Approximation & Estimation (DP IB Maths: AI HL)

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Upper & Lower Bounds

What are bounds?

  • Bounds are the smallest (lower bound, LB) and largest (upper bound, UB) numbers that a rounded number can lie between
    • It simply means how low or high the number could have been before it was rounded
  • 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 could have taken but the upper bound is not

How do we find bounds?

  • The basic rule is “half up, half down”
    • To find the upper bound add on half the degree of accuracy
    • To find the lower bound take off half the degree of accuracy
  • Remember that the upper bound is the cut off point for the greatest value that the number could have been rounded from but will not actually round to the number itself

How do we calculate using bounds?

  • Find bounds before carrying out the calculation and then use the rules:
    • To add UB = UB + UB and LB = LB + LB
    • To multiply UB = UB × UB and LB = LB × LB
    • To divide UB = UB / LB and LB = LB / UB
    • To subtract UB = UB - LB and LB = LB – UB
  • Use logic to decide which bound to use within the calculation
    • For example if you are finding the maximum volume of a sphere with the radius given correct to 1 decimal place substitute the upper bound of the radius into your calculation for the volume

Examiner Tip

  • When in an exam environment it can be easy to make silly errors in questions like this, read the question carefully to determine which parts bounds need to be found for
    • This will normally be any part in the question that has been rounded 

Worked example

A rectangular field has length, L, of 14.3 m correct to 1 decimal place and width, W, of 9.61 m correct to 2 decimal places.

a)
Calculate the lower and upper bound for L and W.
 

ai-sl-1-1-3-bounds-a

b)
Calculate the lower and upper bound for the perimeter, P, and area, A, of the field.
 

ai-sl-1-1-3-bounds-b

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Approximating Values

How do I know what to round my answer to?

  • Unless otherwise told, always round your answers to 3 significant figures (3 s.f.)
    • The first non-zero digit is the first significant digit
    • The first digit after the third significant digit determines whether to 'round up' (greater or equal than5) or 'leave it alone' (<5)
      • where the ‘it’ we are rounding up or leaving alone is the third significant figure
    • Your final answer will have three significant digits and the rest will be zero
      • Any zero after the first significant digit is still significant
      • For large numbers be careful not to change the place value of the significant digits, you will have to fill in any zeros after the third significant figure
      • If your GDC is in scientific mode it may display unnecessary zeros after the decimal point, you do not need to copy these
  • Look out for any questions that ask you to round your answer in a different way
    • Questions often ask for 2 decimal places (2 d.p.)
      • Your final answer will only have 2 digits after the decimal point
      • For 2 d.p. it is the third digit after the decimal place that determines whether to 'round up' (5) or 'leave it alone' (<5)
  • If you are working with a currency you must choose the appropriate degree of accuracy
    • For most this will be a whole number
      • E.g. yen, yuan, peso
    • For others this will be to 2 decimal places
      • E.g. dollars, euro, pounds
    • It will be clear from the question which currency you are using and how you should round your answer
      • The question will state the name of the currency and the symbol you should use as a unit
      • E.g. YEN,

Are there cases when I always have to round up?

  • Yes - there are cases when it makes sense to always round up (or down)
  • These normally involve finding the minimum or maximum number of objects
    • For example consider the scenario: There are 26 people and 5 people can fit in a single vehicle, how many vehicles are needed?
      • 26 over 5 equals 5.2 and normally we'd round to 5
      • However 5 vehicles wouldn't be enough as there would only be room for 25 people
      • In this case we would round up to find the minimum number needed
  • These kind of problems can be solved by inequalities
    • For x greater than k take the smallest value of x at the appropriate degree of accuracy that is greater than k
      • For example: Using 3sf the smallest solution to x > 2.5731... is x = 2.58
    • For x less than k take the biggest value of x at the appropriate degree of accuracy that is less than k
      • For example: The biggest integer solution to x < 10.901... is x = 10

Examiner Tip

  • In the exam you should always give non exact answers correct to 3 significant figures unless otherwise told
    • This means you must round using a higher degree of accuracy within your working to ensure that your final answer is rounded correctly
    • Where possible always use exact values within your working rather than rounding mid way through a question

Worked example

Let T equals fraction numerator b space sin open parentheses 3 a close parentheses over denominator 5 end fraction, where a equals 15 degree and b equals 20.

a)
Calculate the exact value of T.
 

ai-sl-1-1-3-rounding-we-a

b)
Give your answer from part a) correct to two decimal places.


 ai-sl-1-1-3-rounding-b

c)
Give your answer from part a) correct to two significant figures.

 ai-sl-1-1-3-rounding-c

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Percentage Error

What is percentage error?

