Chapter Twenty

WHAT YOU LEARNED AT SCHOOL: SOME BASIC ARITHMETIC REVISION

"How do you choose an accountant?" runs the old joke. Ask each one what two times two makes. Then give the job to the one who answers, 'What number did you have in mind?'

They all know two times two makes four, but you want one who will rearrange the figures to suit you.

I am going right back to the beginning too. Why? To make this book useful to as wide a number of women as possible. Of course, we use calculators. But the batteries run down; machines go on the blink; our fingers slip, and we key in an extra figure. Unless you know the basics, you will not be able to check the answers your calculator offers.

If you are quite happy with arithmetic, you will be encouraged to see that this chapter contains all the maths a chartered accountant needs. So you are cleverer than you thought!

If you have never used a calculator, there is nothing to be afraid of. It is easier than setting a microwave oven or the timer on your cooker. Switch on. Press 2. Press x. Press 2. Press =. The answer comes up on the screen. It is that simple.

ADDITION

Add 123 and 4,564.

Put the figures in columns, one exactly below the other. When you were tiny, you used squared paper to help you, remember? (In France, even adults still do.) Start from the right-hand side. Draw a line beneath the second figure. Then, always starting from the right, add down each column, using your fingers if necessary. Put the answer below:

123   4,564   4,687

Easy. the art is to be neat and orderly, and luckily women excel at this.

Now add 123, 4,564 and 89. Arrange your numbers as before:

123   4,564       89

As soon as you add together the first right-hand column, you see that the total is over ten (in fact 16). You put the 6 in the answers line (below the line you have drawn) and a little ' (to represent the remaining 10) in the next answers position, like so:

123   4,564       89       ' 6

Add up the next column, then add on the 1, to give 17. Again, put the 7 in the answers line, covering the 1, and put another tiny ' in the next position;

123   4,564       89      '76

Add up the next column, add on the 1 and put the answer 7 in the answers line. Finally, add up the last column:

123   4,564       89   4,776

Again, easy. If the total of a column had exceeded 20, you would have put a little 2 where you put the ', or a little 3 if it exceeded 30, and so on.

SUBTRACTION

Is it all coming back now? Let's try subtraction.

Subtract 23 from 645. Again, put the two figures in columns, starting from the right hand side, with a minus sign to show what you are doing:

645 -   23

Starting in the right hand column, take 3 from 5 and put the answer in the answers line. Then deal with the next column, then the next:

645 -   23    622

Easy. Now prove you are correct.   Add the answer, 622, to the number you took away, 23. You should get the number you started with. If not, something is amiss.

Now take 187 from 724. Line up the figures as usual:

724 -  187

You cannot take 7 from 4, so you 'borrow' 10 from the next column. Put a little ' by the 4 to mark the borrowing and a little ' by the 8 to remind you to pay it back, like so:

72'4 -  18'7

The little 1 turns the 4 into 14 and 14 minus 7 is 7, so you put that in the answers line. To pay back the 10 you borrowed, add the little 1 to the 8 to make 9. Now you will try to take 9 away from 2. Of course, this is impossible, so you borrow 10  again from the next column, marking the borrowing and repayment in the same way:

7'2'4 -  1'8'7         7

Now, 12 (the original 2 plus the 10 you borrowed) minus 9 (the original 8 plus the 1 you repaid) equals 3, so put this in the answers line. Then to repay the next borrowing, add the 1 to the 1: 7 minus 2 is 5, so put that in the next answers position.

7'2'4 -  1'8'7    5 3 7

Just to prove you are correct, add the answer to the amount you took away, and you should get back to 724.

Not difficult, so long as you remember to pay back your borrowings and keep your little 1s (it will always be a 1  you borrow) tiny.

MULTIPLICATION

Ready to try multiplication?

For multiplication and division, you need to know your tables. There is no substitute for saying them over and over until they come pay. Einstein said learning his tables was the only difficult part of maths. After that it was all downhill.

To save writing out

1 x 2 = 2

2 x 2 = 4, etc

I have put them into a table. You can read them off by going down each column or by going across if you prefer.

Don't panic. Take a deep breath. Just sit back and look at the figures and you can trace lots of patterns. For a start, 1 x any number is the number itself. So the 1 times table is obvious.

I have not bothered to put it in, but 0 x anything is automatically 0. Why? Well, twice as much nothing is still nothing, isn't it?

When two even numbers (2,4,6,8, etc.) multiply, the answer is always even. So if you think 8 x 8 = 63, you must be wrong. It is 64.

