calculations that arise from using rounded measurements. It involves finding the maximum and minimum possible true values (the upper and lower bounds) for a quantity based on the precision of the numbers used to calculate it.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- To find the bounds of a measurement, take the degree of accuracy, halve it, and then add and subtract this from the stated value. For a value 'x' given to the nearest 'a', the true value lies in the interval x ± a/2.
- The error interval is written as Lower Bound ≤ true value < Upper Bound. Note the strict inequality for the upper bound.
- For addition and multiplication, the logic is straightforward: to find the maximum result, use the maximum inputs (upper bounds). To find the minimum result, use the minimum inputs (lower bounds).
- For subtraction (A - B), the upper bound is found by maximising A and minimising B: UB(A) - LB(B). The lower bound is found by minimising A and maximising B: LB(A) - UB(B).
- For division (A / B), the upper bound is found by maximising the numerator and minimising the denominator: UB(A) / LB(B). The lower bound is found by minimising the numerator and maximising the denominator: LB(A) / UB(B).
Formulae
UB(A - B) = UB(A) - LB(B) To find the maximum possible value of a subtraction.
LB(A - B) = LB(A) - UB(B) To find the minimum possible value of a subtraction.
UB(A / B) = UB(A) / LB(B) To find the maximum possible value of a division.
LB(A / B) = LB(A) / UB(B) To find the minimum possible value of a division.
Definitions
- Lower Bound (LB)
- The smallest possible true value a measurement could have had before it was rounded up to the given value.
- Upper Bound (UB)
- The smallest value that would round up to the next measurement increment. The true value can get infinitely close to the upper bound, but never be equal to it.
- Error Interval
- The range of possible values that a number could have been before it was rounded. For a number 'n', it is expressed as LB ≤ n < UB.
Worked example
A car travels a distance of 60 km, correct to the nearest km. The journey takes 50 minutes, correct to the nearest minute. Calculate the lower bound of the car's average speed in km/h.
- 1
First, state the formula for speed and identify what is needed for the lower bound.
Speed = Distance / TimeTo get the minimum speed, we need the minimum possible distance and the maximum possible time:
LB(Speed) = LB(Distance) / UB(Time) - 2
Find the bounds for the distance.
60 km to the nearest km means an accuracy of 1 km.
The error is 1/2 = 0.5 km.
So, the lower bound for distance is 60 - 0.5 = 59.5 km.
- 3
Find the bounds for the time.
50 minutes to the nearest minute means an accuracy of 1 minute.
The error is 1/2 = 0.5 minutes.
So, the upper bound for time is 50 + 0.5 = 50.5 minutes.
- 4
The speed needs to be in km/h, so convert the upper bound of time to hours.
UB(Time) = 50.5 minutes = 50.5 / 60 hours - 5
Calculate the lower bound of the speed:
LB(Speed) = 59.5 km / (50.5 / 60) h = (59.5 × 60) / 50.5 - 6
Simplify the calculation.
Multiply the numerator and denominator by 10 (or 2) to remove decimals:
(595 × 60) / 505.
Now simplify the fraction by dividing top and bottom by 5:
(119 × 60) / 101.
- 7
Calculate the final answer:
(119 × 60) / 101 = 7140 / 101 km/h.
This fraction cannot be simplified further.
Answer: 7140/101 km/h
Common mistakes
- ×Using the incorrect bounds for subtraction and division. For example, finding the upper bound of A/B by calculating UB(A)/UB(B). Always stop and think: 'How do I make this value biggest/smallest?'
- ×Miscalculating the initial error interval. For a value given to 'n' decimal places, the error is 0.5 × 10^(-n). For a value given to the nearest integer 'k', the error is k/2.
- ×Forgetting to find the bounds of an intermediate calculation. For example, in finding the bounds of Density = Mass / (Area × Height), you must first find the bounds for the volume (Area × Height) before using the division rule.
No-calculator tips
- ✓To handle division with decimals like 59.5 / 50.5, multiply the numerator and denominator by 10 or 2 to work with integers (e.g., 595 / 505), which are easier to simplify.
- ✓Before calculating, use logical reasoning to check your formula. To make a fraction (A/B) as large as possible, you need the largest A and the smallest B. This is a quick way to recall the rules without memorisation.
- ✓Leave answers as exact fractions unless specified otherwise. Trying to convert a fraction like 7140/101 to a decimal without a calculator is time-consuming and unnecessary.