## Some Basic Rules

The $$\div$$ symbol for division tends to be disregarded as we progress to higher levels of mathematics, in favor of a horizontal line separating the two numbers, for example

$\frac{2}{9} \;\;\; \mbox{represents} \;\;\; 2 \div 9$

So, in effect, a division and a fraction are the same thing

When a number is multiplied by zero, the answer is always zero

$0 \times 432 = 0$ $4 \times 0 = 0$ $560\times 0 = 0$

The first expression is fairly self-explanatory - if you have nothing and you multiply it 432 times, you will still have nothing.

The other two expressions follow the same logic, using the idea stated later on, that multiplication is commutative, i.e.

$4 \times 0 = 0 \times 4$

and so on

It is impossible to divide by zero - try it on your calculator and you will get an error. (strictly speaking, I should mention in passing that, at a more advanced higher level, the answer to a division by zero is usually considered to be infinity)

Zero divided by any number equals zero. Applying similar logic as we applied above, if we start off with nothing of something and divide it out among 200 people, everyone gets nothing. We can divide it out among any number of people, and they all still get nothing

$\frac{0}{4} = 0$ $\frac{0}{34} = 0$ $\frac{0}{235} = 0$

## The Order of Operations - Introduction

The order in which you carry out addition or multiplication does not matter, in mathematical terms these operations are called commutative. For example :

$3 \times 6 = 6 \times 3$ $4 + 2 = 2 + 4$ If the expression is more complicated :-

$\mbox{What is } 4 \times 7 \times 8 \times 2 ?$ you don't have to carry out the operation from left to right, you could also go from right to left

$4 \times 7 \times 16$ $= 4 \times 112$ $= 448$ or else proceed as

$4 \times 56 \times 2$ $= 4 \times 112$ $= 448$ Do it in whichever order is most convenient for you - the essence is that whichever way you do it, the answer will be the same. And exactly the same statements hold true for a complicated addition expression. For example

$\mbox{What is }\;\; 4 + 9 + 4 + 3?$ You can go from left to right, or you could also go from right to left

$4 + 9 + 7$ $= 4 + 16$ $= 20$ or else proceed as

$4 + 13 + 3$ $= 4 + 16$ $= 20$ If we include subtraction, the same rules still apply

$4 - 10 + 4 - 5 = ?$ As before you could carry out the operation from left to right, but will still work if you go from right to left

$4 - 10 -1$ $= 4 -11$ $= -7$ or else proceed as

$4 -6 -5$ $= -7$ As an extra indication of how you can carry out the calculation in any order, you could carry out the additions separately from the subtractions, and then sum the two outcomes

$4 + 4 = 8$ $-10 - 5 = - 15$

and finally

$8 - 15 = - 7$ Note, however, that the operations of subtraction and division are not commutative. You should be able to convince yourself of this fairly easily

$8 - 2 \;\;\mbox{is not the same as } 2 - 8$

$4 \;\mbox{divided by} \; 2 \;\;\mbox{is not the same as }\; 2 \; \mbox{divided by } 4$

More about the order in which you tackle a maths problem.

 Quick Quiz     Carry out the following calculations   a)   1 - 6 + 7;   b)  3 + 8 - 12;   c)   4 - 2 - 7;   d)  - 3 + 6 + 4;   e)  7 + 5 + 4 - 11;   f)   - 5 - 6 - 3 + 4;  g)  - 5 + 21 - 45 + 6;  h)  5 + 65 + 12 - 34;  i)   - 34 + 123 - 21 + 56;

## The Order of Operations - Mixture of Operators

When you have a mixture of binary operators, the following rule applies, above all

Multiplication / Division operations are carried out before Addition / Subtraction operations.

So mathematical calculations cannot always be carried out from left to right. In practice, brackets will often be used to make the order of calculation clear (see below).

But in a calculation with no brackets like

$2 + 5\times 6 = ?$

The first step must be the multiplication, giving

$2 + 30 = ?$

$32$

 Quick Quiz     Carry out the following calculations   a)   1 1 × 6 + 7;   b)  16 ÷ 8 - 12;   c)   4 ÷ 2 - 7;   d)  - 3 + 6 × 4;   e)  7 + 5 × 11;   f)   5 × 6 - 3 ;  g)  63 - 45 ÷ 9 ;  h)  55 ÷ 5 + 6;  i)   - 34 - 42 ÷ 7 ;

## Brackets

Brackets are an important means of showing the order in which a calculation can be carried out. In a nutshell -

all calculations within brackets are done first.

