The following is a simple long division method that anyone who can add a number to itself can do. You can use this method to help you build up your skill and understanding of the long division process so you can then move on to the fast long division method. For those of you who are already proficient and confident in doing long division can skip this method and move directly to the fast long division method.
The basis of this simple long division method is the table that you generate as a lookup to help you find the correct answer. The table is simply made up by writing down the divisor with the number 1 next to it, then we add the divisor to itself and write down the total with the number 2 next to it. We continue to add the divisor to each new total and writing the increment number next to it until we get to 9 as the number written next to the total. This table is now a multiplication table for our divisor up to 9 times the divisor.
As we go through our long division we will lookup the values in this table to determine what the answer is for each step. You will be amazed at how simple this actually makes doing the actual long division.
Let's have a look at 9248 divided by 68
You can create the table anywhere but we will show it next to the division equation.
Step 1 - Creating the Table
When creating the table you can decide how much you need to write down, you can write down everything or do some calculation in your head and just write down the new total each time. Do whatever you feel comfortable with. In this example we will write down the totals.
Write down 68 and next to it write down 1.
Next we add 68 to the first 68 which gives us a total of 136, we write a 2 next to this.
Next we add 68 to the 136 which gives us a total of 204, we write a 3 next to this.
Next we add 68 to the 204 which gives us a total of 272, we write a 4 next to this.
Adding 68 to the 272 gives us a total of 340, we write a 5 next to this.
Adding 68 to the 340 gives us a total of 408, we write a 6 next to this.
Adding 68 to the 408 gives us a total of 476, we write a 7 next to this.
Adding 68 to the 476 gives us a total of 544, we write a 8 next to this.
Adding 68 to the 544 gives us a total of 612, we write a 9 next to this.
Now we are ready to begin the long division itself.
Step 2 - The Long Division
The divisor is a two digit number so lets look at the first two digits of the dividend, 92, which is larger than 68 so we can use this.
Checking our table after 68 the next value is 136 which is larger than 92 so the first digit of our answer is 1.
We subtract 68 from the 92 which gives us 24, we then bring down the next digit of the divisor, the 4 which now gives us 244.
Looking down our table we find the first number larger than 244 is 272 so we go back up one to 204. This is the 3rd number in our table so the second digit of the answer is 3.
We subtract the 204 from the 244 which gives us 40, we then bring down the next digit of the divisor, 8 which gives us 408.
Checking our table we find 408 is actually the 6th number in the table so the last digit of our answer is 6.
We subtract 408 from 408 which leaves zero.
The answer to the long division is 9248 divided by 68 is 136 with no remainder.
As you can see once the table is in place you simply refer to the table to get the number of times the divisor divides into the number we are looking at.
We did not do anything except addition and subtraction to solve the long division
Checking your addition
There is an additional process you can follow when making your table to greatly lessen the chance of making a mistake.
That is to check each stage of the addition by casting out nines and using the digit root.
The digit root of 68 is 5.
6 + 8 = 14
1 + 4 = 5
When we added 68 to 68 and came up with 136, we can check the result by adding the digit root of 68 to the digit root of 68 then comparing this with the digit root of 136.
68 + 68 = 136
5 + 5 = 10
1 + 0 = 1 1 (3 + 6 = 9 so is ignored)
For both sides the digit root is one, so we should be correct.
Continuing on with the table we now add 68 to the 136 to get 204
136 + 8 = 204
1 + 5 = 6 2 + 0 + 4 = 6
Adding the digit root of 136 to the digit root of 68 we get 6 which is the same as the digit root for 204 which is also 6. Our addition should be correct.
Adding 68 to 204 we came up with 272.
204 + 68 = 272
6 + 5 = 11
1 + 1 = 2 2 (7 + 2 = 9 so is ignored)
Adding the digit root of 204 which is 6, to the digit root of 68, which is 5, we get 11. This is a two digit number so we add the digits together to get 2.
For 272 the digit root is also 2 as we can ignore the 7 and the other 2 as they add up to nine.
For the rest of the table we can continue to check the the addition in the same manner as we have done so far.
If you wrote just about everything out while creating the table along with the error checking it wouuld look something like this:
Checking your Final Result
You do a check on your final result by doing the following method:
- If there is a remainder subtract it from the dividend then calculate the digit root of the result.
- Multiply the digit root of the answer by the digit root of the divisor
- Compare the results of step 1 and 2, if they are the same the work should be correct
1) From our example above there was no remainder and the dividend was 9248.
2 + 4 + 8 = 14
1 + 4 = 5
2) The answer was 136 so digit root is 1.
The divisor was 68 so the digit root is: 6 + 8 = 14, 1 + 4 = 5
Multiplying the digit root of the answer and the digit root of the dividend we get: 5 x 1 = 5
The digit root in both steps is 5 so our answer should be correct.
So as you can see there is a lot that you are doing when the checking is included but writing everything down is only possibly required when first learning the method then as you get more proficient you can write less and less down, doing more of the problem mentally and therefor faster.
Some people will say that this simple long division method is just too much work but if you don't have a calculator handy and you have to solve the problem and your not very good at long division this is a method that will get you to the correct answer.