When two odd numbers (1,3,5,7, etc.)  multiply, the answer is always odd. So if you think 9 x 7 = 64, you must be wrong. It is 63.

When an odd and an even number multiply, the answer is always even. Like 4 x 3 = 12.

Some tables are very easy. To multiply by 10, just put a 0 on the end. So 6 x 10 = 60.

Five times anything produces a figure which must end in either 0 or 5. So if you think 5 x 7 = 36, you must be wrong. It is 35.

Eleven times anything falls into a regular patter until 100 is reached.

Once you are refreshed on your tables, and can repeat them fast without stopping to think, multiply 136 by 9.

Line up your figures, starting on the right-hand side, with an x to show you are multiplying.

136 x     9

Start multiplying on the right-hand side. 9 x 6 = 54. Put the 4 in the answers line and a little 5 in the next answers position.

136 x     9      54

Now multiply 9 by 3, to get 27. Add on the little 5 you set aside, which gives you 32. Put the 2 in the answers line over the 5 and a little 3 next to it to use later, like so:

136 x     9     324

Now, 1 x 9 = 9. Add on your 3 to get 12. Put the 2 in the answers space over the little 3 and, because you have reached the end, the 1 in the column next to it:

136 x     9 1,224

Not bad, so long as you are strict about your columns. If you find you get muddled, draw vertical lines or use squared paper.

In fact, there is an easier way to do this calculation. Nine is nearly 10, so multiply your number by 10. That is, put 0 on the end to make 1,360. Then take away the original number of 136:

1360 - 126 =  1224. If you get the choice, adding or subtracting is always easier than multiplying or dividing.

Now let's multiply 248 by 39. Remember to lay out your numbers in columns, starting from the right:

248 x   39

Start on the right hand side and multiply by 9. 9 x 8 = 72, so put the 2 in the answers line and a little 7 next to it. 9 x 4 = 36. Add on your 7 to get 43. Put the 3 over the 7 and a little 4 next to it. At this stage, your answer should look like this:

248 x   39    432

Keep going by multiplying 9 x 2 = 18. Add 4 to get 22. Put 2 in the answers column over the little 4 and, because you have finished this part, the other 2 to the left of it. The line should read 2232.

So far, you have multiplied by 9. Now you must multiply by 30. Of course, you don't know your 30 times table. But 30 is only 3 x 10 and you know that to multiply by 10 you add 0.    So, first of all, put 0 in the right-hand column of the next line. Then multiply by 3:   3 x 8 = 24, so put 4 next to the 0 and a little 2 next to the 4, like so:

248 x   39 2,232    240

Carry on multiplying by 3. 3 x 4 = 12. Add on the little 2 to get 14. Put 4 in the answers space to cover the little 2 and a little 1 next to it. 3 x 2 = 6; add the spare 1 to get 7 and put this in the next space, like so:

248 x   39 2,232 7,440

I have put a line under the last row because the final stage is to add the two figures:  2232 is 248 x 9;   7440 is 248 x 30. To get 248 x 39, you add them both together. The answer then is 9,672:

248 x   39 2,232 7,440 9,672

This is not really difficult, just very long-winded. No one will bother to work through the arithmetic when they have a calculator. What they should do is look roughly at the figures, say to themselves, 248 is nearly 250 and 39 is nearly 40. What is 250 x 40? That is easy. Add 0 to the 250 to multiply by 10 and then multiply by 4. 2,500 x 4 = 10,000. So the answer will be roughly 10,000. As we worked out, it is really 9,672. If your calculator produces 967 or 96720 or even 967200, you will know you have made a mistake.

DIVISION

Finally, division: let's divide 8 into 376. Write it down as follows. Leave yourself plenty of space underneath. You will have realized by now that when you know the proper layout you have won half the battle.

_______      8)376

Now, for the first time, you start from the left. Look at the 3 and ask yourself,  does 8 go into 3? Of course not. So look at the first two numbers together. Does 8 go into 37? Yes, 8 x 4 = 32. So put 4 above the 7 and write 32 under the 37, like so.

__4_____      8)376         32

I have put a line under 32 because you are going to subtract it from 37. The answer is 5, and you write this under the 2. If the answer is 8 or more, you have made a mistake.

