Upper and Lower Bounds
Rounded measurements introduce uncertainty into calculations, so finding the maximum and minimum possible true values (the upper and lower bounds) for a quantity depends 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.