Division '÷' | Basics of Arithmetic

See also: Fractions

This page covers the basics of Division (÷).

See our other arithmetic pages, for discussion and examples of: Addition (+), Subtraction (-) and Multiplication (×).

Division

In handwriting the usual sign for division is (÷) on a spreadsheet and some other computer applications the ‘/’ symbol is used to denote division.

Division is the opposite of multiplication in mathematics.

Division allows us to divide or 'share' numbers to find an answer.

 Multiplication gives us a quick way of doing multiple additions and division gives us a quick way of doing multiple subtractions. 

Division is often considered the hardest of the four main arithmetic functions, remember all division calculations can easily be carried out on a calculator. When you think about division think about sharing a number equally by the number of times the sum says. For example 10 sweets divided by 2 children = 5 sweets each.


Some Quick Rules about Division:


  • When you divide 0 by another number the answer is always 0. For example: 0 ÷ 2 = 0. That is 0 sweets shared equally among 2 children - each child gets 0 sweets.
  • When you divide a number by 0 you are not dividing at all (this is quite a problem in mathematics). 2 ÷ 0 is not possible. You have 2 sweets but no children to divide them to. You cannot divide by 0.
  • When you divide by 1 the answer is the same as the number you were dividing.  2 ÷ 1 = 2. Two sweets divided by one child.
  • When you divide by 2 you are halving the number. 2 ÷ 2 = 1.
  • Any number divided by the same number is 1. 20 ÷ 20 = 1. Twenty sweets divided by twenty children - each child gets one sweet.
  • Numbers must be divided in the correct order. 10 ÷ 2 = 5 whereas 2 ÷ 10 = 0.2. Ten sweets divided by two children is very different to 2 sweets divided by 10 children.
  • All fractions such as ½, ¼ and ¾ are division sums. ½ is 1 ÷ 2. One sweet divided by two children. See our page Fractions for more information.

Multiple Subtractions

Division sums can be a useful and quicker way of performing multiple subtraction sums.

For example:

If John has 10 gallons of fuel in his car and uses 2 gallons a day how many days before he runs out?

We can work this problem out by doing a series of subtractions, or by counting backwards in steps of 2.

  • On day 1 John starts with 10 gallons and ends with 8 gallons.  10 - 2 = 8
  • On day 2 John starts with 8 gallons and ends with 6 gallons. 8 - 2 = 6
  • On day 3 John starts with 6 gallons and ends with 4 gallons. 6 - 2 = 4
  • On day 4 John starts with 4 gallons and ends with 2 gallons. 4 - 2 = 2
  • On day 5 John starts with 2 gallons and ends with 0 gallons. 2 - 2 = 0

John runs out of fuel on day 5. 

A quicker way of preforming this sum would be to divide 10 by 2.  10 ÷ 2 = 5. That is, how many times does 2 go into 10.

The multiplication table (see multiplication) can be used to help us find the answer to simple division sums.

In the example above we needed to calculate 10 ÷ 2.  To do this, using the multiplication table locate the column for 2 (the red shaded heading).  Work down the column until you find the number you are looking for, 10.  Move across the row to the left to see the answer (the red shaded heading) 5.

Multiplication Table

× 1 2 3 4 5 6 7 8 9 10
1 1 2 3 4 5 6 7 8 9 10
2 2 4 6 8 10 12 14 16 18 20
3 3 6 9 12 15 18 21 24 27 30
4 4 8 12 16 20 24 28 32 36 40
5 5 10 15 20 25 30 35 40 45 50
6 6 12 18 24 30 36 42 48 54 60
7 7 14 21 28 35 42 49 56 63 70
8 8 16 24 32 40 48 56 64 72 80
9 9 18 27 36 45 54 63 72 81 90
10 10 20 30 40 50 60 70 80 90 100


We can work out other simple division sums using the same method.  56 ÷ 8 = 7 for example. Find 7 on the top row, look down the column until you find 56 find the corresponding row number, 8

If possible you should try to memorise the multiplication table above, it makes solving simple multiplication and division sums much quicker.


Dividing Larger Numbers

Although you can quickly preform division sums on a calculator by learning how to calculate division manually can help in situations when you don't have a calculator to hand or don't want to use one. Division sums can look daunting but in fact, as with most mathematics, they are logical.

