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GCSEMathsAQARevision Notes1. NumberRounding, Estimation & BoundsRounding & Estimation

Rounding & Estimation (AQA GCSE Maths)

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Rounding & Estimation

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 off
    • e.g. To round 1294 to the nearest 100 you would find the 2 digit and then use the 9 to decide how to round 1 bottom enclose 2 circle enclose 9 4
  • 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. the nearest 2 decimal places to 3.497 are 3.49 and 3.50
  • 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
    • If you are rounding to nearest decimal places then make sure you leave your answer with the required amount of decimal places - do not put unnecessary zeros
      • e.g. 1297 to the nearest 100 is 1300
      • e.g. 3.497 to two decimal places (nearest 100th) is 3.50 (exactly two decimal places in answer)

How do I round to significant figures?

  • Rounding to significant figures is the same as 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 the following zeros
      • e.g. 0 is the second significant figure of 3097
      • e.g. 9 is the third significant figure of 3097
  • Use the normal rules for rounding
    • Circle the number to the right
    • Use this to determine whether the given significant figure rounds up or rounds off

What is truncation?

  • Truncation is essentially the same as rounding down to a given degree of accuracy
  • It is usually used in situations where a whole number of items is needed
    • For example if the question is about the maximum number of people that can fit in a taxi and the answer is 4.6, the answer would be truncated to 4, rather than rounded to 5

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
  • 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

Examiner Tip

  • When rounding or truncating to a degree of accuracy greater than ten, remember to add the necessary zeros to ensure the place value of the number is not changed
    • For example to truncate the number 5760 to the nearest hundred, the answer should be 5700, not 57

Worked example

Calculate an estimate for fraction numerator 17.3 cross times 3.81 over denominator 11.5 end fraction. State, with a reason, whether the estimate is an overestimate or an underestimate.

Round each number to 1 significant figure.

17.3 → 20
3.81 → 4
11.5 → 10

Perform the calculation with the rounded numbers.

fraction numerator 20 cross times 4 over denominator 10 end fraction equals 80 over 10 equals 8

An estimate is 8.
This is an overestimate as the numerator was rounded up and the denominator was rounded down.

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    Dan

    Author: Dan

    Expertise: Maths

    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.