Upper & Lower Bounds (Cambridge (CIE) IGCSE Maths: Extended): Revision Note

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, $x$, can be written as $\mathrm{LB}\le x<\mathrm{UB}$

  • Note that the lower bound is included in the range of values $x$ but the upper bound is not

How do I 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

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, $a+b$

  • The upper bound will be when both $a$ and $b$ are at their upper bounds

  • The lower bound will be when both $a$ and $b$are at their lower bounds

• Sometimes you need different bounds in the same question

  • The upper bound of $\frac{a}{b}$ is the upper bound of $a$ divided by the lower bound of $b$

    • Increasing the numerator makes the fraction bigger, $\frac{2}{1}<\frac{3}{1}<\frac{4}{1}<...$

    • But increasing the denominator make the fraction smaller, $\frac{1}{2}>\frac{1}{3}>\frac{1}{4}>...$

• How to find the upper and lower bound for each operation is summarised in the table below

• You will need to use more than one of these if the expression includes multiple operations

  • E.g. consider the upper bound for $\frac{a}{b-c}$

    • You need the upper bound for $a$ and lower bound for $b-c$

    • This means you need the lower bound for $b$ and upper bound for $c$

  • E.g. consider the lower bound for $\frac{a}{b}-c$

    • You need the lower bound for $\frac{a}{b}$ and upper bound for $c$

    • This means you need the lower bound for $a$ and upper bound for $b$

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 $a+b$

  • 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 $\frac{a}{b}$

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 upper bound of 8.33217... and lower bound of s 8.33198...

    • The true value is between 8.33198... and 8.33217...

  • 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 8.3320 and upper is 8.3322)

  • Therefore you know the original value rounds to 8.332 to 4 significant figures