Upper & Lower Bounds (Edexcel GCSE Maths)

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

What is truncation?

• Truncation is essentially the same as rounding down to a given degree of accuracy

• It is usually used in situations where a whole number of items is needed

  • E.g. If the answer to a question about the maximum number of people that can fit in a taxi is 4.6

    • Truncate the answer to 4, rather than round to 5

How do we find the error interval for a truncated number?

• You may be given a question where the number has been truncated

  • E.g. The first 3 digits of an answer, , to a calculation have been written down as 2.95

    • The smallest value that the answer could have been is 2.95

    • The largest value that the number could have been up to (but not equal to) is 2.96 before it was truncated to 2.95

    • The error interval for the size of the number is 

• The truncated value should be the same as the smallest value in the error interval

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

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