Numbers index

 %  Fractions, decimals and percentages  % 


Arabic Numbers --- Introduction --- Add --- Subtract --- Multiply --- Divide --- Practise sums --- Less than 1 --- Types of numbers

Fractions

So far, we have been describing whole numbers. But once you start dividing numbers, you can get part numbers or fractions. Some of these are between zero and one, and they are called proper fractions. Some are more than one, but are between one whole number and the next. These are called improper fractions.

Choose a fraction below to see what it looks like. It shows the fractions of a circle. Try keeping the bottom the same and changing the top. Then keep the top the same and change the bottom.

Choose a fraction:   /  


Compare fractions

The number on the bottom of a fraction is called the denominator. It shows how many pieces a whole is divided into. The number on top of the fraction is called the numerator as it shows how these many pieces there are.

Here is an exercise in comparing fractions. Choose a level of difficulty. If you can't do it, see what the fractions look like by using the exercise above.

    








Reducing fractions

It's sometimes easier to imagine fractions as dividing a chocolate cake between several people. If you divide the cake between two people, each gets half. If you cut a cake into four equal pieces, then they are quarters, or 1/4. If a greedy person eats two of those quarters, then they've eaten half the cake. So 2/4 is the same as 1/2. When we write fractions, it's best to simplify them, writing 1/2 rather than 2/4. We know that 2/4 is not the simplest fraction we can have, since both top (numerator) and (denominator) are both divisible by 2. If both the numerator and denominator of a fraction have the same factor, then divide it into both. This is called reducing the fraction.

Find the factor of the fraction below. If you find it hard, use multiplication tables or read about factors.

    Select factor to reduce fraction:           

 

The factor that you remove when reducing factors is called the Highest Common Factor.


Arithmetic with fractions

It's quite hard to do arithmetic with fractions. I am not going to explain this in detail, but here is the algebraic description of it for anyone who understands algebra. If you don't, then don't worry - just skip it. (I will just mention that the letters stand for numbers, and that ad is short-hand for a x d.) These methods always work, even though it may not be the easiest way.

Assume that you have two fractions: a/b and c/d: Example: the two fractions are 2/3 and 1/4

a/b + c/d = (ad + bc) / bd

2/3 + 1/4 = (2x4 + 1x3) / 3x4 = (8 + 3) / 12 = 11/12

a/b - c/d = (ad - bc) / bd

2/3 - 1/4 = (2x4 - 1x3) / 3x4 = (8 - 3) / 12 = 5/12

a/b x c/d = ab/bd

2/3 x 1/4 = (2x1) / 3x4 = 2/12 = 1/6

(a/b) / (c/d) = ad/bc

(2/3) / (1/4) = (2x4) / (3x1) = 8/3

There is a way of adding fractions which involves finding the Lowest Common Denominator of the two denominators. This is the denominators mutiplied together, and their Highest Common Factor removed. Then you work out the equivalent fractions, which both now have the same denominator, so you can simply add the numerator. The method is rather fiddly. Generations of children struggled with Highest Common Factors and Lowest Common Denominators, and most didn't understand them. You can tell this, as people say "the Lowest Common Denominator of a group of people" as a metaphor, when what they mean is the Highest Common Factor. A Lowest Common Denominator is always higher than a Highest Common Factor, so this way of doing fractions is guarenteed to confuse nearly everyone! If you understand algebra, then the formula above will give the answer, which you might have to reduce. If you don't understand algebra, then stick to decimals.


Decimals

There is another way of writing numbers between zero and one. When you use a calculator and divide 1 by 2, the answer is 0.5 rather than 1/2. This is a decimal rather than a fraction. (The correct name is decimal fraction rather than vulgar fraction like those above.)

As you can see above, it can be quite hard to see whether one fraction is bigger than another. Also, adding fractions can be difficult, and you need two numbers with a line between them to describe them. Decimals are much easier to handle, and you can always get a calculator to do the arithmetic for you. A decimal has a decimal point in it. Everything to the left of the decimal point is a whole number, and everything to the left is less than one. So 1.5 is between 1 and 2. It is, in fact, one and a half.

Probably the first place that people met decimals is when using money. £2.50 is two pounds fifty pence. $3.10 is three dollars ten cents. €8.05 is eight euros five cents. With money, you need two numbers after the decimal point. €8.5 is not eight euros five cents.


Rounding decimals

One problem with decimals is that they can go on for ever. A third is 0.33333333333...... - it never stops! We need a way of writing this decimal. What we do is cut it off at some point. So we say a third is 0.33 to 2 decimal places, or 0.3333 to 4 decimal places. This is not quite enough. If you wanted to write 3.14159265 to 4 decimal places, if you just cut it off, it would be 3.1415. However, the next digit is a 9. In fact, it's far closer to 3.1416 than 3.1415, so we chose the last digit to give the closest digit. This is called rounding. To round to 2 decimal places, you look at the third digit after the decimal point. If this is 4 or below, then you just cut the number off at the second digit after the decimal point - that's good enough. But if the third digit is 5 or above, you add 1 to the second digit and then cut it off after the second digit.

