GCSEMaths: FoundationEdexcelRevision NotesNumberRounding, Estimation & Error IntervalsRounding & Estimation

Rounding & Estimation (Edexcel GCSE Maths: Foundation)

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Naomi C

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Naomi C

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Maths

Rounding to a Given Place Value

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 then 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
  • In money calculations, unless the required degree of accuracy is stated in the question
    • Leave your answer in a whole number of dollars, pounds, etc. if it seems sensible
    • Or round to 2 decimal places

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
    • E.g. If the answer to a question about the maximum number of people that can fit in a taxi is 4.6
      • Truncate the answer to 4, rather than round to 5

Worked example

Round the following numbers to 2 decimal places.

(i)

345.254

(ii)

0.295 631

(iii)

4.998

(i)

Identify the second decimal place (5)
Count in 0.01's, identify the first number below (345.25) and the next number above (345.26)
Circle the digit to the right of the second decimal place (4)

345.25 circle enclose 4

As this digit is less than 5 we will round the number down

345.25 (2 d.p.)

No zeros are required after the second decimal places

   

(ii)
Identify the second decimal place (9)
Count in 0.01's, identify the first number below (0.29) and the next number above (0.30)
Circle the digit to the right of the second decimal place (5)
 
0.29 circle enclose 5 space 631
 
As this digit greater than or equal to 5 we will round the number up

0.30 (2 d.p.)

The zero is important to show we have rounded to two decimal places

  

(iii)

Identify the second decimal place (9)
Count in 0.01's, identify the first number below (4.99) and the next number above (5.00)
Circle the digit to the right of the second decimal place (8)

4.99 circle enclose 8

As this digit greater than or equal to 5 we will round the number up

5.00 (2 d.p.)

Two zeros are needed to show we have rounded to 2 decimal places

Rounding to Significant Figures

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

  • 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 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 and 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 as usual
  • 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

Exam Tip

  • In an exam question check that you have written your answer correctly by considering if the value you have ended up with makes sense.
    • Remember the importance of zeros to indicate place value.
    • E.g. Round 2 530 457 to 3 significant figures, 253 (without the zeros) and 2 530 000 are very different sizes!

Worked example

Round the following numbers to 3 significant figures.

(i)

345 256

(ii)

0.002 956 314

(iii)

3.997

(i)

The first (non-zero) significant digit is in the hundred thousands column (3)
The third significant figure is therefore the value in the thousands column (5)

Count in thousands
Identify the numbers that it could round down to (345 000) or round up to (346 000)

Circle the digit on the right of the third significant figure (2)

345 space circle enclose 2 56

This digit is less than 5 so round down 

345 000 (3 s.f.)

  

(ii)

The first significant digit is in the thousandths column (2)
The third significant figure is therefore in the hundred thousandths column (5)

Count in hundred thousandths
Identify the number it could round down to (0.00295) or round up to (0.00296)

Circle the digit to the right of the third significant figure (6)

0.002 space 95 circle enclose 6 space 314

6 is greater than 5 so we need to round up

0.002 96 (3 s.f.)

 

(iii)

The first significant digit is in the units column (3)
The third significant figure is therefore in the hundredths column (9)

Count in hundredths
Identify the number it could round down to (3.99) or round up to (4.00)
Because the value in the tenths and hundredths columns is 9, it will affect higher place values 

Circle the digit to the right of the third significant figure (7)

3.99 circle enclose 7

This value is greater than 5 so it will round up

4.00 (3 s.f.)

The two zeros are necessary to indicate that it has been rounded to 3 significant figures

Estimation

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

Exam Tip

  • Estimation exam questions often involve small decimals.
    • Avoid rounding to 0, especially if the small decimal is the denominator of a fraction, as dividing by 0 is undefined.
    • Dividing by 0.5 is easy, as it is the same as multiplying by 2 !

Worked example

Calculate an estimate for 7.6 cross times 3.81. State, with a reason, whether the estimate is an overestimate or an underestimate.


Round each number to 1 significant figure

7.6 → 8
3.81 → 4

Perform the calculation with the rounded numbers

8 cross times 4 equals 32

An estimate is 32
This is an overestimate as the both numbers were rounded up

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    Naomi C

    Author: Naomi C

    Naomi graduated from Durham University in 2007 with a Masters degree in Civil Engineering. She has taught Mathematics in the UK, Malaysia and Switzerland covering GCSE, IGCSE, A-Level and IB. She particularly enjoys applying Mathematics to real life and endeavours to bring creativity to the content she creates.