Rounding & Estimation (Cambridge O Level Maths)

How do I round numbers to a given place value?

• Identify the digit in the required place value and circle the number to the right
  • This number will determine whether to round up or round down
  • e.g. To round 1294 to the nearest 100 you would find the digit in the hundreds place (2) and then use the digit to the right of it (9) to decide how to round 
• Identify the two options that the number could round to
  • e.g. the two nearest 100's to 1294 are 1200 and 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
      
      The two options are either 90 or 100 - here we would round to 100 but this will then have a knock-on effect for the 7 and so this would round to 1800
    • For that example rounding to the nearest 100 and the nearest 10 leads to the same result 
• 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
  • You then 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 to decimal places?

• Exactly the same as above but we tend to think in digits rather than in terms of place value
  • e.g.  Round 2.435 123 to 2 decimal places
    Finding the second decimal place and considering the digit to the right of it - 
    The circled number is 5 so round up the 3 up to a 4
    The rounded answer would be 2.44 (2 d.p.)

• When rounding to decimal places make sure you leave your answer with the required amount of decimal places
  • There is no need to put unnecessary zeros
  • e.g. 1297 to the nearest 100 is 1300 but 1.297 to two decimal places (nearest 100th) is 1.30
    • We need one zero as we have been asked to round to 2 decimal places but we do not need a third to replace the original 7 digit (in the thousandths place)
• In money calculations, unless a whole number of dollars, pounds, etc is required or sensible then always round to two decimal places
  • US dollars ($) are split into 100 cents, GB pounds (£) are split into 100 pennies

How do I round to significant figures?

• Rounding to significant figures is similar to rounding to place value
  • You just need to identify the relevant place value
• Find the first significant figure
  • 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
• Start with this number and count along to the right
  • You do count any following zeros
    • e.g. 0 is the second significant figure of 3097 and 9 is the third significant figure of 3097
• Use the normal rules for rounding
  • Circle the number to the right
  • Use this to determinant whether the given significant figure rounds up or rounds off
• With large numbers completing places up to the decimal point with zeros is crucial
  • e.g.  34 568 to 2 significant figures is 35 000 (not 35)
• Similarly, with decimals, zeros at the start are crucial
  • e.g.  0.003 435 to 3 significant figures is 0.003 44 (not 344)

Why do we use estimation?

• We estimate to find approximations for difficult sums
• Or to check our answers are about the right size (right order of magnitude)

How do I estimate?

• We round numbers to something sensible before calculating
  • The exam question will usually tell you what to round each number to before carrying out any calculations
• GENERAL RULE:
  • Round numbers to 1 significant figure
    • 7.8 ➝ 8
    • 18 ➝ 20
    • 3.65 × 10^-4 ➝ 4 × 10^-4
    • 1080 ➝ 1000
• EXCEPTIONS:
  • It can be more sensible (or easier) to round to something convenient
    • 16.2 ➝ 15
    • 9.1 ➝ 10
    • 1180 ➝ 1200
• It wouldn’t usually make sense to round a number to zero

How do I know if I have underestimated or overestimated?

• For addition
  • If you round both numbers up then you will overestimate
  • If you round both numbers down then you will underestimate
• For multiplication
  • If you round both numbers up then you will overestimate
  • If you round both numbers down then you will underestimate
• Subtraction and division are more complicated
• You need to consider the effects of rounding each number 
  • 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 can not 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 can not easily tell if it is an underestimate or underestimate