Upper & Lower Bounds (Edexcel IGCSE Maths A (Modular))

Revision Note

Flashcards
Naomi C

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, x, can be written as LB less or equal than x less than UB

    • Note that the lower bound is included in the range of values x 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, l, is given as l equals 3.6 space km, correct to 1 decimal place.

Find the lower and upper bounds for l.

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 3.55 space less or equal than space l space less than space 3.65

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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, a plus 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 bare at their lower bounds

  • Sometimes you need different bounds in the same question

    • The upper bound of a over b is the upper bound of a subtract the lower bound of b

      • Increasing the numerator makes the fraction bigger, 2 over 1 less than 3 over 1 less than 4 over 1 less than...

      • But increasing the denominator make the fraction smaller, 1 half greater than 1 third greater than 1 fourth greater than...

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

Upper Bound

Lower Bound

a plus b

Upper + Upper

Lower + Lower

a minus b

Upper - Lower

Lower - Upper

a cross times b

Upper × Upper

Lower × Lower

a over b

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 a plus b

    • For example

      • To find the minimum speed (speed = distance divided by time)

      • Divide the lower bound of the distance by the upper bound of the time

      • This is like finding the lower bound of a over 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 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 = fraction numerator 6.05 over denominator 0.45 end fraction equals 13.444...

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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Naomi C

Author: Naomi C

Expertise: Maths

Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.