The Four Rules of Arithmetic

The Four Rules of Arithmetic (in brief):
 -  Addition
 -  Addition of decimals
 -  Subtraction
 -  Subtraction of decimals
 -  Subtraction other methods 
 -  Multiplication 
 -  Division
 -  Long division 
 -  Number relationships 

1: ADDITION
Addition tells us the TOTAL of things.

Q1: What is 23 + 140?
The best start is to correctly line up the digits in HUNDREDS, TENS, and UNITS.
START ON THE RIGHT HAND SIDE and add up the numbers IN COLUMNS

H T U
  2 3
1 4 0
 
The first column is 3 and 0, so your first addition is 3 + 0.
Write your answer DIRECTLY UNDERNEATH the column you’ve just added.
Now move to the next column:
And finally the last column:
Once you’ve done the last column you are finished!
So, 23 + 140 = 163.

Q2: What is 123 + 189?
Again, first of all line the digits up according to place value:
And again, you start on the RIGHT HAND SIDE.
The first column is 3 + 9, but 3 + 9 = 12. There’s only have room for one number underneath!
So, you “carry over” the 1 in 12, and write the 2 from the 12 underneath:
You now move on the next column. It used to be 2 and 8.
Now though, you have 2, 8, and the 1 you carried over before.
So now, you do 1 + 2 + 8. This is 11.
Write “1” down and “carry over” the other 1 in the 11.
Now you do the last column: 1 + 1 + the 1 you carried:
You’ve done the last column, so you have finished.
So, 123 + 189 = 312.

Addition of decimals

Q3: What is 5.5 + 0.98?
Don’t let the decimals bother you! You still start in the same way by lining up the digits according to place value.
The easiest way to do this is: MAKE SURE THE DECIMAL POINTS LINE UP!
The first column is just 8 on its own. 8 plus... nothing!
Now the second column is 5 + 9, and 5 + 9 = 14.
“Carry over” the 1, and write the 4 underneath.
You must line up the decimal points, so put in another point directly beneath the others:
Then you finish off by doing the last column as usual:
So, 5.5 + 0.98 = 6.48

Subtraction

2: SUBTRACTION
Subtraction is the reverse of addition!
If you add something to a number, then take it back off, you’re back to where you started:
2 + 2 = 4  4 – 2 = 2.
7 + 31 = 38  38 – 31 = 7

Subtraction can tell you quite a few useful things:
 The difference between two numbers;
 The RANGE of different values (see later!);
 What I need to add to a smaller number to get to the bigger one;
 How much change I can expect back from my £10 note!
So how do you do it?

Q1: What is 98 – 44?
What you must ALWAYS do is PUT THE NUMBER YOU ARE TAKING AWAY ON THE BOTTOM.
You are taking away 44, so this goes on the bottom. You line up the digits according to place value just like you do with addition:
Again, like addition, you start on the RIGHT HAND COLUMN.
You do THE TOP NUMBER MINUS THE BOTTOM NUMBER, which here is 8 – 4:
Then the same for next column, 9 – 4:
That was the last column so you are finished. So, 98 – 44 = 54.

Q2: What is 109 – 38?
Again, you start in the same way. Line up the digits according to place value and put the number you are taking away on the bottom:
Your first column is 9 – 8:
Now the next column is 0 – 3. How can you do that?
To make it possible, YOU BORROW 1 FROM THE LEFT AND MOVE IT OVER.
The digit you borrow from becomes smaller by 1, and the 1 you take is placed just beside the number on the right:
The 0 has become 10 after this “borrowing”.
The 1 you borrowed from becomes 1 smaller, so it becomes 0.
This now means the last column has gone altogether – there’s nothing there.
So all you need to do is the next column, 10 - 3:
You have now done all columns. So, 109 – 38 = 71.

Q3: What is 102 – 45?
Line up the digits according to place value and put the number you’re taking away on the bottom:
The first column is 2 – 5, which you can’t do. Can you borrow?
The next number across from 2 is 0, so there’s NOTHING THERE TO BORROW!
You need to borrow from the very end, then move the 1 across gradually:
Now the “0” has become “10” so you CAN now borrow from the middle column.
You borrow 1, which makes the 10 smaller by 1:
Now the “2” has become “12”. You can now carry on and subtract in the usual way.
Start with the right column, 12 – 5:
Then 9 – 4:
And you are now finished because the 1 on the left was moved over.
So, 102 – 45 = 57

Subtraction of decimals

Q4: What is 5 – 2.37?
You set out the subtraction calculation in the same way as before.
However, one number is a decimal and the other is not!
If 5 had a decimal point, it would be 5.0, or even 5.00.
There are 0s after the decimal, because there is no decimal part to 5 – it is a whole number.
So, you can think of the question as 5.00 – 2.37:
You can’t do 0 – 7, so you need to borrow.
You can’t borrow from the next digit, 0, so you need to borrow from the 5:
The middle 0 has become 10. Now you borrow AGAIN: You can now subtract column - by - column as normal.
First column: 10 – 7 : Then 9 – 3...
Now you need to add a decimal point below, so all decimal points line up. Just like you had to do for addition with decimals:
And finally, the last column, 4 – 2:
So, 5 – 2.37 (which is the same as 5.00  -  2.37) = 2.63

Subtraction – other methods

Other subtraction methods
Like most maths, there are several methods for any one problem.
You can use subtraction to find the difference between two numbers, or to find out what you need to add to the smaller number to get to the bigger one.
Using that idea, you can calculate subtraction problems by using addition.