__4_____      8)376         32          5

So you are left with 5. Take the 6 from above and rewrite it next to the 5 to give you 56. Does 8 go into 56? Yes, 8 x 7 = 56. So write 7 next to the 4 in the answers space. Write 56 under the 56 just to finish the thing off and underline it. You take 56 from 56 and end up with 0, which shows that you have your complete answer, 47:

__47____      8)376         32          56          56            0

Again, in practice, you would say to yourself, 8 is nearly 10. If  I divide 376 by 10, what do I get? It is easy to divide anything by 10, you just ignore the last number. This gives you  37. So, when you are working out the real answer with your calculator, you know that if the answer comes up   479 or 4, you have made a mistake.

One last exercise: suppose you have to divide 3,956 by 43. Nobody knows their 43 times table, so you just have to keep guessing.

_______    43)3,956

Because 43 will not go into 3, you look at the first two numbers; 43 will not go into 39 either, so you look at the first three numbers. It must go into 395, but how many times? You scribble on another piece of paper 43 x 10 = 430. Too big. You work out 43 x 8 = 344. Is that big enough? Add another 43 to 344 and the answer is 387. So 43 x 9 = 387. Great. Write 9 over the 5, and write 387 under the 395 and take one from the other.

____9__    43)3,956         3 87             8

Take the remaining 6 and put it next to the 8 to make 86. Does 43 go into 86? Yes. 43 x 2 = 86. Write 2 next to the 9 and take one from the other. This leaves 0, which shows nothing is left and the answer is complete.

____92__    43)3,956         3 87             86             86               0

By now you are probably thanking heaven for your calculator. Don't worry; this is the most complicated sum we shall do. In practice, you would say to yourself, 3956 is nearly 4,000 and 43 is close to 40. How many times would 40 go into 4000? Easy, about 100. So the answer will be around 100. In fact it was 92. If your machine produces 922 or 9 or 9222, you will know you have slipped up.

If you have kept pace with me so far, I suggest you take a break before we tackle fractions and decimals - not because they are difficult, but to approach them feeling fresh...

Ready for the next stage? So far we have looked at complete numbers. Fractions and decimals are bits of numbers.

FRACTIONS AND DECIMALS

A fraction of something is a part of it. Suppose your house has three floors and you occupy two of them, or two floors out of three. You occupy two-thirds of the house. That is written 2/3 .

The bottom number shows how many parts (or floors) there are altogether. There are three. The top number shows how many you occupy; that is two.

If you live in a block of flats with a flat on each floor and there are twelve floors, you occupy 1/12  or one-twelfth of the building.

We use fractions in our everyday speech. We all know that one-half means 1/2, one-third means 1/3, one-quarter means 1/4  and so on.

Any number put over 1, say 6/1 is simply an ordinary complete number; it is just another way of writing 6. 6 & 4/5 is six complete numbers plus 4 fifths of another. You could also write it 34/5. Each whole number has 5 fifths; 5 x 6 = 30/5. Then there are four left over, so 34 fifths in all.

Virtually nobody in business ever needs those sums you practised at school with lowest common denominators and highest common factors. So I will not bother you with them here. If we want to add, subtract, multiply or divide bits, or complete numbers with bits in like 7 & 1/2; , we change the fractions into decimals. Then we add, subtract etc. their decimal equivalents. This is far easier and it suits our calculators better. If we really need the answer to be in fractions, we just change the decimal answer back to a fraction at the end. How?

To convert a fraction to a decimal, simply divide the bottom number into the top. Of course, it will not go straight away (otherwise it would not be a fraction).

Suppose you want to convert a half, that is 1/2 into a decimal. Remember your division?

___     2)1

Two into one will not go, so you put a 0 in the answer space, followed by a decimal point (a dot). Put another decimal point underneath it, behind the 1. These decimal points show that you have reached the end of the complete numbers and that what comes after is bits. Any number followed by .0 is simply the same complete number; 1.0 is just another way or writing 1. So put 0 after the 1.

Now look at the two numbers 1 and 0 together as though the decimal point had disappeared and they read 10; 2 into 10 goes 5, so write 5 in the answer space. Your answer becomes 0.5:

_ 0.5_     2)1

Decimal bits are measured in tenths. (The Latin word for ten was decem.)* You should not be surprised  that a half turns out to be five-tenths. What is new is to see it written 0.5

If there are two numbers after the decimal point, they are hundredths. You meet this every day. £3.25 is 3 complete pounds and 25 pence. There are 100 pence in a pound, so 25 pence represents 25 hundredths of a pound.

