Rounding & Estimation (CIE IGCSE Maths: Core)

How do I round a number to a given place value?

• Identify the digit in the required place value
• Identify the two options that the number could round to
  • Count in the units you are rounding to,
  • Find the first value below and the next value above the number you are rounding 
    • E.g. Round 1294 to the nearest 100, count in 100's
    • The number will round down to 1200 or up to 1300
  • Be careful if your digit is a 9 and the next number up will affect the higher place values
    • E.g. Rounding 1798 to the nearest 10, count in 10's
    • The number will round down to 1790 or up to 1800
• Circle the number to the right of the required place value
  
  • If the circled number is 5 or more then you round to the bigger number
  • If the circled number is less than 5 then you round to the smaller number
  • Put a zero in any following place values before the decimal point
    • E.g. 1567.45 to the nearest 100 would be 1600

How do I round a number to a given decimal place?

• Identify the position of the decimal place you are rounding to
• Identify the two options that the number could round to
  • E.g. Round 7.82741 to 3 d.p., count in 0.001's
  • The number will round down to 7.827 or up to 7.828
• Circle the number to the right of the required decimal place
  • If the circled number is 5 or more then you round to the bigger number
  • If the circled number is less than 5 then you round to the smaller number
    
    • E.g. 2.435123 to the nearest 2 d.p. would be 2.44
• When rounding to decimal places make sure you leave your answer with the required amount of decimal places
  • Do not put any zeros after the position of the decimal place you are rounding to
    • E.g. 1267 to the nearest 100 is 1300
    • But 1.267 to two decimal places (nearest 100th) is 1.27 not 1.270
  • If asked for a certain number of decimal places, you must give an answer with that number of decimal places
    • E.g. 2.395 to two decimal places is 2.40 (do not write 2.4)

How do I round a number to a given significant figure?

• To find the first significant figure when reading from left to right, find the biggest place value that has a non-zero digit
  • The first significant figure of 3097 is 3
  • The first significant figure of 0.006207 is 6
    • Zeros before 6 are not significant
    • Zeros after 6 are significant
• Count along to the right from the first significant figure to identify the position of the required significant figure 
  • Do count zeros that are between other non-zero digits
    • E.g. 0 is the second significant figure of 3097
    • 9 is the third significant figure of 3097
• Use the normal rules for rounding
  • Count in units determined by the place value of the significant figure
    • If the second significant figure is in the 10's column, count in 10's
  • Identify numbers it could be rounded up or down to
  • Circle the number to the right of the significant figure
  • Use this value to determine which number it rounds to
• For large numbers, complete places up to the decimal point with zeros
  • E.g.  34 568 to 2 significant figures is 35 000
• For decimals, complete places between the decimal point and the first significant figure with zeros
  • E.g.  0.003 435 to 3 significant figures is 0.003 44

How do I know what degree of accuracy to give my answer to?

• If a question requires your answer to be an exact value
  
  • You can leave it as a simplified fraction
    • E.g.
  • You can leave it in terms of or a square root
    • E.g. , or
  • If it is an exact decimal up to and including 5 s.f., you can write it out without rounding it
    • E.g. 0.9375, or 850.25

• If the answer is not exact, an exam question will often state the required degree of accuracy for an answer
  • E.g. Give your answer to 2 significant figures

• If the degree of accuracy is not asked for, use 3 significant figures 
  
  • All working and the final answer should show values correct to at least 4 significant figures
  • The final answer should then be rounded to 3 significant figures
• In money calculations, unless the required degree of accuracy is stated in the question, you can look at the context
  • Round to 2 decimal places
    • E.g. $64.749214 will round to $64.75
  • Or to the nearest whole number, if this seems sensible (for example, other values are whole numbers)
    
    • $246 029.8567 rounds to $246 030
• When calculating angles, all values should be given correct to 1 decimal place
  • • An angle of 43.5789 will round to 43.6
    • An angle of 135.211... will round  to 135.2°

Why do I need to estimate?

• Estimation can be used to find approximations for difficult calculations
• You can estimate a calculation to check your answers
  • You can identify if there is a mistake in your working out if your answer is much bigger or smaller than your estimated value

How do I estimate?

• Round each number in the question to something sensible then perform the calculation
  • The exam question will usually tell you what to round each number to before carrying out any calculations
• The general rule is to round numbers to 1 significant figure
  • • 7.8 ➝ 8
    • 18 ➝ 20
    • 3.65 × 10^-4 ➝ 4 × 10^-4
    • 1080 ➝ 1000
• In certain cases it may be more sensible (or easier) to round to something convenient
  • • 16.2 ➝ 15
    • 9.1 ➝ 10
    • 1180 ➝ 1200
• Avoid rounding values to zero

How do I know if I have underestimated or overestimated?

• For addition a  + b
  • If you round both numbers up then you will overestimate
  • If you round both numbers down then you will underestimate
• For multiplication a  x b
  • If you round both numbers up then you will overestimate
  • If you round both numbers down then you will underestimate 
• For subtraction a  - b
  • Increasing a and/or decreasing b will increase the answer so you will overestimate
  • Decreasing a and/or increasing b will decrease the answer so you will underestimate
  • If both numbers are increased or both are decreased
    • then you cannot easily tell if it is an underestimate or underestimate
• For division a  ÷ b
  • Increasing a and/or decreasing b will increase the answer so you will overestimate
  • Decreasing a and/or increasing b will decrease the answer so you will underestimate
  • If both numbers are increased or both are decreased
    • then you cannot easily tell if it is an underestimate or underestimate