What is Division?
There are four basic operations in math: addition, subtraction, multiplication, and division. Division is the process of splitting numbers into equal parts or equal shares. Suppose 25 students have to be sorted into basketball teams containing 5 players each. To find the number of basketball teams, we divide 25 by 5 = 5. That means the students can be sorted into 5 teams equally. Here, 25 is known as the dividend, 5 is known as the divisor, and the result, 5, is known as the quotient.
What are Dividend, Divisor, Quotient, and Remainder?
The general form of division is \(\text{Dividend}\div \text{Divisor}=\text{Quotient}\)
Division can also be written as \(\frac{\text{Dividend}}{\text{Divisor}}=\text{Quotient}\)
The dividend is the number that is being divided in the division operation. The divisor is the number that divides the dividend. The quotient is the result of the division operation.
In many cases, the divisor may not fully divide the dividend. In such cases, we get a remainder after the division operation. A remainder is a left-over value that we get after a division operation.
Suppose you want to distribute 15 cookies equally among 4 friends. You can only distribute 12 cookies among your friends if you want each of them to get the same share. The remaining 3 cookies represent the remainder. The result of the division operation can be written as 3 R 3, where 3 is the quotient and 3 is the remainder.
In case there is a remainder, then
\(\text{Dividend}=\text{Divisor}\times \text{Quotient}+\text{Remainder}\)
Example of Remainder:
It is important to note that the divisor cannot be 0. In such cases the division operation is undefined. That is, we can’t perform the division, and hence we won’t get a result.
Division by 1-digit Numbers
Division by 1-digit numbers can be performed using the long division table. Suppose you want to divide \(2048\div 5\).
Follow these steps.
We first find an estimate \(2000\div 5=400\)
Here, the dividend is 2048, the divisor is 5, the quotient is 409 and the remainder is 3.
To check the result we can multiply the quotient with the divisor, add the remainder, and compare it with the dividend. If both values are equal, that means the division operation is correct.
\(\text{Divisor}\times \text{Quotient}+\text{Remainder}=5\times 409+3\)
\(=2045+3\)
\(=2048\)
\(=\text{Dividend}\)
Hence, the division operation is verified
Division of 3-digit Numbers by 2-digit Numbers
Dividing 3-digit numbers by 2-digit numbers is similar to the division we performed earlier with 1-digit numbers. In this case, the only difference is that the dividend will be a 3-digit number and the divisor will be a 2-digit number.
Let’s see how we can divide
\(369\div 12\).
We first find an estimate so \(300\div 10=30\)
In this case, the dividend is 369, the divisor is 12, the quotient is 30, and the remainder is 9.
To check the result, we can multiply the quotient with the divisor, add the remainder, and compare it with the dividend. If both values are equal, that means the division operation is correct.
\(\text{Divisor}\times \text{Quotient}+\text{Remainder}=12\times 30+9\)
\(=360+9\)
\(=369\)
\(=Dividend\)
The division operation is verified.
Division of 4-digit Numbers by 2-digit Numbers
Division of a 4-digit number by a 2-digit number is also similar to the division operations that we performed earlier. Let’s divide 4156 by 41.
We first find an estimate so \(4000\times 40=100\)
In this example, the dividend is 4156, the divisor is 41, the quotient is 101 and the remainder is 15.
To check the result, we can multiply the quotient with the divisor, add the remainder, and compare it with the dividend. If both values are equal, that means the division operation is correct.
\(\text{Divisor}\times \text{Quotient}+\text{Remainder}=41\times 101+15\)
\(=4141+15\)
\(=4156\)
\(=Dividend\)
The division operation is verified.
Solved Examples
Example 1: Divide 418 ÷ 4 and verify the result.
Solution:
Let’s find an estimate \(400\div 4=100\)
We can verify the answer using the following equation.
\(\text{Divisor}\times \text{Quotient}+\text{Remainder}=4{\times} 104+2=416+2=418=\text{Dividend}\)
Example 2: William divided \(2149\div 7\), and he got the quotient as 306 and the remainder as 6. Is this answer reasonable?
Solution:
If the division operation is correct, it should satisfy the following condition.
\(\text{Divisor}\times \text{Quotient}+\text{Remainder}=\text{Dividend}\)
In this case, the dividend is 2149, the divisor is 7, the quotient is 306, and the remainder is 6.
Substitute the values in the equation:
\(7\times 306+6=2142+6=2148\)
\(2148\neq 2149\)
Therefore, we can conclude that William made an error in the division operation.
Example 3: A period of two weeks is known as a fortnight. How many fortnights are there in a year? (1 year = 365 days)
Solution:
1 year = 365 days
1 fortnight = 14 days
To find the number of fortnights in a year, we need to divide the number of days in a year by the number of days in a fortnight. That means we need to find \(300\div 10=30\).
We first find an estimate \(300\div 10=30\)
So, there are 26 full fortnights in a year. The extra 1 that we got as the remainder can be counted as an extra day. Hence, there are 26 fortnights and an extra day in a year.
Here, the dividend is 365, the divisor is 14, the quotient is 26, and the remainder is 1. To verify the answer, we can use the following equation.
\(\text{Divisor}\times \text{Quotient}+\text{Remainder}=\text{Dividend}\)
\(14\times 26+1=364+1=365=Dividend\)
Hence, the division operation is verified
Example 4: Franklin owns a corn field and has 14 workers who work on the field. He handed over $6200 to Cooper who is also a farmworker. Franklin asked Cooper to divide the money equally and pay the wages to all workers. If there is any money remaining after splitting the wages equally, Franklin wants Cooper to buy some refreshments for everyone. What is the daily wage of each worker? How much money is left for purchasing refreshments?
Solution:
Total wage = $6200
Number of workers = 14
To find the daily wage of each worker, we need to find \(6200\div 14\). The amount of money for refreshments will be obtained as the remainder of this division operation.
Let’s first find an estimate \(6000\div 15=400\)
Here, the dividend is 6200, the divisor is 14, the quotient is 442, and the remainder is 12.
That means the farmworkers will get $442 each, and they have $12 to purchase refreshments.
The remainder is the left-over value that we get after performing a division operation. In many cases, a dividend may not completely divide a divisor. We get a remainder in such division operations.
For a division operation to be correct, it should satisfy the following condition.
\(\text{Divisor}\times \text{Quotient}+\text{Remainder}=\text{Dividend}\)
If we don’t get the dividend when we add the remainder to the product of the divisor and the quotient, that means the division operation went wrong.
We are able to check the result of a division operation using multiplication because the multiplication operation is the inverse of the division operation. For example, let’s divide 1 by 2.
\(1\div 2=\frac{1}{2}\)
Now, let’s multiply the result by 2.
\(\frac{1}{2}{\times}2 =1\)
By performing multiplication with the same number, we got the original number back. That’s because multiplication is the inverse of division. We use the same logic to check the result of a long division operation.
We use division in our day-to-day life for various purposes. Division is used while splitting the bill at a restaurant, sharing snacks equally among friends, allotting time for studying different subjects, and so on.