Examples

$\frac{24}{( 5 + 3 )} = \frac{24}{8} = 3$

$( 6 + 2 ) \times 3 = ( 8 ) \times 3 = 24$

$\left( \frac{25}{5} \right) + 3 = (5) + 3 = 8$

$( 5 - 3 )\times ( 13 + 5 ) = ( 2 ) \times ( 18 ) = 36$

$( 3\times 6 ) - ( 5 + 4 ) = ( 18 ) - ( 9 ) = 9$

$\frac{( 9 - 3 )}{( 5 - 2 )} = \frac{6}{3} = 2$

I'll have to make a comment on the last example, at the risk of introducing confusion - the same problem could be expressed equivalently as $\frac{ 9 - 3}{ 5 - 2 }$ If you were to think in terms of trying to do the division first (in line with the rule I stated in the last section) then you would have problems. Essentially, there are implied brackets here - the implication is so strong that they can be omitted without ambiguity

Note

• Multiplication signs are often omitted between brackets

$( 3 \times 9 ) × ( 3 + 6 ) = ( 3 \times 9 ) ( 3 + 6 )$

or between a coefficient in front of a bracket and the bracket itself

$2 \times ( 3 + 9 ) = 2 ( 3 + 9 )$

• Brackets have to be used to remove any ambiguity in operations involving a mixture of multiplications and divisions. For example

$3\times 4\div 9 \times 5$

would obviously produce a different answer if you did the multiplication first, to the answer you would get if you did the division first.

Strictly speaking, there is a rule that the division must be done first (the BODMAS rule for those who have heard of it) but, in practice, if you don't use brackets in a situation like this then you are asking for trouble.

Quick Quiz

Carry out the following calculations

 1.   (6 - 1) × 7;   2.   (3 × 8) ÷ 12;   3.   4 - (7 - 2);   4.   (3 + 6) × (4 + 2);   5.   (7 + 53) ÷ (4 × 5);   6.   (7 - 6) × (4 × 2);  7.   (5 + 21) - (45 ÷ 5);  8.   (65 ÷ 5) - (14 - 12);  9.   (24 ÷ 2) ÷ (8 ÷ 4);

## Addition / Subtraction of Decimal Numbers

This is an extension of adding and subtracting integers. Just make sure that the decimal point is 'lined-up' properly.

Example : $$3940.61 + 21.49$$ $\begin{array}{ccccccc} 3 & 9 & 4 & 0 &. & 6 & 1\\ && 2 &1 &. & 4 & 9 \\ \hline 3 &9& 6& 2 &. & 1 & 0\end{array}$

This would be stated as 3962.1

Example : $$274.34 + 69814.9$$ $\begin{array}{cccccccc} & & & 2 & 7 & 4 &. & 3 & 4 \\ &6& 9 &8& 1 &4 &. & 9 & 0 \\ \hline & 7&0 &0& 7& 9 &. & 3 & 4\end{array}$

Example : $$3427.891 - 469.32$$ $\begin{array}{cccccccc} & 3 & 4 & 2 & 7 &. & 8 & 9 &1 \\ & &4& 6 &9 &. & 3 & 2 &0 \\ \hline &2 &9& 5& 8&. & 5 & 7 &1\end{array}$

## Multiplication by Ten and Multiples of Ten

Stated simply, multiplication by 10 or 100 or 1000 (or any multiple of ten) involves purely shifting the decimal point to the right - 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 'multiplier'. For example,

$\mbox{multiplication by}\; 10 \rightarrow \mbox{shifts the decimal point by 1 place} \; e.g. \; 23.42\times 10 = 234.2$ $\mbox{multiplication by}\; 100 \rightarrow \mbox{shifts the decimal point by 2 places} \; e.g. \; 434.2312\times 100 = 43423.12$ $\mbox{multiplication by}\; 1 000 000 \rightarrow \mbox{shifts the decimal point by 6 places} \; e.g. \; 47.42765434\times 1 000 000 = 47427654.34$

You need to realise that a number like

$$2.34$$

can be written as

$$2.340000000$$

and actually the number of 'trailing' zeroes can be any number you like.

With this information you can see that

$2.4\times 100 = 240$ $34.23\times 10 000 = 342300$ $2.71\times 1000 000 = 2710000$

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

$$7$$

can be written as

$$7.0000000$$

So therefore

$2\times 100 = 200\;\;\; ( 2.00\times 100 )$ $36\times 10\; 000 = 360\; 000 \;\;\; ( 36.0000\times 10\; 000 )$ $71\times 1\; 000\; 000 = 71\; 000\; 000 \;\;\; ( 71.000000\times 1\; 000\; 000 )$

## 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,

$\mbox{division by}\; 10 \; \mbox{ shifts the decimal point by}\; 1\;\mbox{place}$
e.g.
$\frac{23.42}{10} = 2.342$

$\mbox{division by} \; 100\; \mbox{shifts the decimal point by} \; 2 \; \mbox{places}$

e.g.
$\frac{434.2312}{100} = 4.342312$

$\mbox{division by} \; 1\; 000\; 000 \; \mbox{shifts the decimal point by} \; 6\; \mbox{places}$

e.g.
$\frac{47427654.34}{1\; 000\; 000} = 47.42765434$

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

$\frac{2.4}{100} = 0.0240$ $\frac{34.23}{10\; 000} = 0.00342300$ $\frac{2.71}{1\; 000\; 000} = 0.000002710000$

