Basic Arithmetic

Some Basic Rules

2.4×100 = 240
34.23×10 000 = 342300
2.71×1000 000 = 2710000

Additionally, you need to realize that a number written without a decimal point can actually be written with a decimal point. For example,

7

can be written as

7.0000000

So therefore

2×100 = 200        ( 2.00×100 )
36×10 000 = 360 000       ( 36.0000×10 000 )
71×1 000 000 = 71 000 000       ( 71.000000×1 000 000 )

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Division by Ten and Multiples of Ten

Stated simply, division by 10 or 100 or 1000 (or any multiple of ten) involves purely shifting the decimal point to the left - the figures themselves do not change.

The number of places that the decimal point is shifted is directly related to the number of zeroes in the 'divisor'. For example,

division by 10 shifts the decimal point by 1 place e.g. 23.42 / 10 = 2.342
division by 100 shifts the decimal point by 2 places e.g. 434.2312 / 100 = 4.342312
division by 1 000 000 shifts the decimal point by 6 places e.g. 47427654.34 / 1 000 000 = 47.42765434

You need to realise that a number like

2.34

could be written as

0000002.34

for the purposes of carrying out a division (the number of 'leading' zeroes can be any number you like).

With this information you can see that

2.4 / 100 = 0.0240
34.23 / 10 000 = 0.00342300
2.71 / 1 000 000 = 0.000002710000

Additionally, you need to realize that a number written without a decimal point can actually be written with a decimal point. For example,

7

can be written as

7.0

So therefore (along with the information in the previous section), you can see that

2 / 100 = 0.02
36 / 10 000 = 0.0036
71 / 1 000 000 = 0.000071

Additional Note
Division by ten is equal to multiplication by 0.1
Division by 100 is equal to multiplication by 0.01

etc.

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Long Multiplication of Integers

References

Example : 76×8

76
8
------
608

Example : 462×34

462
34
------
1848
13860
------
15708

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Long Multiplication - Decimal Numbers

Method

  1. Multiply as though you were multiplying two integers (disregarding the decimal point)
  2. Sum the decimal places in the original numbers (i.e. the numbers to the right of the decimal point), e.g.

    2.94×329.625

    has a total of 5 decimal places (the 94 from the first number and the 625 from the second number)

  3. Shift the decimal place in the product from Stage 1 by the number of places found from stage 2, e.g. if the product from Stage 1 was

    2957867

    the final answer would be

    29.57867

    (Note that the number 2957867 from above has an implied decimal point at the end, which is obviously normally omitted)

Example : 6.94×29.3

694
293
------
2082
62460
138800
------
203342

This becomes 203.342

Example : 0.29×1.4

29
14
---
116
290
---
406

This becomes 0.406

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Long Division of Integers

References

Proceed by example

     126
   ______
6 | 756

  1. 6 divided into 7 = 1 with a remainder of 1. Therefore 1 goes above the 7, and a 1 is placed in front of the 5 to make 15
  2. 6 divided into 15 = 2 with a remainder of 3. The 2 goes above the 5, and the 3 is placed in front of the 6 to make 36
  3. 6 divided into 36 = 6 with a remainder of 0. The 6 goes above the 6. The remainder of 0 means that the calculation is finished, producing an exact number of 126 for the division.

The answer is 126

       9.5
   ______
8 | 76

  1. 8 divided into 7 = 0 with a remainder of 8. Therefore 0 goes above the 7 (usually omitted), and a 7 is placed in front of the 6 to make 76
  2. 8 divided into 76 = 9 with a remainder of 4. The 9 goes above the 6, and the 4 is placed in front of the 0 to the right (which must be now be written in) to make 40. There is a decimal point between the 6 and this 0, and another decimal point must also be inserted directly above this decimal point, in the answer line.
  3. 8 divided into 40 = 5 with a remainder of 0. The 5 goes above the 0 (not shown here). The remainder of 0 means that the calculation is finished, producing an exact number of 9.5 for the division.

The answer is 9.5

         13.58
     ______
34 | 462

  1. 34 divided into 4 = 0 with a remainder of 4. Therefore 0 goes above the 4 (usually omitted), and a 4 is placed in front of the 6 to make 46
  2. 34 divided into 46 = 1 with a remainder of 12. The 1 goes above the 6, and the 12 is placed in front of the 2 to make 122
  3. 34 divided into 122 = 3 with a remainder of 20. The 3 goes above the 2, and the 20 is placed in front of the 0 to the right (which must be now be written in) to make 20. There is a decimal point between the 2 and this 0, and another decimal point must also be inserted directly above this decimal point, in the answer line
  4. 34 divided into 200 = 5 with a remainder of 30. The 5 goes above the 0 (not shown here). and the 30 goes in front of the next zero to the right (which must now be written in) to make 300
  5. 34 divided into 300 = 8 with a remainder of 28. The 8 goes above the 0 (not shown here). And the remainder of 28 goes in front of the next zero to the right to make 280.
  6. This procedure can be repeated until you get an exact answer.

    But here it is beginning to look as though the calculations could go on forever, so usually some decision is made as to where to stop. Here we have gone as far as 13.58

The answer based on our calculations is 13.6 (to one place of decimals) - see Rounding if necessary, to show why we just stated the answer to one place of decimals in this instance

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Long Division of Decimal Numbers

Do not confuse the procedures for long division with those for long multiplication

The division will be in the form

dividend / divisor

Procedure

  1. Make the divisor into an integer by multiplying both dividend and divisor by the same multiple of ten (or, equivalently, shift the decimal point by the same number of places). There is no need to make the dividend into an integer.

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Past Exam Questions

A business has a bank balance of £300 in January. The balance sheet for the first 8 months of the year is shown below, but some of the amounts are missing.


1.    What were the sales in June?

  • A    £327
  • B    £378
  • C    £397
  • D    £401

2.    What were the expenses in April?

  • A    £353
  • B    £378
  • C    £397
  • D    £401

3.    What was the profit or loss in March?

  • A    -£99
  • B    -£92
  • C    £99
  • D    £156

4.    What was the bank balance in July?

  • A    -£99
  • B    -£92
  • C    £99
  • D    £156

1. Caroline wants to know the weight of a chicken so that she can work out how long to cook it. First she puts a dish on the scales and records the weight of the dish. Then she weighs the dish with the chicken in it.

How heavy is the chicken?

  • A     2.2kg
  • B     2.3kg
  • C     2.4kg
  • D     2.7kg

A store cafeteria is providing packed lunches for 54 employees. This will include 2 filled rolls each. The requests are for 50 ham, 20 cheese and 38 tuna rolls.

All the rolls wil be spread with butter and include lettuce and tomato as well as the requested filling.

The table below shows how many rolls a given ingredient will fill.

Copy the table and complete the end column to show the ingredients needed to meet the requests. Give your results to appropriate levels of accuracy.

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