Fast Long Division

We will be looking at fast long division using two digit divisors here, but before we get into the fast long division, we need to explain some term that we will be using when going through the fast long division method.

Number, Tens and Units

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In the fast long division method, we will be dealing a lot with two digit numbers. The left hand digit of the number is the tens digit and the right hand digit is the units digit. We will refer to the tens digit with "T" and the units digit as "U" and we will refer to the whole two digit number as "N" for the number.

We will also be dealing with pairs of two digit numbers and to create the NT (Number - Tens) product we take the whole number (N) of one two digit number and add it to the tens (T) digit of the other two digit number.

The N number comes from multiplying the tens digit of the divisor by the first digit of the answer. The T number comes from multiplying the units digit of the divisor by the first digit of the answer. The N and T numbers are then added together to give us our NT value.

We will also refer to a "partial dividend" which is a number will lead us to the next digit of the answer by dividing it by the first digit of the divisor. The partial dividend will also lead us to the tens digit of the next working figure.

The "working figure" is a number whose purpose is to find the next partial dividend.

In explaining the fast division method, we will be writing everything down and to show where numbers are copied we include arrows. In practice when doing this method, not everything needs to be written down. At first, you may find it necessary to write down everything, but as you become accustomed to the method, you will find it easy to start to omit writing down some of the working figures. Your goal should be to become proficient enough with the fast division method that you do not need to write anything down except the answer.

Here is a video on Fast Division using two digit divisors.

 

Division With Two Digit Divisors

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Let's use 4712 divided by 31 as our example as we explain the method.

Step 1
Divide the first figure of the dividend by the first figure of the divisor and the result is the first figure of the answer.

There are some points to remember for this step:

• If the first figure of the dividend is smaller than the first figure of the divisor then use the first two figures of the dividend.
• If the second digit of the divisor is 8 or 9 then add 1 to the value of the first figure of the divisor and divide the dividend by this new figure.
• Always ignore any remainder as we are only interested in the whole number value from the division.

The first figure of the dividend is 4, and the first figure of the divisor is 3. Dividing 4 by 3 we get 1 as the first figure of our answer.
The 4 is also our first partial dividend, so we will copy it down to what will be our partial dividend row below the dividend

   

Step 2
Calculate the NT product by multiplying the divisor by the first figure we have for the answer.

We subtract this NT product from the first partial dividend from step 1.

   

We will write the 1 just below and to the left of the next figure in the dividend, the 7. This 1 will be the tens digit of our first working figure.

Step 3
We bring down the next figure of the divisor, the 7, down next to the 1 to create our first working figure, 17.

We now calculate the U product by multiplying the units figure of the divisor by the first figure we have for the answer.

We subtract the U product from the working figure to get the second partial dividend.

   

So 16 is our next partial dividend, and we will write it below the 17 on our partial dividend row.

Step 4
We now divide the second partial dividend, the 16, by the first figure of the divisor to get the second figure of our answer.

   

The following steps are just repeating steps 2, 3 and 4 using the latest found figure of the answer.
Step 5
Calculate the NT product by multiplying the divisor by the second figure we have for the answer.

We subtract this NT product from the second partial dividend from step 3.

   

We will write the 1 just below and to the left of the next figure in the dividend. This 1 will be the tens digit of our second working figure.

Step 6
We bring down the next figure of the divisor, the 1, down next to the 1 to create our second working figure, 11.

We now calculate the U product by multiplying the units figure of the divisor by the second figure we have for the answer.

We subtract the U product from the working figure to get the third partial dividend.

   

So 6 is our next partial dividend, and we will write it below the 11 on our partial dividend row.

Step 7
We now divide the third partial dividend, the 6, by the first figure of the divisor to get the third figure of our answer.

   

We have the answer at this point the remaining steps are to see if there is any remainder.
Remainder Step 1
Calculate the NT product by multiplying the divisor by the third figure we have for the answer.

We subtract this NT product from the third partial dividend from step 6.

   

We will write the 0 just below and to the left of the last figure in the dividend. This 0 will be the tens digit of our third working figure.

Remainder Step 2
We bring down the next figure of the divisor, the 2, down next to the 0 to create our third working figure, 02.

We now calculate the U product by multiplying the units figure of the divisor by the third figure we have for the answer.

We subtract the U product from the working figure to get the fourth partial dividend.

   

Our fourth partial divisor is zero which means we have no remainder in this example.
The answer to 4712 divided by 31 is 152 with no remainder.

You May Need To Backtrack

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The process of division, unlike multiplication, is not always exact. We make a choice which we hope will get us close to the answer. Sometimes we make the wrong choice, and we need to backtrack a few steps and make a different choice, but this time we have the knowledge of what went wrong with our first choice. We will look at an example of a problem where we need to backtrack a few steps.