Dave’s car needs new tyres, all four driving tyres and the spare need to be replaced. 

Dave has had a quote from a local garage for £480 to include the tyres, fitting and disposal of the old tyres.  How much does each tyre cost?

The problem we need to calculate here is 480 ÷ 5.  This is the same as saying how many times will 5 go into 480?

Traditionally division sums are written like:

5 4 8 0

We start by dividing 5 by 4 and immediately hit a problem.  5 does not divide by 4 to leave a whole number, as 5 is greater than 4.  We are only interested in whole numbers and as 5 does not go into 4 we put a 0 in the first (hundreds) column. For help with the hundreds, tens and units columns see our page on numbers.

Hundreds Tens Units
0    
5 4 8 0

The next step is to use the next number to see how many times 5 goes into 48.

5 does go into 48 as 48 is greater than 5, however we need to know how many times it goes.

If we refer to our multiplication table we can see that 9 × 5 = 45 and 10 × 5 = 50.

48; the number we’re looking for falls between these two values.  We take the lower value as it is smaller than 48 and therefore will go.  5 goes into 45 9 times, but not exactly as it leaves a remainder of 3.  The remainder what is left when we subtract the number we have found from the number we are dividing into: 48 - 45 = 3.

So 5 × 9 = 45, + 3 to get 48.

We can enter 9 as our answer for the second part of the sum and bring our remainder in front of our last number.  Our last number becomes 30.

Hundreds Tens Units
0 9  
5 4 8 30

We now divide 30 by 5 or find out how many times 5 goes into 30.  Using our multiplication table we can see the answer is exactly 6, with no remainder.  5 × 6 = 30.

Hundreds Tens Units
0 9 6
5 4 8 30

As there are no remainders left we have finished the sum and have the answer 96.  Dave’s new tyres are going to cost £96 each.  480 ÷ 5 = 96 and 96 × 5 = 480.


Recipe Division

Our final example of division is based on a recipe.  Often when cooking, recipes will tell you how much food they are going to make, enough to feed 6 people, for example.

The ingredients below are needed to make 24 fairy cakes, however, we only want to make 8 fairy cakes. We have modified the ingredients slightly for the benefit of this example, original recipe at: http://www.bbc.co.uk/food/recipes/fairycakes_93711). 

The first thing we need to establish is how many 8's there are in 24 – use the multiplication table above or your memory.  3 × 8 = 24 – if we divide 24 by 8 we get 3.  Therefore we need to divide each ingredient below by 3 in order to have to right amount of mixture to make 8 fairy cakes.

Ingredients

  • 120g butter, softened at room temperature
  • 120g caster sugar
  • 3 free-range eggs, lightly beaten
  • 1 tsp vanilla extract
  • 120g self-raising flour
  • 1-2 tbsp milk

The amount of butter, sugar and flour are all the same, 120g it is only necessary to work out 120 ÷ 3 once, the answer will be the same for those three ingredients.

3 1 2 0

As before we start in the right column and divide 3 by 1. However 3 ÷ 1 doesn’t go as 3 is greater than 1.  Next we look at how many times 3 goes into 12.  Using the multiplication table if needed you can see that 3 goes into 12 exactly 4 times with no remainder.

0 4 0
3 1 2 0

120g ÷ 3 is therefore 40g. We now know that we’ll need 40g of butter, sugar and flour.

The next sum is simple.  The original recipe calls for 3 eggs, we again divide by 3.  So 3 ÷ 3.  This is the same as saying how many 3s are there in 3.  The answer is obviously 1.  Therefore one egg is needed.

Next the recipe calls for 1tsp (teaspoon) of vanilla extract.  We need to divide one teaspoon by 3.  A fraction is written like a division sum, 1 ÷ 3 is the same as ⅓ one third.  You’ll need ⅓ of a teaspoon of vanilla extract – in reality it may be difficult to accurately measure ⅓ of a teaspoon!

Next the recipe calls for 1-2 tbsp of milk.  That is between 1 and 2 tablespoons of milk.  We have no definitive amount and how much milk you add will be dependent on your mixture consistency.

We already know that 1 ÷ 3 is ⅓; therefore 2 ÷ 3 is ⅔.  ⅓ - ⅔ of a tablespoon of milk is needed to make 8 fairy cakes.

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