Division is a key component of mathematics and is one of the four basic operations that students are introduced to at an early age.
Division questions are often dreaded by students as they can appear harder than addition, subtraction, or multiplication questions.
However, there are a number of division methods you can use to help you make the division process much simpler.
We are going to take a look at division, some of the key methods and concepts behind it, and walk you through some division problems to ensure you have no fear when you are next faced with a division question.
Division is the process of calculating how many times one number is contained within another. Along with addition, subtraction, and multiplication, it is one of the four basic operations of arithmetic.
Although division can appear hard at first, there are a number of techniques that you can use to make it as simple as possible.
So let's jump in and take a look at some of the key definitions we use when talking about division.
When we use division, there are a few terms and definitions that are useful to know about as they can often come up in exams.
- Dividend. The dividend is the number that is being divided. For example, in the question 20 ÷ 5, the dividend is 20.
- Divisor. The divisor is the number that we are dividing by. So in the same example of 20 ÷ 5, the divisor is 5.
- Quotient. The quotient is the answer to a division question. 20 ÷ 5 = 4. So 4 is the quotient.
- Integer. An integer is just another word for a whole number. 3 is an integer, 3.5 is not.
- Quotition. Quotition is a form of division that considers how many parts of a whole there are.
- Partition. Partition is a form of division that asks how big is each part that forms the whole.
Let's take a closer look at these last two terms - quotition and partition - and find out exactly what we mean by them.
The best way to think of division is to think of it in terms of "quotition" and "partition".
Quotition division asks, how many equal parts or equal groups of a whole are there? Whereas, partition division asks, how big is each part?
For example, if you had 20 apples, quotition division would find that you could divide those apples into groups of 2, 4, 5, and 10.
Partition would look at how many groups there are of each part. For example, 2 groups of 10 apples.
Both quotition and partition will ultimately yield the same answers, but they look at the same question from different angles.
There are various ways you can write out division using division signs.
The most commonly used symbol is known as the obelus (÷). When using the obelus, the dividend is placed first, followed by the obelus, and then the divisor. For example, 20 ÷ 5.
Division can also be depicted using a slash, with the dividend first, then the slash, followed by the divisor. For example, 20 / 5.
You may also find division written using a horizontal bar known as a fraction bar. In this instance, the dividend is on top of the bar and the divisor below it.
When working out complicated division questions, you will often need to use a division bar (厂), which we will now look at in more detail.
In maths, the remainder is the amount left over after a calculation has been completed. Remainders are used when you want to find integers (whole numbers).
- 23 ÷ 5 = ?
- 23 cannot be divided by 5 and arrive at a whole number. 20, on the other hand, can.
- 20 ÷ 5 = 4
- 23 - 20 = 3
- Therefore, 23 ÷ 5 = 4 remainder 3.
Confusingly, when we divide by decimal numbers we arrive at a larger value than the original dividend.
- 15 ÷ 0.2 = 75
This is because there are 5 lots of 0.2 in 1. So there are 15 x 5 lots of 0.2 in 15.
Dividing decimals is a question that many students dread. However, there is a simple method of dividing using decimals that you can use to make the process a lot easier:
- to begin with, you need to convert your decimal number into a whole number. You can do this by multiplying it by 10.
- 0.2 x 10 = 2
- you then need to do the same with your dividend, which is 15 in our example.
- 15 x 10 = 150
- now you can return to the original question but with the new values.
- 150 ÷ 2 = 75
- as you can see, we have arrived at the same answer, even though we were no longer using the decimal number.
This method works whether the decimal point is the divisor or the dividend and can make dividing with decimals a lot less scary!
In this method, you work your way back down the number line until you either reach 0 or you can no longer subtract the divisor.
- 18 ÷ 3 = ?
- Subtract 3 from 18 until you cannot subtract it anymore.
- 18 - 3 = 15
- 15 - 3 = 12
- 12 - 3 = 9
- 9 - 3 = 6
- 6 - 3 = 3
- 3 - 3 = 0
- Then count up how many times you were able to subtract the divisor (3), which is 6 in our example.
- So, 18 ÷ 3 = 6.
The chunking method is similar to the long division method, but it uses a bit more mental arithmetic to arrive at the answer quicker.
The chunking method encourages you to use your multiplication knowledge to complement your division skills.
For example, let's take the question 378 ÷ 18 = ?
- We know that 18 x 10 = 180. So we can write down 10 as one chunk.
- 378 - 180 = 198.
- So then we can figure out the chunks of 198.
- We can again do 18 x 10 = 180. So we can write down 10 as another chunk, meaning our total chunks are now 20 (10 + 10).
- 198 - 180 = 18.
- Well, we obviously know that 18 goes into 18 only once. So we add a chunk of 1 to our total chunks.
- This means our total chunks are 21 (10 + 10 + 1).
- Therefore, 378 ÷ 18 = 21.
The division formula, or the division equation, refers to the rule that you can find the dividend by multiplying the divisor and the quotient.
- Dividend ÷ Divisor = Quotient
- Quotient x Divisor = Dividend
- 100 ÷ 5 = 20
- This means that...
- 20 x 5 = 100
Here we will take a look at some of the key divisibility rules for the numbers 1 to 10 that can help you work out division questions as quickly as possible.
Every number is divisible by 1. Whenever you divide a number by 1, it will give the same number you started with because it has not been split.
100 ÷ 1 = 100
Any number with an even last digit, including 0, can always be divided by 2 to arrive at an integer.
2468 ÷ 2 = 1234
A number is completely divisible by 3 if the sum of its digits is wholly divisible by 3.
For example, take the number 516.
- 5 + 1 + 6 = 12
- 12 is divisible by 3
- Therefore 516 is divisible by 3.
If the final two digits of a number are divisible by 4, then the whole number is divisible by 4.
- The final digits of 2316 are 16
- 16 is divisible by 4
- Therefore 2316 is divisible by 4.
Any number that ends with a 0 or a 5 is always divisible by 5, no matter how big.
7635218195 ÷ 5 = 1527043639
Numbers that are divisible by both 2 and 3 are divisible by 6. So if the last digit of the number is even and the sum of the digits in the number totals a multiple of 3, then the number is divisible by 6.
- 630 ends in an even digit
- 6 + 3 = 9 and 9 is a multiple of 3
- Therefore, 630 is divisible by 6.
The 7 divisibility rule is a bit complicated.
If a number is divisible by 7, then the difference between twice the unit digit of the given number and the remaining part of the given number should be a multiple of 7 or it should be equal to 0.
- Is 798 divisible by 7?
- What is twice the unit digit? 8 x 2 = 16
- What is the remaining part of the number? 79.
- What is the difference between twice the unit digit (16) and the remaining part of the number (79)?
- 79 - 16 = 63
- 63 is divisible by 7, therefore 798 is divisible by 7.
If the last three digits of a number are divisible by 8, then the whole number is divisible by 8.
23,400 is divisible by 8 because 400 is divisible by 8.
Similar to 3, if the sum total of the digits of a number equals a multiple of 9, then the original number can be divided by 9.
- Is 189 divisible by 9?
- 1 + 8 + 9 = 18
- 18 is divisible by 9
- Therefore, 189 is divisible by 9.
Any number that ends with 0 is divisible by 10.
Division is a key component of mathematics, but it is also used in our everyday lives. So it is important to understand exactly how it works!
For example, a waiter in a restaurant may need to divide a bill for a group of people eating together and ensure that the cost is split equally amongst them.
Or you may receive a certain amount of money a month and you want to make sure you spend an equal amount each week. So you will need to divide your monthly earnings into 4.