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

$\frac{2}{100} = 0.02$ $\frac{36}{10\; 000} = 0.0036$ $\frac{71}{1\; 000\; 000} = 0.000071$

## Long Multiplication of Integers

Example : $$76 \times 8$$ $\begin{array}{ccc} & 7 & 6 \\ & & 8 \\ \hline 6 & 0 & 8\end{array}$

Example : $$462\times 34$$ $\begin{array}{ccccc} & &4 & 6 & 2 \\ & & & 3 & 4 \\ \hline & 1& 8 & 4 & 8\\ 1 & 3 & 8 & 6 & 0\\ \hline 1 & 5 & 7 & 0 & 8\end{array}$

The $$1848$$ is obtained by multiplying $$462$$ by $$4$$,

...while the $$13860$$ is obtained by multiplying $$462$$ by $$3$$ after first inserting a $$0$$ to the right.

The results of the two multiplications are then added together.

If you were to multiply by a 3-digit number then you would have three separate multiplications, one for each digit. The process for the first two digits would proceed as above, but the process for the third digit would involve first inserting two zeroes to the right.

Example : $$394\times 121$$ $\begin{array}{ccccc} & &3 & 9 & 4 \\ & & 1 & 2 & 1 \\ \hline & & 3 & 9 & 4\\ & 7 & 8 & 8 & 0\\ 1 & 2 & 1 & 0 &0\\ \hline 2 & 0 & 3 & 7 & 4\end{array}$

This method generalizes in the expected way if you were ever to multiply by a number with more than three digits.

## Long Multiplication - Decimal Numbers

Method

• Multiply as though you were multiplying two integers (disregarding the decimal point)

Example : $$329.625 \times 2.94$$

$\begin{array}{cccccccc} & & 3 & 2 & 9 &6 & 2 & 5 \\ & & & & & 2 & 9 & 4 \\ \hline &1&3 &1 & 8 & 5 & 0 & 0\\ & 2&9 &6 & 6 & 2 & 5 & 0\\ 6 & 5 &9 & 2 & 5 & 0 & 0 &0\\ \hline 9 &6 &9 & 0 & 9 & 7 & 5 & 0\end{array}$

• Sum the decimal places in the original numbers (i.e. the numbers to the right of the decimal point),

... so here we have a total of five decimal places (the $$625$$ from the first number and the $$94$$ from the second number).

• Correspondingly, shift the decimal place in the answer above by five places to the left, so

$$96909750$$

becomes

$$969.09750$$

(Note that the number $$96909750$$ has an implied decimal point at the end, which is obviously normally omitted. In other words it can be considered as $$96909750.0$$ )

Example : $$6.94 \times 29.3$$

$\begin{array}{cccccc} & & & 6 & 9 &4 \\ & & & 2 & 9 &3 \\ \hline & & 2 & 0 & 8 &2 \\ &6 & 2 & 4 & 6 &0 \\ 1 & 3 & 8 & 8 & 0 &0 \\ \hline 2 & 0 & 3 & 3 & 4 &2 \\ \end{array}$

This becomes 203.342

Example : \(0.29 \times 1.4\$

$\begin{array}{ccc} & 2 & 9 \\ & 1 & 4 \\ \hline 1 & 1 & 6\\ 2 & 9 & 0 \\ \hline 4 & 0 & 6 \\ \end{array}$

This becomes 0.406

## Long Division of Integers

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.

 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.

 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

If we were to state the answer to one place of decimals, we would give the answer as 13.6 - see Rounding if necessary, to show how we arrived at this answer from 13.58 to two places of decimals

I complete this section by looking an example where the dividend (the number to be divided) is not an integer. The procedure just follows on logically from the previous examples

 3.27      ______ 12 | 39.32

1. 12 divided into 3 = 0 with a remainder of 3. Therefore a 3 is placed in front of the 9 to make 39
2. 12 divided into 39 = 3 with a remainder of 3. The 'first' 3 goes above the 9, and the 'second' 3 is placed in front of the 3 to the right to make 33.
3. There is a decimal point between the 9 and the 3 to the right of it, and another decimal point must also be inserted directly above this decimal point, in the answer line.
4. 12 divided into 33 = 2 with a remainder of 9. The 2 goes above the 3 while the 9 goes in front of the 2 to make 92.
5. 12 divided into 92 = 7 with a remainder of 8. The 7 goes above the 2 and the 8 goes in front of the (assumed) 0 to the right of the 2 in the dividend to make 80.
6. This procedure can be either continued or cut short here if you consider you have sufficient decimal places in the answer already

## 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

$\frac{\mbox{dividend}}{\mbox{divisor}}$

I have already mentioned above the procedure when the dividend is not an integer, but the divisor is.   I extend the description here to when the divisor is not an integer.

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). Then just proceed as described in the last example in the section above.

Note 1 : There is no need to make the dividend into an integer, although you will still get the correct answer if you choose to do this (the divisor will be correspondingly larger if you were to choose to do this).

Note 2 : You do not adjust the answers after the calculation - as I have just said, avoid confusing these procedures with those for long multiplication

## 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