Long Division to Decimal Places

Long Division to Decimal Places

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 c 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 (ie 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 as 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 be perceived as 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 forward to 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 make 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 and it is written above the last digit of those selected from the dividend (see first step above)
  • Any remainder is written below the digits selected from the dividend (aligning with the right hand digits)
  • The next digit/s from the dividend are then copied down (alongside any remainder) to form a new number (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
  • For an exact result. Continue until you have used all of the none zero digits in the dividend and there is no remainder from the last stage you performed. (Place zeros in any spaces between the digits of the result up to the decimal point)
  • 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 the result has produced a “”overall”” remainder. See below for producing an approximate result

If divisor does not divide exactly into the dividend (i.e. the final stage produces a remainder) 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 to the dividend after the decimal point and carry out further stages using these digits until you have produced the required number of decimal points in the result.
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.

When we are given a long division to do it doesn’t always work out to a whole number. Sometimes there are numbers left over. We can use the long division process to work out the answer to a number of decimal places.

The secret to working out a long division to decimal places is the ability to add zeros after the decimal point.

Example: 150 is the same as 150.00

We can add as many zeros as we wish after the decimal point without altering the numbers value.

We will use the example below. It works out neatly to one decimal place

435 ÷ 25

If you feel happy with the process on the long division page you can skip the first bit.

 

4 ÷ 25 = 0 remainder 4 The first number of the dividend is divided by the divisor.
The whole number result is placed at the top. Any remainders are ignored at this point.
25 × 0 = 0 The answer from the first operation is multiplied by the divisor. The result is placed under the number divided into.
4 – 0 = 4 Now we take away the bottom number from the top number.
Bring down the next number of the dividend.
43 ÷ 25 = 1 remainder 18 Divide this number by the divisor.
The whole number result is placed at the top. Any remainders are ignored at this point.
25 × 1 = 25 The answer from the above operation is multiplied by the divisor. The result is placed under the last number divided into.
43 – 25 = 18 Now we take away the bottom number from the top number.
Bring down the next number of the dividend.
185 ÷ 25 = 7 remainder 10 Divide this number by the divisor.
The whole number result is placed at the top. Any remainders are ignored at this point.
25 × 7 = 175 The answer from the above operation is multiplied by the divisor. The result is placed under the number divided into.
185 – 175 = 10 Now we take away the bottom number from the top number.
Now we have reached the end of the whole numbers we add a decimal place and the first zero. Notice the decimal point which has appeared on the answer line and by the dividend. It does not appear anywhere else.
Bring down the next number of the dividend.
100 ÷ 25 = 4 remainder 0 Divide this number by the divisor.
The whole number result is placed at the top. Any remainders are ignored at this point.
25 × 4 = 100 The answer from the above operation is multiplied by the divisor. The result is placed under the number divided into.
100 – 100 = 0 Now we take away the bottom number from the top number.
The subtraction has given zero. We stop when this happens. The answer is 17.4

As long as the subtraction gives a number above zero the long division can carry on to as many decimal places as we wish.

 

Answer: 435 ÷ 25 = 17.4