  • Percentage error is how far away from the actual value an estimated or rounded answer is
    • Percentage error can be calculated using the formula

epsilon equals open vertical bar fraction numerator v subscript A minus v subscript E over denominator v subscript E end fraction close vertical bar cross times 100 percent sign

      • where v subscript E is the exact value and v subscript A is the approximate value of v
      • The | | is the absolute value meaning if you get a negative value within these straight brackets, you should take the positive value
    • This formula is in the formula booklet so you do not need to remember it
  • The further away the estimated answer is from the true answer the greater the percentage error
  • If the exact value is given as a surd or a multiple of π make sure you enter it into the formula exactly as you see it
  • Percentage error should always be a positive number

Examiner Tip

  • In the exam percentage error will usually be a part of a bigger question on another topic, make sure you know how to find the formula for it in the formula book so that you are prepared to answer these questions

Worked example

Let P equals x cos open parentheses 2 y close parentheses, where  y equals 15 degree and x equals 4.

a)
Calculate the exact value of P.
 

ai-sl-1-1-3-percentage-error-a

b)
Calculate the percentage error if an estimate for P was 3.5.
 

ai-sl-1-1-3-percentage-errorb

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Accuracy & Estimation

What are exact values?

  • Exact values are forms that represent the full and precise value of a number
    • For example,space pi spaceis an exact value and 3.14 is an approximation using 3 significant figures
  • If a number has an infinite number of non-zero digits after the decimal point then you can use three dots to signal that the decimal representation goes on for example
    • For example, √2 = 1.414...
  • Exact values can involve
    • Fractions:  2 over 7 
    • Roots: square root of 3fifth root of 7 
    • Logarithms: ln 2log subscript 10 5
    • Mathematical constants: straight pistraight e
  • Your GDC might automatically give your answer as an exact answer
  • If your GDC does not do this then you may need to evaluate parts of the expression separately and use algebra
    • For example: If space f left parenthesis x right parenthesis equals straight e to the power of x left parenthesis 2 plus square root of x stretchy right parenthesis then your GDC will probably not give you the exact value of space f left parenthesis 2 right parenthesis
    • You would insist evaluate it without a GDC to get the exact value: space f left parenthesis 2 right parenthesis equals straight e squared left parenthesis 2 plus square root of 2 right parenthesis 

Why use estimation?

  • We estimate to find approximate answers to difficult sums
  • Or to check our answers are about the right size (order of magnitude)
    • For example, if the question is to find a length the answer cannot be negative
    • or if we are looking for the mean age of some people an answer of 150 must be incorrect
  • Estimating an answer before carrying out a calculation will help you know what you are looking for and determine if your answer is likely to be correct or not
  • In real life estimation skills are used every day in many activities

How do I choose the correct answer?

  • Sometimes a mathematical argument will lead to more than one answer
    • This is common with problems involving quadratics, you will usually have two solutions
    • If you have more than one solution after you have solved a problem, always check to see if they are both valid
  • Most of the time you can simply use logic to choose the correct answer
    • If the problem involves length or area and one of the answers is negative, the true solution will be the positive answer
  • Occasionally you will need to see if an answer can be valid
    • If one of your answers is cos space x greater than 1for example, x will not give a true solution

Examiner Tip

  • Be aware that your GDC will not always give you an answer as an exact value, this means that you will need to find the exact value by hand

Worked example

A rectangular floor has an area of 40 m2 to the nearest square metre. It is going to be tiled using square tiles with side length 39.8 cm.

a)
Use estimation to find the number of tiles needed to cover the whole area.
 

ai-sl-1-1-3-accuracy--estimation-a

b)
Given that there are 15 more tiles places length-wise than width-wise, find the approximate length and width of the floor.
 

ai-sl-1-1-3-accuracy--estimation-b

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.