Upper & Lower Bounds (Edexcel IGCSE Maths A (Modular))
Revision Note
Written by: Naomi C
Reviewed by: Dan Finlay
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Bounds & Error Intervals
What are bounds?
Bounds are the values that a rounded number can lie between
The smallest value that a number can take is the lower bound (LB)
The largest value that a number must be less than is the upper bound (UB)
The bounds for a number, , can be written as
Note that the lower bound is included in the range of values but the upper bound is not
How do we find the upper and lower bounds for a rounded number?
Identify the degree of accuracy to which the number has been rounded
E.g. 24 800 has been rounded correct to the nearest 100
Divide the degree of accuracy by 2
E.g. If an answer has been rounded to the nearest 100, half the value is 50
Add this value to the number to find the upper bound
E.g. 24 800 + 50 = 24 850
Subtract this value from the number to find the lower bound
E.g. 24 800 - 50 = 24 750
The error interval is the range between the upper and lower bounds
Error interval: LB ≤ x < UB
E.g. 24 750 ≤ 24 800 < 24 850
Examiner Tips and Tricks
Read the exam question carefully to correctly identify the degree of accuracy
It may be given as a place value, e.g. rounded to 2 s.f.
Or it may be given as a measure, e.g. nearest metre
Worked Example
The length of a road, , is given as , correct to 1 decimal place.
Find the lower and upper bounds for
The degree of accuracy is 1 decimal place, or 0.1 km
Divide this value by 2
0.1 ÷ 2 = 0.05
The true value could be up to 0.05 km above or below the given value
Upper bound: 3.6 + 0.05 = 3.65 km
Lower bound: 3.6 - 0.05 = 3.55 km
Upper bound: 3.65 km
Lower bound: 3.55 km
This could also be written as f
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Calculations using Bounds
How do I find the bounds of a calculation?
To find the upper bound of a calculation, consider how the result can be made as large as possible
To find the lower bound of a calculation, consider how the result can be made as small as possible
E.g. For an addition,
The upper bound will be when both and are at their upper bounds
The lower bound will be when both and are at their lower bounds
Sometimes you need different bounds in the same question
The upper bound of is the upper bound of subtract the lower bound of
Increasing the numerator makes the fraction bigger,
But increasing the denominator make the fraction smaller,
How to find the upper and lower bound for each operation is summarised in the table below
Upper Bound | Lower Bound | |
---|---|---|
Upper + Upper | Lower + Lower | |
Upper - Lower | Lower - Upper | |
Upper × Upper | Lower × Lower | |
Upper ÷ Lower | Lower ÷ Upper |
How do I use upper and lower bounds in contexts?
Questions often give real-life contexts and ask about bounds
For example
To see if two cars will fit on the back of a truck
Use the upper bounds of the lengths of the two cars
This is like finding the upper bound of
For example
To find the minimum speed (speed = distance time)
Divide the lower bound of the distance by the upper bound of the time
This is like finding the lower bound of
How can bounds help with calculations?
You can use bounds to determine the level of accuracy of a calculation
E.g. If a value has a lower bound of 8.33217... and upper bound of s 8.33198...
The true value is between 8.33217... and 8.33198...
Find the level of accuracy for which both bounds round to the same number
This happens at 4 sf (rounding to 8.332)
To 5 sf they are different (lower is 8.3322 and upper is 8.3320)
Therefore you know the original value rounds to 8.332 to 4 significant figures
Worked Example
A room measures 4 m by 7 m, where each measurement is made to the nearest metre.
Find the upper and lower bounds for the area of the room.
Find the bounds for each dimension, you could write these as error intervals, or just write down the upper and lower bounds
As they have been rounded to the nearest metre, the true values could be up to 0.5 m bigger or smaller
3.5 ≤ 4 < 4.5
6.5 ≤ 7 < 7.5
Calculate the lower bound of the area, using the two smallest measurements
3.5 × 6.5
Lower Bound = 22.75 m2
Calculate the upper bound of the area, using the two largest measurements
4.5 × 7.5
Upper Bound = 33.75 m2
Worked Example
David is trying to work out how many slabs he needs to buy in order to lay a garden path.
Slabs are 50 cm long, measured to the nearest 10 cm.
The length of the path is 6 m, measured to the nearest 10 cm.
Find the maximum number of slabs David will need to buy.
Find the bounds for each measurement
As they have been rounded to the nearest 10 cm, the true values could be up to 5 cm bigger or smaller
Change quantities into the same units
Length of the slabs: 45 ≤ 50 < 55 cm
or in metres: 0.45 ≤ 0.5 < 0.55 m
Length of the path: 5.95 ≤ 6 < 6.05 m
The maximum number of slabs needed will be when the path is as long as possible (6.05 m), and the slabs are as short as possible (0.45 m)
Maximum number of slabs =
Assuming we can only purchase a whole number of slabs, round up to nearest integer
The maximum number of slabs to be bought is 14
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