Number Operations

Addition & Subtraction

What kind of addition or subtraction questions could I be asked?

Adding and subtracting could be a part of any question within your GCSE course, some examples are
- Adding areas or volumes together in geometry questions
- Working with estimated mean or other averages
- Any problem solving questions in a variety of contexts
- Simply showing that you have the skills to carry out these calculations without a calculator
A variety of vocabulary can be used to imply you must add or subtract values
- For adding the word plus may be used or you could be asked to find the total or find the sum
- For subtracting the word minus or take away may be used or you could be asked to find the difference
You must be confident with non calculator methods for adding and subtracting large numbers and decimals

How do I add large numbers?

Write one number above the other in a column, making sure that all the ones, tens, hundreds, and so on, are lined up in the same columns
- - For example, 382 + 943

Add the numbers in each column, beginning with the units and working through the columns to the left
- Write each answer as a single digit below the line
- If your sum is a two digit number, split it into ones and tens, writing the value of the ones below the line, and the value of the tens at the top of the next column
  - For example, in the sum 382 + 943, the sum of the middle column is 12
  - Put the 2 below the line and the 1 above the 3

Repeat this for all columns, until complete
If the sum of the last column is a 2 digit number, you can write the entire number below the line
- This is effectively the same as writing it at the top of the next column, but it would be the only number in that column
- For example when adding the 3 and 9 in the sum below

How do I subtract large numbers?

Write one number above the other in a column, making sure that all the ones, tens, hundreds, and so on, are lined up in the same columns
- - For example, 435 − 183

Subtract the bottom number in each column from the top number in the same column, starting with the ones (units)
If your subtraction produces a negative number “borrow” a ten from the column to the left
- For example in the middle column of the difference between 435 and 183, the difference between 3 and 8 is negative, so borrow a ten from the 4 in the hundreds column

Repeat this for each of the next columns, until complete
- For example

If you need to borrow from the next column, but the next column is a zero, you can borrow a ten from the column to the left of this
- This will turn the 0 into a 10, which you can then borrow from
- For example, in the problem 303 – 56

- the units column cannot borrow from the tens as it is a zero, so the tens must borrow from the hundreds first

How do I add or subtract with decimals?

If the numbers involve decimals, make sure the decimal points are lined up in a column of their own and keep the decimal point in the answer in this column too
- You may need to add zeros before or after the decimal point to keep everything in line
- Again, estimate the answer by rounding to sensible values to check your answer makes sense

Exam Tip

Before performing the calculation, estimate the answer
- This will help you spot any errors you might make with place value
- 4321-284 is roughly 4300-300 so the answer should be approximately 4000
After finding the answer to a subtraction, turn it into an addition to check your answer
- If you find 303-56=247, then 247+56 should be 303

Worked example

(a)

Find the sum of 3985 and 1273.

Notice that the word sum requires you to add the numbers together.
Begin by estimating the answer.

4000 + 1000 = 5000
so the answer should be a little more than 5000

Write one number above the other.

Add the digits in the ones column.

Add the digits in the tens column, writing the 1 above the next column.

Add the digits in the hundreds column, including the extra 1.

Add the digits in the thousands column, including the extra 1.

Check the final answer is similar to your estimate.

5258

(b)

Find the difference between 506 and 28.

Notice that the word difference requires you to subtract the second number from the first.

Begin by estimating the answer.

500 – 30 = 470
so the answer should be about 470

Write one number above the other, be careful to line up the columns correctly.

6 – 8 would be negative, so we need to borrow from the next column.
However the next column is 0, so we will borrow from the column to the left of it.
This turns the 0 into a 10.

We can then borrow from the tens column.

So we can now find 16 – 8 for the ones column.

For the second column (tens) we can do 9 – 2 = 7 and finally in the hundreds column, 4 – 0 = 4.

Check that your answer is similar to your estimate.

478

(c)

Without using your calculator, calculate 32.5 – 1.74.
You must show all your working.

Begin by estimating the answer.

32.5 is about 33 and 1.74 is about 2
33 – 2 = 31
So the answer should be about 31

Write one number above the other, be careful to line up the columns correctly, putting the decimal point in a column of its own and filling in the spaces with zeros.

Consider the hundredths column; 0 – 4 is negative, so we need to borrow from the next column.

Consider the tenths column, 4 – 7 would be negative, so we need to borrow from the next column.

We can now do 14 – 7 = 7, and continue subtracting in the other columns.
Remember to put the decimal point in line in the answer with those in the question.

Check that this is similar to your original estimate.

30.76

Multiplication

What kind of multiplication questions could I be asked?

Multiplying two or more numbers could be a part of any question within your GCSE course, some examples are
- Finding areas or volumes in geometry questions
- Finding a distance given a speed and a time
- Working with probabilities
- Problem solving questions in a variety of contexts
A variety of vocabulary can be used to imply you must multiply two or more values
- The word times or multiply may be used or you could be asked to find the product
Although you may have a calculator to carry out these sums on, you could be asked to do this in the non calculator paper so you need to be confident with a non calculator method
- You should be especially confident showing methods for multiplying large numbers and decimals

How do I multiply two or more numbers without a calculator?