If you want to see this in action, then use the arithmetic calculator. You can enter a number, then display it fully, as a whole number (rounded) or as 2 decimal places (rounded). The complete number stays in the calculator, so you don't loose any precision when you do arithmetic.


Multiply / divide decimals by ten

Decimal arithmetic is the same as whole number arithmetic, except that you have to understand where the decimal point goes. Here is an exercise to explain how a decimal point works.

       

Why does it do this? When we write an Arabic number, we don't write all of it. We can't, because it would go on for ever! Every number that we write down has infinitely more zeroes both at the front and back. Of course, it would be silly to write this. So we don't bother about any of the zeroes on the left or right, until we get to a non-zero number, or the decimal point. In fact, sometimes we leave an extra zero in. We say $4.50 rather than $4.5 (although your calculator may say 4.5 if you do a sum about money). We say 0.25 rather than .25 - that's probably to make sure that the decimal point gets noticed in the middle of the number rather than stuck on the front. Decimal points on the end tend to get dropped. We say 57, not 57. or 57.0 but they all mean the same thing.

Once you understand about the zeroes on the front and back of all numbers, you can see why the multiplication and division of ten works in the way that it does. Multiply by ten, and the decimal point shifts one place to the right. Divide by ten, and the decimal point shifts one place to the left. If there isn't any number left, then put in a zero. Don't forget to add a decimal point at the end if there isn't one already. Here is the same number as above, but with the zeroes left in.

       

You must realise that it's not really the decimal point that moves, but all the digits which shift one place to the left or right, past the decimal point. However, it looks as if it's the decimal point moving.


Significant figures of decimals

When numbers get very large or very small, they are difficult to write down or understand. So there is a short-hand way to do it. For these extreme numbers, we are only interested in a few figures, which are called the significant figures. In these examples, there are four significant figures. The rest of the number is just zeroes, and of course the important decimal point. Now you can write lots of zeroes as a power of ten.

103 = 1000    102 = 100    101 = 10    100 = 1    10-1 = 0.1    10-2 = 0.01    10-3 = 0.001   

So you can write down the significant figures of the number, then multiply by the correct power of ten. Normally, you would only do this for large and small numbers, but the example below shows it for the complete range so you can see what's going on.

       

Scientific calculators use this method if numbers get too large or small. But they don't have room in their display to show the power of ten as a superscript (this is what the small number above the ten is called). So they use this notation instead:

2.5 x 1025 = 2.5e+25     3 x 10-18 = 3.e-18    


Decimal point

The decimal point separates the whole part of the number from the part that is less than one. In Britain and America, it is written as a full-stop, like this "." so

one and a half is written as 1.5      two and a tenth is written as 2.1

However, in mainland Europe, the decimal point is written as a comma, like this "," so

one and a half is written as 1,5      two and a tenth is written as 2,1

What is worse, sometimes English speaking countries divide up big numbers with commas like this

a million is 1,000,000

and occasionally, Europeans use full stops for this so

a million is 1.000.000

As you can see, this can lead to problems! A better convetion for big numbers is the space:

a million is 1 000 000

although that can can look like more than one number. It is a good idea to be aware that these different convetions exist, and make it clear which you are using. I use a full-stop as a decimal point throughout this website. All this helps to show that all number systems are conventions. 2 is not two, it's just the convention that we write that particular number that way. (Yes, I know, two is not the number either. It's just another convention, writing numbers as words.)


Percentages

Numbers between zero and one are often described as percentages. 'Percent' comes from 'per centum' which is Latin for 'per hundred'. So a percentage is a special sort of fraction, where the denominator (the number underneath the line) is always 100. 23% means 23/100.

Percentages are often used in financial products. It is very important that you understand them, otherwise you might lose money! Here is a simple calculator for working out savings and debts using compound interest. You can change the amounts if you want. Enter the number of years, or click on Next year to see the savings or debt increase year by year.

You save £    at %    for years           You now have

You borrow £    at %    for years           You now owe


Comparison of fractions, decimals and percentages

Fractions, decimals and percentages are different ways of writing the same number. We tend to talk of fractions if we divide something into simple amounts, "There are four of us, so we can have a quarter of the cake each." Decimals are useful for arithmetic, as that is how a calculator works. Percentages are thought to be easier to understand than decimals as they are whole numbers (although they are really a fraction, since you divide them by a hundred to use them).

Enter the top and bottom of a fraction to see what it looks like as a decimal or a fraction. Both the decimal and the percentage may be rounded.

FractionDecimalPercentage