Division Correction - Answer Too Small

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The partial dividend can inform us that the latest figure of the answer is too small. If the partial dividend is equal to or greater than the divisor, then the latest figure of the answer is too small.

This time, we will look at 18096 divided by 29.

Step 1
We need to divide the first figure of the dividend by the first figure of the divisor.

Firstly, the first figure of the dividend is 1 while the first figure of the divisor is 2 and we cannot divide 1 by 2 so we must use the first two figures of the dividend which is 18.

Secondly, we are dividing by 29, so the second figure of the divisor is 9 so instead of dividing by 2 we instead divide by 3.

   

So 6 is the first figure of the answer.

We used 18 divided by 3 above, we will copy 18 down as our first partial dividend.

Step 2
We now calculate the NT product by multiplying the divisor by the first figure we have for the answer, which is 6.

Multiplying 2 by 6 we get the N value of 12 and multiplying 9 by 6 we get the T value of 5, adding these together we get the NT value of 17.

We subtract the NT value from our partial dividend, 18 - 17 = 1, so we put the 1 up in the working figure row below the next figure in the dividend, the zero.

Step 3
We bring down the next figure in the dividend, the zero, and we have our first working figure of 10.

We now calculate the U product by multiplying the units figure of the divisor, the 9, by the first figure we have for the answer, the 6.

The U-value is the unit figure of the result of the multiplication, in this case, 4. We subtract the U value from the working figure.

   

The 6 is our second partial dividend.

Step 4
We now divide the second partial dividend by 3. Remember, although the first figure of the divisor is 2 because the second figure is 9 we add 1 to the value of the first figure of the divisor and use that in the division. 6 divided by 3 is 2 so 2 is the next figure of our answer.

We repeat steps 2 to 4 but now use the second figure of the answer in the calculations.
Step 5
Calculate the NT product by multiplying the divisor by the second figure we have for the answer.

Multiplying 2 by 2 we get the N value of 04 and multiplying 9 by 2 we get the T value of 1, adding these together we get the NT value of 05.

We subtract the NT value from our partial dividend.

   

So we put the 1 up in the working figure row below the next figure in the dividend, the 9.

Step 6
We bring down the next figure of the divisor, the 9, down next to the 1 to create our second working figure, 19.

We now calculate the U product by multiplying the units figure of the divisor by the second figure we have for the answer.

We subtract the U product from the working figure to get the third partial dividend.

   

So 11 is our next partial dividend, and we will write it below the 19 on our partial dividend row.

Step 7
We now divide the third partial dividend, the 11, by 3 to get the third figure of our answer.

   

Although we have the answer at this point it is not confirmed, the remaining steps besides checking if there is any remainder will also be used to confirm our result.
Remainder Step 1
Calculate the NT product by multiplying the divisor by the third figure we have for the answer.

Multiplying 2 by 3 we get the N value of 06 and multiplying 9 by 3 we get the T value of 2, adding these together we get the NT value of 08.

We subtract the NT value from our partial dividend.

   

So we put the 3 up in the working figure row below the next figure in the dividend, the 6.

Remainder Step 2
We bring down the next figure of the divisor, the 6, down next to the 3 to create our third working figure, 36.

We now calculate the U product by multiplying the units figure of the divisor by the third figure we have for the answer.

We subtract the U product from the working figure to get the fourth partial dividend.

   

We have a problem. Our partial dividend, which in this case is also the remainder, is 29, the same value as the divisor. If the remainder is the same or larger than the divisor, it means that the last figure of our answer is too small and should be increased.

In this case, it is easy to see that if the remainder is the same as the divisor, our final result should be 624 instead of 623.

However, it may not always be so clear, the best thing to do is to go back to step 7 and add one to our figure.
Step 7 Try 2
We divided the partial dividend by 3, and we got 3

   

However, we now know this is too small so we will use 4 instead.

We will redo the remaining steps to confirm our result.
Remainder Step 1 Try 2
Calculate the NT product by multiplying the divisor by the third figure we have for the answer.

Multiplying 2 by 4 we get the N value of 08 and multiplying 9 by 4 we get the T value of 3, adding these together we get the NT value of 11.

We subtract the NT value from our partial dividend.

   

So we put the 0 up in the working figure row below the next figure in the dividend, the 6.

Remainder Step 2 Try 2
We bring down the next figure of the divisor, the 6, down next to the 0 to create our third working figure, 06.

We now calculate the U product by multiplying the units figure of the divisor by the third figure we have for the answer.

We subtract the U product from the working figure to get the fourth partial dividend.

   

Now the last partial dividend is zero so we have no remainder, and so we have confirmed that our result is 624.

Practice is the best way to get familiar with this method and there are a number of long division worsheets available to practice with. All worsheets contain answers.

 

 
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