# Long division and why it works

This is a complete lesson with instruction and exercises for fourth or fifth grade, explaining why long division works. We compare long division to the repeated subtraction process. The steps in these two procedures are actually the same, just written out in a different format!

For clarity's sake, we will initially write out the subtracted numbers with all their zeros. Also, for clarity and for easy comparison, we will write the parts of the quotient above each other.

As an example, let's study  789 ÷ 3.  You can think of it as 789 apples that you are bagging in bags of 3 apples, wanting to know how many bags you need.

First write the dividend 'inside' the corner, and the divisor outside:
 3 7 8 9
Then let's divide!

Continued subtraction

 Dividend (the apples) Quotient (the bags) 789 -  600 200 189 - 180 60 9 3 -  9 0 263
 2 0 0 3 7 8 9 - 6 0 0 1 8 9

 6 0 2 0 0 3 7 8 9 - 6 0 0 1 8 9 - 1 8 0 9
 3 6 0 2 0 0 3 7 8 9 - 6 0 0 1 8 9 - 1 8 0 9 - 9 0
The frst step is the hundreds, finding out which multiple of 300 will fit into 789.  It is 600.  In terms of apples and bags, one has now used 200 bags to bag 600 apples. Second step, the tens.  Which multiple of 30 will fit into 189?  That is 180, meaning one uses 60 bags to bag 180 apples.  So 60 is added to the quotient. Lastly look at the ones. There are still 9 apples left, which means one needs 3 bags more.  Add 3 to the quotient.

The last step is to check the division by multiplication:
whether 3 × 263 is 789.
 263 ×  3 789

## Why it works

Comparing the division to the continued subtraction probably has already let you see why it works.  In the conventional way of writing the long division, it is not so easy to see the process.  The key is that in each step, one does NOT actually divide by the actual divisor but by a multiple of it.  Just like in the apples/bags examples, you don't start out by subtracting 3 apples each time, but first 'hit it hard' by subtracting multiples of 300 apples if possible, then multiples of 30, then 3.  In essence, you first divide by 300, then by 30, then by 3.

Also, in the conventional long division, you only place one digit into the quotient in each step, not with all the zeros.  The digits shown in gray are not usually written out in the conventional long division algorithm.

Hundreds
"How many 3's in 7?"
(How many 300's in 789?)
Tens
"How many 3's in 18?"
(How many 30's in 189?)
Ones
"How many 3's in 9?"
 2 0 0 3 7 8 9 - 6 0 0 1 8 9

 2 6 0 3 7 8 9 - 6 0 0 1 8 9 - 1 8 0 0 9

 2 6 3 3 7 8 9 - 6 0 0 1 8 9 - 1 8 0 0 9 - 9 0

To get the hundreds digit in the quotient, one asks the question:  "How many times does 300 go into 789", or the division 789 ÷ 300!  You are not dividing by 3 because you try to 'hit it hard' and subtract as many multiples of 300 as possible.  Since 300 is a whole hundred, the tens and ones digits in the 789 won't matter when you are finding how many times 300 goes into 789.  So the thing can be done easier by calculating  7 ÷ 3, or thinking "How many times does 3 go into 7".
The remainder from first step (what is left after subtraction) is in reality 189.  But since the ones digit (9) won't be important in the next step (which deals with the tens digit), in the traditional way, you only subtract 7-6 and then you 'drop' down the tens digit 8 from the dividend.

To get the tens digit, similarly one asks the question: "How many times does 30 go into 189", or does the division  189 ÷ 30.  Again, since you're dividing by a multiple of ten, the ones digit '9' in the 189 does not affect the division at all.  The important thing is to look at the whole tens in the number 189, which is 180. So to find the answer to the division 189 ÷ 30, you can think of the division 180 ÷ 30, which is the same as thinking 18 ÷ 3:  "How many times does 3 go into 18?"

The last step is simple since it is dealing with ones digits, how many times does 3 go into 9.

## Examples of long division

These examples show how long division is done, including the dropping down of digits and such.  It is important to keep the rows and columns lined up.

850 ÷ 2 = ?

 4 2 8 5 0 - 8 0
 4 2 2 8 5 0 - 8 0 5 - 4
 4 2 2 8 5 0 - 8 0 5 - 4 1 0

Drop down the 0 of the 850 next to the 1. Then divide 2 into 10.

In the hundreds digits, divide 2 into 8. Ask, 'How many 2's in 8?" That is EXACTLY 4 times. Multiply 4 × 2 = 8 and subtract that from 8 to find the remainder which is of course 0. Then drop down the tens digit 5 and divide 2 into 5.  2 goes into 5 two times but the division is not exact.  So multiply 2 × 2 = 4, place 4 underneath the 5 and subtract to find the remainder.
Then multiply 5 × 2 = 10 and place the result under the 10 and subtract.  Since the result is zero and there are no more digits to drop from the dividend, the division is over.
 4 2 5 2 8 5 0 - 8 0 5 - 4 1 0 - 1 0 0
 1    4 2 5 ×  2 8 5 0

Check the division
by multiplication.

Study also the following examples with your teacher.

Thousands digit Hundreds digit Tens digit Ones digit
How many 7's in 1? How many 7's in 15? How many 7's in 11? How many 7's in 42?
 (0) 7 1 5 1 2

 2 7 1 5 1 2 - 1 4 1

 2 1 7 1 5 1 2 - 1 4 1 1 - 7 4

 2 1 6 7 1 5 1 2 - 1 4 1 1 - 7 4 2 - 4 2 0

1.  Divide using long division.  Check by multiplication.

a.
 5 8 6 0

×         5

c.
 2 3 7 8

×         2

e.
 4 6 3 2

×         4

g.
 6 7 5 0

×         6