Q1: What is 512 – 149?
Thinking of this in terms of adding, you could also ask
“WHAT DO I NEED TO ADD TO 149 TO GET TO 512?
The trick now is to start at 149 and get to the next “nice” number to make it easy on yourself.
You could ADD 1 to 149 to get 150.
You could ADD 50 to 150 to get 200.
You could ADD 300 to 200 to get 500.
You could ADD 12 to 500 to get 512!
In total, you added 1, 50, 300 and 12:
This tells you that 512 – 149 = 363, because 149 + 363 = 512
This method comes into its own when you may otherwise have to “borrow” from the other side again, and again, and again!

Q2: What is 50 001 – 49 999?
The standard way would be tedious:
You can’t do 0 – 9, so you must “borrow”, but the next THREE digits are all 0!
You would have to borrow 1 all the way from one end to the other.
OR...
You could ADD 1to 49 999 to get 50 000.
You could ADD 1 to 50 000 to get 50 001.
In total you just added 1 and 1, which is 2.
So, 50 001 – 49 999 = 2.
Note: Both ways of subtraction will always work. Use whichever you feel most confident with.

Multiplication

3: MULTIPLICATION
Multiplication can tell you lots of things, including:
 The AREA of a 2D shape;
 The VOLUME of a 3D shape.
In some cases, it is like a faster way of doing addition:
9 + 9 + 9 + 9 + 9 + 9 + 9 + 9 (Slow)  9 x 8 (Quick!)
10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 (Slow)  10 x 10 (Quick!)

Q1: What is 17 x 9?
You put the smaller number on the bottom:
Then multiply the bottom number by the numbers above,
MOVING RIGHT TO LEFT.
First, do 9 x 7. This equals 63. Like addition, you “carry over” the 6.
Then do 9 x 1, and remember to add on your carried 6:
9 x 1 equals 9, plus the carried 6 gives 15:
So, 17 x 9 = 153.
Note: With multiplication ALWAYS PUT THE FIRST NUMBER DIRECTLY BENEATH THE DIGIT YOU ARE MULTIPLYING WITH.

This general method also works when larger numbers are on the bottom.
Start with the bottom - right digit, then multiply it by the digits above, moving FROM right TO left.
When you’re done with that number move on to the next number to the left.
Here’s some examples for the pattern you would follow when multiplying with larger numbers:
Example 1: 15 x 15

Note: With multiplication ALWAYS PUT THE FIRST NUMBER DIRECTLY BENEATH THE DIGIT YOU ARE MULTIPLYING WITH.
With the first number – 5 – the first digit would go below the 5.
With the second number – 1 – the first digit would go below the 1:

Just in case, here is 15 x 15 done STEP - BY - STEP:
Start in the bottom - right, the 5, and multiply with the digits above moving FROM right TO left. First is 5 x 5, which is 25:
Next you do 5 x 1, PLUS the carried 2:
Now move to the next number, the 1. Again, you multiply from the top - right and move left, so first calculation is 1 x 5.
THE ANSWER IS PUT DIRECTLY BENEATH THE 1 WHICH YOU ARE MULTIPLYING:
Next you do 1x1: And you have finished.
THE FINAL STEP IS TO ADD UP THE MULTIPLICATION ANSWERS:
So, 15 x 15 = 225.

Example 2: 108 x 356
Again, it is exactly the same method. I would advise you try this one yourself, then check back here to see if you got it right!
Then finally, add the totals together:
So, 108 x 356 = 38 448.
Note: There are other ways of doing multiplication (the Lattice Method for example). Use the way you are most comfortable with.
This way is just a personal suggestion.

4: DIVISION
Division is the opposite of multiplication.
Because of this, division can tell you what you need to multiply a number by to get another number:

Example
6 x ? = 18. What is ?
In plain English:
6 multiplied by SOMETHING equals 18. What is that something?
(Hopefully) you know that 6 x 3 = 18, so that something MUST BE 3.
You could have worked this using division: 18 ÷ 6 = 3

Other Examples
2x ? = 10 ? = 5 10 ÷2 = 5
5x ? = 15 ? = 3 15÷5 = 3
10x ? = 20 ? = 2 20÷10 = 2  -  see a pattern?
These mystery numbers may have been quite easy to work out WITHOUT thinking of division, but...if the question was “7 x ? = 1477”, division comes into its own!

Division can tell you:
 What you need to multiply one number by to get the bigger number;
 How much of something – maybe money – each person receives;
 The FRACTION of a value.

And those are just a few.
 

Division with smaller numbers: mental methods
For smaller numbers, division can often be done without using LONG DIVISION.

Q1: What is 24