1.04 Long division.

Division is based on the repeated subtraction of one number (a) from another (b). The result of a division gives the number of times a can be subtracted from b and may also include a remainder. The remainder is any amount left after a has been subtracted the maximum number of times from b.

e.g.

4/2 = 2 (with no remainder) .
5/2 = 2 remainder 1 .

Terminology.

The number being divided is called the dividend.
The number the dividend is being divided by is called the divisor.
The number of times the divisor can be subtracted from the dividend is called the quotient.
The amount that is "left over" after the divisor has been subtracted the maximum number of times is called the remainder.

As an alternative to expressing the result as a quotient with a remainder, we can also express the result as a fractional quotient with no remainder.

e.g. 5/2 = 2.5 (i.e. two "fits into" five two and a half times).

Note, when expressing quotients this way (i.e.with the use of decimal fractions) some answers will be approximate. This is because you can't express all fractions exactly in decimal. e.g. 1/3 would be expressed as 0.33. However 3 x 0.33 = 0.99 (but 3 x 1/3 = 1). Therefore 0.33 is only approximately equal to a third.

Procedure for carrying out long division.

The steps involved in long division are quite simple, but as it involves a series of repeated steps, it can appear to be quite complicated. The basic principal is to break the task into stages, with a simple division at each stage that gives a single digit result. Any remainder is carried over into the next stage. The final answer is made up of the digits produced from each individual stage.

You may need to read through the procedure below several times while stepping through example problems to master this method.

First, select enough digits from the left hand side of the dividend to form a number larger than the divisor. (This will require either two or three digits in the examples generated below.) This number is then divided by the divisor.
The result of this division gives the first digit of the answer. This is written above the dividend, above the last of the digits used to form the number that we have just divided. (see first step above.)
Any remainder is written below the number we divided, (with their right hand digits aligned).
More digits from the dividend are then copied down (alongside any remainder) until we form a new number which is larger than the divisor.
This new number is divided by the divisor in the same way as before and the process is repeated until the final result is obtained.
If an exact result will be obtained, then you will achieve this by continuing with the steps described above, until you have used all of the none zero digits in the dividend without the final step producing a remainder. Finally for each remaining trailing 0 that remains in the dividend (i.e. which was never copied down to form a number in any step of the division)upto the decimal point, insert a trailing zero into the quotient.
If you have used up all of the digits in the dividend (including any zeros) and still have a remainder from the last stage, then this simply means that the dividend cannot be divided exactly by the divisor and this final remainder is the a overall remainder from the division. (See below for producing an approximate result with no remainder).

If divisor does not divide exactly into the dividend (i.e. the final stage produces a remainder), then you can produce an approximate result which is either rounded down or rounded up. You can decide how many decimal points to calculate the answer to, before rounding the final value. To do this add zeros after the decimal point to the dividend and carry out further steps of division using these digits, until you have produced the required number of decimal points in the quotient.
In the example below two zeros have been placed after the decimal point in the dividend. They will be used when required to produce an answer rounded down to two decimal points.

Note the calculation of the quotient and the remainder at each step are shown below. This is not normally shown in the working out for a question. However the subtraction to determine the remainder, usually is included within the main calculation as illustrated below. (The amount of working out included with the answer, makes the process look even more complicated!)


	LONG DIVISION

Press step to step through the calculation

Press reset to clear and set new calculation

divided by = remainder

e.g. x =

- =

Note:
The app above always rounds down the answer. In practice you should round up or down depending on the value of the third decimal place.
e.g. 3.252 should be rounded down to 3.25 and 3.256 should be rounded up to 3.26 .

Also the script is not "intelligent" enough to spot when it has found and exact answer early in the procedure and so carries on until every digit in the divisor as been processed.
e.g. 2800/ 14 just requires 1 step for the solution (28/14 = 2 with no remainder. So you would insert the remaining zeros to give a final value of 200).
The app however will continue to check the remaining zeros to "see" if they can be divided!