Different methods work for different people, and some are better depending on the size of number you are dealing with
We recommend the following 3 methods depending on the size of number you are dealing with
- If in doubt all methods will work for all numbers!

1. Lattice method

This method is 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 multiply two numbers of 2 or more digits, write the longer number across the top of the grid and the shorter one to the right of it
- Use diagonal lines to separate the tens and the units in each part of the answer
- Add the digits together by following the lines through the diagonals, starting from the right and working towards the left
- If any of the diagonals add to a 2 digit number, carry the value in the tens across to the left and add to the next section
- For example, to multiply 3516 by 23, write the longer number across the top of the grid and the shorter one to the left

Lattice Complete, IGCSE & GCSE Maths revision notes

So, 3516 × 23 = 80 868

2. Partition method

This method is useful for adding a large number by a one or two digit number
It works by separating the larger number into the value that each digit represents
- So with 3516 you would use 3000, 500, 10 and 6
This method is not suitable for two larger numbers as you can end up with a lot of zero digits that are hard to keep track of

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 calculations
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
- eg. 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 with decimals?

These 3 methods can easily be adapted for use with decimal numbers
You ignore the decimal point whilst multiplying but put it back in the correct place in order to reach a final answer
- eg. 1.3 × 2.3
- Ignoring the decimals this is 13 × 23, which from above is 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
Remember to always estimate the rough answer in the exam to check your answer
- e.g. 4.2 × 7.9 is similar to 4 × 8 so will be about 32

Exam Tip

Always start the question by estimating the answer in your head, this will give you an idea of the answer you should get and let you know at the end if your answer is likely to be correct
If you have time at the end of the exam, go back and check your answer by working backwards
- For example if the question was a subtraction, add your answer to one of the original numbers to see if you get the other

Worked example

Without using a calculator, multiply 2879 by 36. You must show all your working.

Start by estimating the answer.

2897 is about 3000 and 36 is near(ish) 40

3000 × 40 = 120 000 and both were over estimations so your answer should be a little less than this

As you have a 4-digit number multiplied by a 2-digit number then the lattice method is the best choice.

Start with a 4×2 grid.

Lattice Ex1, IGCSE & GCSE Maths revision notes

Notice the use of listing the 8 times table at the bottom to help with any you may have forgotten. Remember to check your answer is similar to your original estimate.

103 644 is less than 120 000, but it has the same place value so it is likely to be correct.

2879 × 36 = 103 644

Division

What kind of division questions could I be asked?

Dividing two numbers could be a part of any question within your GCSE course, some examples are
- Converting a fraction to a decimal
- Finding a speed given a distance and a time
- Finding a length or a width given an area
- Problem solving questions in a variety of contexts
A variety of vocabulary can be used to imply you must divide one number by another
- The words divide, quotient, share or per may be used
You could be asked to show a non calculator method or need to do a division in the non calculator exam
- You should be especially confident showing with methods for dividing small or large numbers and decimals

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

Most students will have seen short division (bus stop method) and long division and there is often confusion between the two
Fortunately, you only need one – so use short division
While short division is best when dividing by a single digit, for bigger numbers you need a different approach
You can use other areas of maths that you know to help
- eg. cancelling fractions, “shortcuts” for dividing by 2 and 10, and the repeated addition (“chunking”) method covered in Multiplication

1. Short division (bus stop method)

Apart from where you can use shortcuts such as dividing by 2 or by 5, this method is best used when dividing by a single digit

eg. 534 ÷ 6

Bus Stop Completed Example, IGCSE & GCSE Maths revision notes

So, 534 ÷ 6 = 89

2. Factoring & cancelling

This involves treating division as you would if you were asked to cancel fractions
You can use the fact that with division, most non-calculator questions will have only integer answers
The only thing to be aware of is that this might not be the case if you’ve been asked to write a fraction as a mixed number (but if you are asked to do that it should be obvious from the question)
- eg. 1008 ÷ 28 1008 ÷ 28 = 504 ÷ 14 = 252 ÷ 7 = 36
You may have spotted the first two values (1008 and 28) are both divisible by 4 which is fine but if not, divide top and bottom by any number you can
To do the last part (252 ÷ 7) you can use the short division method above

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

This is a case of moving digits along the place value columnseg.
- 380 ÷ 10 = 38 45 ÷ 100 = 0.45

How do I divide a number by a decimal?

If the number you are dividing by is a decimal, use a multiple of ten to change it to an integer before carrying out the division
Always change both parts of the problem, chunking may be used to change the value to an easier number to divide by
- This may be easier to see by writing the problem as a fraction
- For example, to divide 512 by 1.6

Without using a calculator, divide 568 by 8. You must show all your working. 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: