Division is one of the four main operations in mathematics. It's used frequently in various walks of life, from cutting up a cake to creating a financial budget. It goes hand in hand with multiplication, which is the opposite operation of division, as it involves increasing one number by the other instead of reducing them.
Although division is taught to children from a young age, not everyone finds division problems simple. There may be some terms in relation to division that you are unfamiliar with or you might struggle to know the best method to calculate the answer in a complex equation.
It's helpful to understand the various components and terminology of mathematical operations so that you can understand how you reached the final answer. This can make it easier to spot any mistakes that you may have made and could help you to learn different methods of calculating the answer to equations.
Division can seem complex, so we've compiled this guide that breaks down the basic components of these mathematical operations, including the terminology that describes the formula, such as a quotient definition. We'll also explain two different methods that you can use to find a quotient from a division equation.
The quotient is the term used to describe the final answer when you divide one number by another. It can be an integer (full number) if the numbers can be divided equally or a decimal number if they cannot. For example, the quotient of the equation 8 ÷ 2 is 4, while the quotient of the equation 7 ÷ 2 = 4.5 or 3 r 1 (where 'r' refers to the remainder). It's important to remember that the quotient and divisor will always be smaller than the dividend.
You can use the formula Dividend = Divisor x Quotient to check if you have calculated the right quotient. In some instances, you may have a number left over, so you will need to use the equation Dividend = Divisor x Quotient + Remainder instead.
The table below features the terms related to division that you will encounter in these types of equations. The values in the final column are in reference to the quotient example from the equation 7 ÷ 2.
| Terms | Descriptions | Values |
| Dividend | The total number of objects, pieces etc | 7 |
| Divisor | The amount that the total number of groups will be equally split into | 2 |
| Quotient | The number of objects or pieces in each group (the final answer) | 3 |
| Remainder | The leftover number that does not cleanly fit into any of the groups | 1 |
Continue reading to find out the different methods that you can use to calculate the dividend, along with the various forms that the quotient may take.
The quotient can either be an integer or a decimal number, depending on the dividend and divisor. When the dividend is completely divisible by the divisor, the subsequent quotient will be a full number. However, when the dividend is not completely divisible by the divisor, it will leave a remainder, which can be written as a decimal. Alternatively, the remaining amount can be added to the full quotient number after the letter 'r'.
For example,
15 ÷ 3 = 5, which is a full number
However, 15 ÷ 2 = 7.5 or 7 r 1
In the above example, the quotient is 7, and the remainder is 1. This is because 2 fits into 15 seven times, but there is one remaining number that doesn't completely fit. Although both the decimal number and the quotient with the remainder are the correct answer, you may decide to calculate the quotient in one form over the other. This could be because the project you are working on requires the answer in decimal points or vice versa. You will also find that most calculators will automatically give the quotient as a decimal number.
The quotient is the resulting answer after one number has been divided by another. Below is a breakdown of the various methods that you can use to calculate the answers.
Subtraction
You can calculate the quotient without directly using division. To do this, you need to repeatedly subtract the same number from the dividend until you cannot subtract the number anymore without making a minus number. For example, you may be trying to calculate what 32 ÷ 8 equals. To find the quotient using subtraction, you need to subtract from 32 in groups of 8 until you cannot subtract 8 anymore. This would look like the following:
32 - 8 = 24
24 - 8 = 16
16 - 8 = 8
8 - 8 = 0
In the above example, 8 was subtracted from 24 on four different occasions, which means that 32 ÷ 8 = 4. In the same way, to calculate what 32 divided by 4 is, you would need to subtract 4 from 32 until you aren't left with anything. You would end up subtracting 4 from 32 on eight different occasions, which means that 32 ÷ 4 = 8.
This process will take longer if you are subtracting smaller amounts from the dividend as you will likely have to subtract more groups. This does mean that it's easier and quicker to calculate the quotient if the divisor is a larger number.
The method of repeated subtraction is largely used by people who aren't as confident with division. It is easy to work out on paper or through the use of physical objects. For example, a child could try to calculate 12 ÷ 3 by taking 12 toys and splitting them into groups by separating three toys from the original group at a time.
Long division
While repeated subtraction can be easily used to divide small numbers, the long division method is a better option when dividing larger numbers. The process of long division takes five steps:
- Divide
- Multiply
- Subtract
- Bring Down
- Repeat/ Remainder
These five steps form the foundation for the following process using a 'division house' or 'bus stop' as it is sometimes called:
1. First, you need to see if the divisor is equal to or lower than the first digit of the dividend. If the divisor isn't equal or lower, you will need to combine the first and second digits of the dividend together.
2. You should then divide the dividend by the divisor and write the first digit of the quotient at the top of the division house
3. After you have subtracted the result from the digit, you should write the difference below the original number
4. Next, bring the next digit of the dividend down
5. Repeat the same process until you have your final answer
Calculator
If all else fails, you can always rely on your trusty calculator to give you the quotient of a division equation. All you need to do is type in the first number, the divide symbol and the number that you are trying to divide the first number by. You can also use online calculators if you don't have a portable calculator or your mobile phone with you.
What is the difference between a quotient and a product?
The quotient is the answer when one number is divided by another. A product, on the other hand, is the answer when one number is multiplied by another. Quotients are always smaller than the dividend and divisor. Products are always larger than the numbers that have been multiplied together.
How can I check whether the quotient is correct?
The opposite operation from division is multiplication, so it is to be expected that you must perform a multiplication equation to check the quotient. You can use the formula (Divisor × Quotient) + Remainder, which should equal the dividend if the quotient is the correct answer to the division problem.
In the example equation of 25 ÷ 5, the answer is 5. To confirm that the quotient is correct, you can multiply 5 x 5, which will give you the answer 25, which was the dividend in the original equation.
A quotient is the term used to describe the answer after one number (the dividend) has been divided by another (the divisor). In perfect divisions, the quotient will be an integer, which is a full number. This is because the dividend can be completely separated into the divisor. However, you may be left with a decimal or a remainder if the dividend cannot be fully separated into the divisor.
There are two main ways to calculate the quotient in an equation: through long division or repeated subtraction. Long division is usually the preferred method when dividing large numbers, whereas repeated subtraction is a simpler method that can quickly divide small numbers.
You can check whether the quotient is correct by multiplying it by the divisor and adding on any remainders. The product of this equation should be the same as the dividend in the original division equation.
