Number Operations (Edexcel GCSE Maths: Foundation)

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Jamie W

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Jamie W

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Number Operations

What are number operations?

  • Addition, subtraction, multiplication and division are the four basic number operations
  • For the non-calculator paper, a written method will be required
  • Different vocabulary may be used for each of the operations
    • Addition may be phrased using plus, total or sum
    • Subtraction may be phrased using difference or take away
    • Multiplication may be phrased using lots of, times or product
    • Division may be phrased using quotient, share and per

Addition & Subtraction

How do I add large numbers without a calculator?

  • There are a variety of written methods that can be used to add large numbers
    • You only need to know one method, but be able to use it confidently
    • The method described below is commonly known as the column method
    • The order in which numbers are added is not important
  • STEP 1
    Write one number above the other in two rows, making sure that the place values are lined up in the same columns
    e.g.  9789 + 563 would be written as   
  • STEP 2
    Add the digits in the ones (units) column
    Write the answer below the line in the ones column
    However, if your sum is a two digit number, split it into ones and tens
    Write the value of the ones below the line in the current column, and the value of the tens at the top of the next (to the left) column
    e.g. for the units column, 9+3=12
       
  • STEP 3
    Repeat STEP 2 for each column, in increasing place value order (i.e. moving a place to the left each time)
    e.g.  For the tens column, 1+8+6=15
       
  • If the sum of the last (highest place value) column is a two digit number, you can write the entire number below the line
    • This is effectively the same as writing the tens digit at the top of the next column, but it would be the only digit in that column
  • The final answer for this example would look like this:

How do I subtract large numbers?

  • A variety of written methods exist, but you only need to know one
    • The method described below is the (subtraction) column method
    • The order in which two numbers are subtracted is important so ensure the calculation is the right way round
  • STEP 1
    Write the number that is being subtracted below the original amount, making sure that the place values are lined up in the same columns
    e.g.  392 - 28 would be written as   
  • STEP 2
    In the ones (units) column, subtract the bottom number from the top number, writing the answer below the answer line
    If the top number is smaller than the bottom number then "borrow ten" from the next column (to the left) then continue as above
    e.g.   
  • If the column being borrowed from is a zero, "borrow ten" from the next column again, turning the zero into a ten, which can then be borrowed from
  • STEP 3
    Repeat STEP 2 for each column, in increasing place value order (i.e. moving a place to the left each time)
    e.g.     
  • If you find you need to borrow in the last (highest place value) column, an error in one of the previous steps may have been made, so check your working carefully

How do I add or subtract with decimals?

  • If the numbers involve decimals, the column methods for addition and subtraction are the same as above
    • In STEP 1, line the place value columns up carefully, and make sure the decimal points are all in the same column
      •  Writing zeros (often called place value holders) can help keep everything in line
    • e.g.  2.145 + 13.02 would be written as   

Examiner Tip

  • Start questions involving large, long or awkward numbers by estimating using approximations
    • e.g. Estimate 32 870 ÷ 865 by mentally calculating 30 000 ÷ 1 000 giving 30 as an estimate for the answer
      (The actual answer is 38) 
  • Remember that only one method for each operation is needed; learn one and make sure you are confident with it

Worked example

(a)

Find the sum of 3985 and 1273.

Notice that the word sum is used but this means add
Quickly estimate the answer

4000 + 1000 = 5000

STEP 1
Write the numbers in two rows and columns aligned

STEP 2
Start with the ones (units) column, writing the answer below the line but in the same column

STEP 3
Move on to the tens (next on the left) column
The sum is 15 so the 5 (ones) is written below the line and the 1 (tens) 'carries over' to the next (hundreds) column 

Next is the hundreds column which again results in a two-digit answer

Finally add the digits in the thousands column

Check the final answer is similar to your estimate; 5000 and 5258 are reasonably close

3985 + 1273 = 5258

 

(b)

Find the difference between 506 and 28.

Notice that the word difference is used; this means subtract
Quickly estimate the answer

500 - 30 = 470

STEP 1
Write the numbers in two rows, column aligned, ensuring the top number is the number being subtracted from

STEP 2
In the ones (units) column, 6 is smaller than 8, so borrow from the next column (tens)
The tens column is 0, so borrow from the column to the left of that (hundreds)

This turns the 0 (in the tens column) into a 10 which we can then borrow from (for the ones column)

16 - 8 can be now be calculated in the ones column

STEP 3
Move onto the tens column which is 9 - 2 = 7
There is nothing to subtract in the last (hundreds) column (4 - 0 = 4)

Check the final answer is similar to the estimate; 470 and 478 are reasonably close

506 - 28 = 478

(c)

Johnny has £32.50 and spends £1.74.  Calculate how much money Johnny has left.

This is a subtraction question as Johnny has spent money
Quick estimate

33 - 2 = 31

STEP 1
Align the digits by place value, ensuring £32.50 is the top number and using the decimal points as a starting point
Using place value holding zeros is optional

STEP 2
The column furthest right is the hundredths column
0 is smaller than 4 so borrow from the next (tenths) column

STEP 3
Next is the tenths column, but again 4 is smaller than 7 so borrow 10 from the ones column

 

14 - 7 = 7 in the tenths column and continue working 'right to left'

Check the final answer is similar to the estimate; £31 and £30.76 are reasonably close

Johnny has £30.76 left

Multiplication

How do I multiply two numbers without a calculator?

