Fractions, decimals and percentages: Explanation.  
3rd year of secondary education.  
Explanation.  
When we say "I've eaten half a sandwich" we're actually using a fraction: I've divided the sandwich into two equal parts, 2 (called the denominator), and I've eaten one of them, 1 (called the numerator)  
We are going to use the following window to represent fractions. In this window there is one complete unit which is represented by the red rectangle. Follow the instructions given in the margin to represent a fraction, e.g. 1/2.  
 
If you change the value of the denominator to 2 and then you drag the point to the right until the numerator equals 1 you will see the fraction appear below. The shaded area of the rectangle is The rest of THE UNIT which is unshaded is the other Then again, if we divide 1 between 2 we get 0.5, which is also indicated in the window. This is how we represent the fraction as a decimal number. Also, as this fraction represents 50% of the unit we can work out a percentage by multiplying the decimal number by 100.  
You can use this
window to see lots of fractions, find out their decimal form and what percentage
of the unit they represent. Their form is always
We make a fraction by dividing the unit into as many parts as we want, as represented by the denominator d (use the button under the diagram) and selecting a certain number of these parts, as represented the numerator n (move point A slowly). In this example the numerator will always represent a number of parts equal to or less than the number of parts represented by the denominator. This is because we are only working with one unit, but we will look at other examples later on.  
Exercise 1 Represent the following fractions
in the window above. Click on the start button each time you want to change fraction and make a note of the fraction represented by the unshaded section of the unit necessary to complete the whole unit. Use a calculator to work out the decimal number and percentage of each fraction and check that they correspond to those given under the diagram. You can use this window to represent other fractions too, as long as the numerator is less than or equal to the denominator.

Equivalent fractions  
 
 
1) In
this window represent a fraction, e.g. 1/3 (you know how to do it
now. Make the denominator =3
using the button underneath and the numerator =1
dragging point A carefully.)  
4)
Click on the denominator once again (d=6)
The unit is now divided into six
equal parts (each one is 1/6) but the shaded
area coincides exactly with two of these parts. Therefore the
fraction 2/6 represents the same quantity as 1/3 that we had
earlier on. 
 
 
You should have noticed that the three fractions represent the same quantity. These are what we call equivalent fractions:
 
If we multiply the numerator and denominator of the first fraction by 2 you can see that we get the second fraction and if we multiply the numerator and denominator of the first fraction by 3 we get the third fraction.
If we have the fraction we can work out an equivalent fraction by multiplying or dividing the numerator and denominator by the same number. Therefore:
 
Exercise 3. Find two equivalent fractions for and illustrate them in the previous window to check that they do represent the same quantity.  
However, how do we know if two fractions are equivalent when we can't
illustrate them in this way?  
Are the following pairs of fractions equivalent?and &. Check your results in the window above.  
Two fractions are equivalent if they satisfy the following relation:

Finding a common denominator.  
The
aim of this section is the following: "When we have a pair of fractions
find another pair of fractions which are equivalent to the first pair but which
both have the same denominator".
Later on this will allow us to compare fractions and carry out operations with fractions more easily.
We are going to change the fractions so that they have a common
denominator. We need to follow this procedure:
Now we have two fractions which are equivalent to the first ones but have the same denominator as each other.  
 
Exercise 4
Change these fractions so that they have a common denominator:
Repeat the exercise with different pairs of fractions of your choice. 
Comparing fractions.  
Introduce the fractions 3/9 and 7/9 ,which have the same denominator, into the following window. (The denominators are introduced using the buttons underneath and the numerators by dragging points A and B). Which fraction is the biggest?  
Now introduce these fractions that have the same numerator: 3/4 and 3/7. Which fraction is bigger?  
When two fractions have the same denominator the
bigger fraction has a greater numerator. When two fractions have the same
numerator the bigger fraction has a smaller
denominator. If you want to compare two fractions that don't have the same numerator or denominator change them so that they have a common denominator and then compare them using the rule above. Another way of doing it is to change both fractions into decimal numbers and compare the results. We will be coming back to this later on. 
Ángela Núñez Castaín  
Spanish Ministry of Education. Year 2001  
Except where otherwise noted, this work is licensed under a Creative Common License