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.

 First select the denominator using the controls in the lower part of the window. Then, drag the point in the left hand corner of the window carefully to the right until the numerator reaches the desired value.

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

 This window is similar to the one above but this time the numerator is changed by dragging the point in the bottom left hand corner (A). Also, the numerator can be expressed as a decimal. The numerator control point is more sensitive this time and so you have to drag it across the diagram carefully to make sure the shading is uniform.

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.)

2) Click on the denominator button once so that
d=4 and you will see how the unit is divided into 4 equal units (each one is equal to 1/4). However, the shaded area does not coincide with any of the four divisions and we have a decimal number as a numerator which WE DO NOT WANT.

3) Click on the denominator button again (d=5) so that the unit is divided into five equal parts (each one is 1/5) and we can see that the same thing happens.

4) Click on the denominator once again (d=6) THIS IS INCREDIBLE! 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.
This means that 1/3=2/6 and therefore they are
EQUIVALENT FRACTIONS.

In the following window you can represent three fractions at the same time.
E.g. represent these fractions:

 Use the buttons below the diagram to select the denominator and drag points A, B and C with the mouse until you see each one written down underneath each diagram. You can see now that fractions can be bigger than one unit (this window includes four units)

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.

We can also work this out the other way round. If we divide the numerator and denominator of the third fraction by 3 we get the first fraction and if we divide the numerator and denominator of the second fraction by 2 we also get the first 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?
That's easy! For example, if we want to know whether we just need to check that 12x10=15x8=120.

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:

1. We have to find the lowest common multiple (LCM) of 4 and 6 = 12. This is going to be the common denominator. Then, we divide this common denominator by each denominator and multiply each part of the fraction by the result.

2. 12÷4 = 3; We have to multiply by 3;
3. 12÷6 = 2; We have to multiply by 2;

Now we have two fractions which are equivalent to the first ones but have the same denominator as each other.

 Introduce two different fractions into the following window by using the controls a, b, c and d. Then, calculate the lowest common multiple of the denominators in your exercise book and change the fractions so that they have a common denominator. Once you have done this, introduce the value for the lcm using the relevant buttons and change the solution button to 1 to check your answer.

Exercise 4

Change these fractions so that they have a common denominator:

a) 5/4 & 7/18             b) 7/3 & 8/27         c) 4/7 & 5/14       d) 3/100 & 5/4

Check your results in the window above.

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