  • There are a variety of written methods that can be used to add large numbers
    • You only need to know one method, but be able to use it confidently
    • Three common methods are described below, but there are many other valid methods

1. Lattice method

(Best for numbers with two or more digits)

  • This method allows you to work with individual digits
  • So in the number 3 516 you would only need to work with the digits 3, 5, 1 and 6 
  • To find 3516 × 23 :

Lattice Complete, IGCSE & GCSE Maths revision notes

So, 3516 × 23 = 80 868

2. Grid method

  • This method keeps the value of the larger number intact
  • So with 3516 you would use 3000, 500, 10 and 6
  • This method may take longer with two larger numbers as you can end up with many terms to add up
    • Be wary of large numbers with lots of zeros; line up the columns carefully in the final step
  • To find 3516 × 7 :

Partition Complete, IGCSE & GCSE Maths revision notes

Partition Lined Up, IGCSE & GCSE Maths revision notes

So, 3516 × 7 = 24 612

3. Repeated addition method

  • This is best for smaller, simpler cases
  • You may have seen this called ‘chunking’
  • It is a way of building up to the answer using simple multiplication facts that can be worked out easily
    • To find 13 × 23 :

1 × 23 = 23

2 × 23 = 46

4 × 23 = 92

8 × 23 =184

    • So, 13 × 23 = 1 × 23 + 4 × 23 + 8 × 23 = 23 + 92 + 184 = 299

How do I multiply decimals without a calculator?

  • These 3 methods can be easily adapted for use with decimal numbers
  • Ignore the decimal point whilst multiplying but put it back in the correct place for the final answer
  • eg. 1.3 × 2.3
    • Ignoring the decimals this is 13 × 23, which can be worked out as 299
    • There are two decimal places in total in the question, so there will be two decimal places in the answer
      • So, 1.3 × 2.3 = 2.99
      • Be careful that 1.30 is the same as 1.3, so 1.30 is still only 1 decimal place
    • Estimating is a good check too
      • 1.3 × 2.3 is approximately 1.5 × 2, which is 3
      • So we know the answer should be close to 3, rather than 0.3 or 30

Examiner Tip

  • Estimating an answer first, by rounding to 1 significant figure, is a really quick and effective check

Worked example

Multiply 2879 by 36.

As you have a 4-digit number multiplied by a 2-digit number then the lattice method is a good choice
Start with a 4×2 grid.…

Lattice Ex1, IGCSE & GCSE Maths revision notes

Notice the use of listing the 8 times table underneath to help with some of the multiplication within the lattice

Use an estimate to check your answer; 3000×40 is equal to 120 000

2879 × 36 = 103 644

Division

How do I divide a number by another without a calculator?

  • The most common written method for division is short division (or "the bus stop method")
    • There are other methods such as long division, but short division is generally the most efficient
  • While short division is best when dividing by a single digit, for bigger numbers you may need a different approach
  • You can use other number skills to help
    • eg. cancelling fractions, “shortcuts” for dividing by 2 and 10, and the repeated addition (“chunking”) method covered in Multiplication

Short division (bus stop method)

  • Unless you can use simple shortcuts such as dividing by 2 or by 10, this method is best used when dividing by a single digit

To find 174 ÷ 3

  • Starting from the left; 3 fits into 1, 0 times, with a remainder of 1
    • Carry the remainder of 1 over to the next digit, which forms 17
  • 3 fits into 17, 5 times, with a remainder of 2
    • Carry the remainder of 2 over to the next digit, which forms 24
  • 3 fits into 24, 8 times, with no remainder
  • So, 174 ÷ 3 = 58

Factoring & cancelling

  • This involves treating division as you would if you were asked to simplify fractions
    • For example, 1008 ÷ 28 can be written as 1008 over 28
    • This can then be simplified
      • 1008 over 28 equals 504 over 14 equals 252 over 7
      • 252 ÷ 7 can then be calculated using short division; the answer is 36

Dividing by 10, 100, 1000, … (Powers of 10)

  • This is a case of shifting digits along the place value columns

    For example

    • 380 ÷ 10 = 38.0 (shifts by 1 column)

    • 45 ÷ 100 = 0.45 (shifts by 2 columns)

    • 28 ÷ 1000 = 0.028 (shifts by 3 columns)
      • For cases like this, it can help to add leading zeros
      • 0028 ÷ 1000 may be easier to visualise

How do I divide a decimal without a calculator?

  • Use a similar approach to when multiplying decimals
  • Ignore the decimal point whilst carrying out the division, but put it back in the correct place for the final answer
  • e.g.  4.68 ÷ 6
    • Ignoring the decimal, this is 468 ÷ 6, which can be worked out as 78
    • There are two decimal places in total in the question, so there will be two decimal places in the answer
      • So, 4.68 ÷ 6 = 0.78
      • Be careful if 6 is written as 6.0 instead, this is still zero decimal places
    • Estimating is a good check too
      • 4.68 ÷ 6 is approximately 5 ÷ 6, which is between 1 and 0.5
      • So we know the answer should be 0.78, rather than 7.8 or 0.078

Worked example

Divide 568 by 8. 

This is division by a single digit so short division would be an appropriate method
If you spot it though, 8 is also a power of 2 so you could just halve three times

Using short division, the bus stop method:

Ex1 Short Divison, IGCSE & GCSE Maths revision notes

8 fits into 5, 0 times, with a remainder of 5
8 fits into 56, 7 times exactly
8 fits into 8, 1 time exactly

Use an estimate to check your answer; 600 ÷ 10 is equal to 60

568 ÷ 8 = 71 

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Jamie W

Author: Jamie W

Expertise: Maths

Jamie graduated in 2014 from the University of Bristol with a degree in Electronic and Communications Engineering. He has worked as a teacher for 8 years, in secondary schools and in further education; teaching GCSE and A Level. He is passionate about helping students fulfil their potential through easy-to-use resources and high-quality questions and solutions.