If you want to add or subtract numbers with decimals in them it is easy. You follow the same rules as for complete numbers. So 5.5 plus 7.1 is calculated as follows:

5.5     7.1   12.6

Parts of numbers can borrow and repay just like complete numbers. So 8.3 - 6.8 is worked out as below.

8.13   - 61.8     1 .5

First, 3 borrowed 10 from 6, so that 13 minus 8 gave 5. Then it repaid, so that 8 minus 7 (6 + 1) gave 1.

When you come to multiplication, be careful. Whenever you multiply complete numbers, the answer is always much bigger than the numbers you multiply. When you multiply by a bit (a fraction or a decimal), the answer is always smaller! Why?

There are 7 days in a week, right? So 5 weeks covers 35 days. Agreed? Put another way 7 x 5 = 35.

In half a week there are only 3 1/2  days. Half is 5/10 in fractions or 0.5 in decimals. Put another way, 7 x 0.5 = 3.5

Written out in full the sum is:

7.0 x   0.5      3.5 days

When you multiply by a decimal, you put the first number of your answer to the right hand side of the decimal point, to show it is a decimal you are dealing with.

If the decimal you are multiplying by is even smaller (say, 0.05, which is 5 hundredths, instead of 0.5, which is 5 tenths), you put the first number two places to the right-hand side, not one. Five hundredths is the same thing as one twentieth. One twentieth of a week is far smaller than one-half:

7.0 x  0.05     0.35 or roughly an afternoon

If you want to convert a decimal back to a fraction, it is easy. The first figure is tenths, so 0.6 is 6/10. The first two figures are hundredths, so 0.54 becomes 54/100. The first three are thousandths, and so on.

There is no need to know more about decimals than this. Use arithmetic basically to check your calculator. If you have to multiply 245.789 by 97.753, just ignore the decimals. Say to yourself, 245 is nearly 250 and 97 is nearly 100. What is 250 x 100? Easy; it is 25,000. This is roughly what you expect the answer to be.

PERCENTAGES

Percentages are really easy. Per cent means 'out of a hundred'. So, 1 per cent (often written 1%) means one out of a hundred, the same as 1/100 or 0.01 or one-hundredth.

So if you want to write 65 per cent as a fraction, simply put 100 under it: 65/100.

If you want to write it as decimals, it is 0.65.

Suppose your employers have agreed to consider a creche. They tell you that 75 per cent of your colleagues have one pre-school child, and they wonder how many children to cater for. You know that 3,600 women work in your factory. Multiply 3,600 by 75 per cent. You could do it by 3600 x 0.75, but most calculators have a % (percentage) button that you can press instead. Either way, the answer will be 2,700 children, and your employer may think again!

Perhaps you are considering borrowing £375 at 8 per cent interest APR and want to know how much extra the interest will cost you a year. Multiply 375 by 8 per cent, or, if you don't use the % button, multiply 375 by 0.08. Either way, the answer is £30.

Maybe you are worried after learning that 100 planes were hijacked last year. Perhaps you even talk of cancelling your holiday. Then you discover that there were 15 million flights last year worldwide. So what percentage was hijacked? 100 divided by 15,000,000 gives you 0.0000066 per cent. This percentage is so tiny that you should be reassured. If the newspaper had quoted it this way in the first place, you would never have panicked.

Perhaps a friend offers to lend you £1,000 provided you pay them back £1,100 in three month's time. You think £100 sounds very reasonable. What is it as a percentage?

100/1000 = 1/10; or 10 per cent, which seems fair. Then you remember that you are to repay after three months, or one-quarter of a year; 10 per cent x 4 = 40 per cent annual interest (in fact, a bit more even than that) - far too high to accept.

AVERAGES

Averages are easy to calculate too. Jane tells you she went off with her caravan for the weekend three times last year. Mary tells you she went four times. You went four times, and three other friends tell you they went twice, not at all and five times. You want to know what the average is for all of you. Add together the number of times:

3 + 4 + 4 + 2 + 0 + 5  = 18

Now divide by the number of people:  18 divided by 6 is 3, so the answer is 3 times. Now you know the average and also that you are above average.

Unfortunately, averages don't necessarily mean very much. You would have had the same result if two of you had gone nine times and the rest not at all, or if one of you had gone all eighteen times and the rest of you never. So take all those averages bandied about with a pinch of salt. They sell newspapers, delight television producers and worry people.

There you are. This is all the maths you need to be an accountant or to handle your own money matters. Not too terrible, was it?

*There used to be only ten months in the year, so the last one was called December.

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