How do you do long division?

Answer 1

In long division, you physically do the subtractions of each multiple from the remainder at each step, bringing down additional digits one at a time. In some forms of long division, you will stop at the whole number and indicate a remainder (R).

(* see the related link for a graphic illustration of the process)

In short division, you have to get the most appropriate multiple at every stage. In long division, it doesn't matter what multiple you use as you add them all up at the end.

1. Arrange dividend and divisor as you would in short division.
2. Pick a multiple of the divisor - its only restriction is that it has to produce a number less than or equal to the dividend.
3. Write the number down in the far left column.
4. Write the multiple down under the dividend
5. Perform the subtraction.
6. Keep repeating steps 2 to 5 until you have reached a point of solution (the remainder is zero) or 'far enough' to satisfy you.
7. Add up all the numbers in the far right column.

146 divided by 2

• 2)146 |
• 40 |20
• _____|
• 106 |
• 100|50
• _____|
• 6|
• 6|3
• ____
• 73

A:

Let's start with an easier one,

___

8) 36 - 36 is the number your dividing. 8 is the number of pieces you're dividing it into.

___

8) 36 - First how many times the 8 go into 36?

4

___

8) 36- 8 goes into 36 four times to put the 4 on top.

4

* ___

8) 36 - Then 8 times 4 and that is thirty-two. So put that below 36 and subtract.

32

__4_R4

8) 36

- 32

----

4- 4 is a reminder so the answer is 4 remainder 4! This is how you write it 4R4!

---

The term short division is usually applied to the process of representing the remainder as a digit inserted within the number being divided.

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I have come across two versions of long division.

• The Spanish version is "backward" to the English version I was taught. It was shown to me once when I noticed an adult native Spanish speaker at the numeracy class at which I help read divisions "backward" (swapping the dividend and divisor as normally read in English) but getting the answer correct. It suffers in that when demonstrated to me it seemed very difficult to extend from whole numbers into decimals;
• The English version is also known as the "Bus stop" method as it looks a bit like a bus shelter; it is a neater, shorthand way of writing the "chunking method" whereby "chunks" of the divisor (which are multiples of the divisor) are subtracted from the dividend until no more can be subtracted.

The Dividend is the number being divided

The Divisor is the number doing the division.

The Quotient is the result of the division - it is often used to mean just the whole part of the result, but it can include any fractional (decimal) part.

For example, if you had 100 chocolate bars to divide between 50 people, there dividend would be 100 and the divisor 50. It would be written as 100 ÷ 50 and read as "100 divided by 50". When written as a long division (in the English version using the "bus stop method" as I'll describe below) it can be read as "50 divide into 100".

English itself is not a very good language to describe mathematics as it is "sloppy". For example "10 divided into 2" from a mathematical point of view means exactly "2 ÷ 10", but in English it could also mean dividing the 10 into 2 piles, ie the mathematical "10 ÷ 2"

I'll explain the English version (with which I am most familiar), working with decimal numbers:

1. If the divisor contains a decimal point multiply BOTH the divisor and dividend repeatedly by 10 until the divisor no longer has a decimal point;
2. Write the dividend under a bus stop shelter and the divisor to the left outside the shelter;
3. If there is a decimal point in the dividend put a decimal point in the quotient directly above it. Now continue, ignoring the decimal points;
4. Start at the left of the dividend with the first digit which forms the "number under consideration"
5. Find the largest multiplier of the divisor which is less than or equal to the "number under consideration" (known as "how many times the divisor goes into the number under consideration"). Put this multiplier in the quotient over the last digit of the number under consideration.
6. If it is not zero, multiply the divisor by this multiplier, write it under the number under consideration and subtract to form a new "number under consideration". If the multiplier is 0, there is nothing to subtract;
7. Bring the next digit of the dividend down to append it (write it on the end) to the result of the subtraction to form the new "number under consideration"
8. Repeat from step 5 until you run out of digits in the dividend.
9. If the result of the final subtraction is zero the division is complete.
10. Otherwise there is a choice of how to continue:
11. If the dividend was a whole number, it can be seen as a whole number quotient and remainder which can be written as a remainder, or with the remainder over the divisor as a fraction (and simplified)
12. Otherwise, if the dividend was a whole number, the division can be extended to a decimal by putting a decimal point after the last digit of the dividend (and above it in the quotient) and then:
13. If there is a decimal point in the dividend, zeros can be appended (written on the end) to the dividend and the division continued until either a zero subtraction occurs or enough accuracy (as many decimal places as required) has been achieved.
14. If the result of the last subtraction is greater than or equal to half the divisor, the result can be rounded up, otherwise it should be rounded down.

In Short Division the subtraction of step 6 is done mentally and the remainder written in the dividend before the next digit, thus "collapsing" the subtractions which make it a "long" division.

It is easier to show than describe:

(I have to use underscores to try to keep the digits aligned; think of them like lines on a page.)

For example 70819 ÷ 3

____2 3 6 0 6

__--------------

3 | 7 0 8 1 9

____6 _________←2 × 3 = 6, so put 2 in quotient over the 7, write the 6 and subtract

___--- _________

____1 0 _______←Bring down the next digit, the 0 to make 10; highest multiple of 3 not greater than

______9 _______← 10 is 3×3 = 9, so write the 3 over the 0 and put 9 below 10 and subtract

___----- _______

______1 8 _____ ←Bring down next digit, the 8 to make 18; 3 goes into 18 6 times, write 6 over 8 and

______1 8 _____ ← write 3×6 = 18 below the 18 and subtract

______-----_____

________0 1 ___ ←Bring down the next digit, the 1 to make 01 which is the same as 1 (leading zeros

________________←make no difference to the number); 3 does not go into 1, ie the highest multiplier of

________________←3 not greater than 1 is 0, so write 0 above the 1 and bring the next digit down

________0 1 9 __←The 9 would just be appended to the line which already has the 01 above, but as I

________________← needed to explain what is going I have repeated it; 3 goes into 19 6 times, so write

__________1 8 __← 6 above the 9, 18 below and subtract

_______-------___

___________ 1 __←Run out of digits in quotient.

The result can now be considered as 70819 ÷ 3 = 23606 r 1 or 70819 ÷ 3 = 23606⅓

Or the division can be extended into a decimal by continuing on after putting a decimal point in the quotient and dividend and appending a few zeros:

____2 3 6 0 6 ⋅ 3 3 3

__------------------------

3 | 7 0 8 1 9 ⋅ 0 0 0

____6 ________________

___--- ________________

____1 0 ______________

______9 ______________

___----- ______________

______1 8 ____________

______1 8 ____________

______-----____________

________0 1 9 ________

__________1 8 ________

_______-------_________

___________ 1 _ 0 ____←bring down the first appended 0 (I've put an extra space to try to line up the 0)

_______________9 ____←3×3 = 9 as before, and subtract

___________-------_____

_______________1 0____←bring down the next digit etc...

________________ 9____

_______________----____

_________________1 0__

__________________ 9__

________________-----__

___________________1_ ←this is going to repeat for ever, so probably better to stop now.

→ 70819 ÷ 3 = 23606.333... ≈ 23606.333