Chapter 5.2 - Equivalent fractions

In the previous sections we saw how fractions help us to measure small distances. In this section we will discuss about Equivalent fractions.

Consider the following situation: A customer walks into a prepay counter of a bake shop. He pays for 1/5 (one fifth) of a cake and gets the token. But when he arrives at the despatch section, he finds that the cake is already cut. It has been cut into to 15 pieces. He was hoping to ask the baker to cut it into 5 equal pieces so that he could take 1 out of 5. But now there are 15 pieces. But the baker tells the customer not to worry. “I shall give you 3 pieces from among these 15. It will be equivalent to the 'one piece out of five' that you have paid for”. How did the baker arrive at such a conclusion? Let us find out:

In the animation in fig.5.12 below, (a) shows the cake divided into 5 equal pieces. The customer was hoping to divide the cake in this way. So that he could take one out of the five. But the baker has already cut the cake into 15 pieces as shown in (b).

Fig.5.12

Fig.(c) shows one piece from (a). Fig. (d) shows 3 pieces from (b) put together. We find that The single piece in (c) has the same size as the '3 pieces put together' in (d). 

So the calculation made by the baker turns out to be correct. One piece out of 5 is indeed equal to 3 pieces out of 15. We can write:
1/5 = 3/15 - - - (1)
The customer can take the three pieces and happily go home. But we will stay for a while and see if there are any other possibilities:

The customer was hoping to take one out of 5. But he had to take 3 out of 15 because, the cake was already cut into 15 pieces. What if the cake was cut into 10 equal pieces instead of 15 ? In this case also, the baker has to do some calculations. Let us see the fig.5.13 below:

Fig.5.13

From the fig., we see that, one out of five is equivalent to 2 out of 10. So we can write:
1/5 = 2/10  - - - (2)
Combining this information with (1), we can write:
1/5 = 3/15 = 2/10  - - - (3)

In (3) above, there are 3 fractions. 1/5, 3/15 and 2/10. Each of them appears to be different. But they represent the 'same part' of the whole. That is., all three have the same value. Such fractions are called Equivalent factions.

Let us explore some more possibilities:
The customer had paid for 1/5 of the cake at the prepay counter. Let us suppose he had paid for 3/5 of the cake. So he would hope to take 3 pieces out of 5. But the baker had already cut it into 15 pieces. How many pieces out of 15 would the baker give him? Let us see the fig.5.14 below:

Fig.5.14

From the fig., we can see that 3 out of 5 is equivalent to 9 out of 15. So we can write:
3/5 = 9/15 - - - (4) 
If the cake was cut to 10 pieces instead of 15, then also we can do calculations using a fig:

Fig.5.15

From the fig., we see that, 3 out of 5 is equivalent to 6 out of 10. So we can write:
3/5 = 6/10 - - - (5)
Combining this information with (4) we can write:

3/5 = 9/15 = 6/10 - - - (6) 
In (6) above, 3/5, 9/15 and 6/10 have the same value. So they are Equivalent fractions.

We have discussed about equivalent fractions by dividing an object (a cake) into different equal parts. Instead of objects, we can use distances also for the discussion. Consider the fig.5.16 below:

Fig.5.16

Two points A and B are marked on a straight line on the ground. (The exact distance between A and B is not important for our present discussion). A point is marked at half way distance between A and B. This is shown in the fig.(a).    

Next, The line AB is reproduced exactly to the same measurement. This new line is drawn parallel to the original. This is shown in fig (b). It is divided into 4 equal parts. Take the first two of these four parts. They will have a total length of 2/4. From the yellow dotted line we see that this 2/4 is equal to 1/2.

Fig.(c) shows another parallel line. It is divided into 6 equal parts. Take the first three of these six parts. They will have a total length of 3/6. From the yellow dotted line we see that this 3/6 is equal to 1/2. In the same way we find that in fig.(d), 4/8 is equal to 1/2. And in fig.(e), 5/10 is equal to 1/2.

So we can write:
1/2 = 2/4 = 3/6 = 4/8 = 5/10 - - - (7)
These are all equivalent fractions of 1/2

Let us see one more example: The fig.5.17 below shows an equivalent fraction of 3/4

Fig.5.17

In fig.(a) the line AB is divided into 4 equal parts. In fig.(b), it is divided into 12 equal parts. Take the first 3 from (a). The three parts will together give a length 3/4 of AB. From the yellow dotted line, we can see that this 3/4 is equal to 9/12 in fig (b). So we can write:
3/4 = 9/12 - - - (8)
Thus 9/12 is an equivalent fraction of 3/4.

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So we have discussed the basics of equivalent fractions. In the next section we will see more of their features.

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