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 nonzero digit is the first significant digit
 The first digit after the third significant digit determines whether to 'round up' (5) 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
 Your final answer will have three significant digits and the rest will be zero
 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)
 Questions often ask for 2 decimal places (2 d.p.)
 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, ￥
 For most this will be a whole number
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?
 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
 For example consider the scenario: There are 26 people and 5 people can fit in a single vehicle, how many vehicles are needed?
 These kind of problems can be solved by inequalities
 For 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 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
 For take the smallest value of x at the appropriate degree of accuracy that is greater than k
Exam 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 , where and .
a)
Calculate the exact value of .
b)
Give your answer from part a) correct to two decimal places.
c)
Give your answer from part a) correct to two significant figures.
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, , can be written as
 Note that the lower bound is included in the range of values 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 or multiply UB = UB + UB and LB = LB x LB etc
 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
Exam 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, , of 14.3 m correct to 1 decimal place and width, , of 9.61 m correct to 2 decimal places.
a)
Calculate the lower and upper bound for and .
b)
Calculate the lower and upper bound for the perimeter, , and area, , of the field.
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


 where is the exact value and is the approximate value of
 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
Exam 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 , where and .
a)
Calculate the exact value of .
b)
Calculate the percentage error if an estimate for was 3.5.
Accuracy & Estimation
What are exact values?
 Exact values forms that represent the full value of a number
 For example,is an exact value and 3.14 is an approximation using 3 significant figures
 If a number has an infinite number of nonzero 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:
 Roots: ,
 Logarithms: ,
 Mathematical constants: ,
 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 then your GDC will probably not give you the exact value of
 You would insist evaluate it without a GDC to get the exact value:
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 for example, will not give a true solution
Exam 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 m^{2} 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.
b)
Given that there are 15 more tiles places lengthwise than widthwise, find the approximate length and width of the floor.