Chapter 3 - Direct Proportions

In the previous chapter we completed the discussion on Graphs. In this chapter we will discuss about 'Direct proportions'. Consider the following situation:

We want to apply paint to a fence. For this, we take the paint in a container and add some turpentine to it. Suppose that for 2 litres of paint, we add 250 ml of turpentine. When this mixture is applied on the fence, we find that the finish is excellent. So we decide to paint more portion of the fence. This time we take 6 litres of paint. We want the same finish that we obtained before. We added 250 ml of turpentine for 2 l of paint. The same 250 will not be sufficient for 6 litres of paint. How do we calculate the exact quantity of turpentine required?

6 l is 3 times of 2 l. So we need 3 times of 250 ml also. Thus we get 750 ml of turpentine. Similarly, if we take 7 l of paint, we will need  875 ml of turpentine. The calculation steps by which 875 ml was obtained are given below:
• For 2 l of paint we added 250 ml of turpentine
• 7 l is 3.5 times of 2 l
• So we need 3.5 times of 250 also.
• 3.5 x 250 = 875 ml

In the above examples we took increasing quantities of paint. If we take a lesser quantity of paint, there will be a decrease in the quantity of turpentine also. For example, if we take 1 l of paint, the quantity of turpentine required is 125 ml. The calculation steps are given below:
• For 2 l of paint we added 250 ml of turpentine
• 1 l is 0.5times of 2 l
• So we need 0.5 times of 250 also.
• 0.5 x 250 = 125 ml 
The following fig.3.1 shows the quantity of turpentine required for various quantities of paint:

Fig.3.1

So we find that, there indeed is ‘variation’. That is., when one quantity (the quantity of paint) changes, the other quantity (the quantity of turpentine) also changes. We can give the notations of variables for these two varying quantities. The usual notations for variables are x and y. And we can form a table instead of the above fig.3.1. This is shown as table 3.1 below:

Table 3.1

Let us see another example: We will see the purchase of a commodity, say rice. We are in a Super market. If we take 3 kg of rice and take it to the billing counter, we will get a bill for Rs. 114.00. If we take 6 kg, the bill will be for Rs. 228.00 The calculation is shown below:

• Cost for 3 kg is Rs 114.00
• 6 kg is 2 times of 3kg
• So we need to pay 2 times of 114 also
• 2 x 114 = 228.00

If we take 5 kg, the bill will be for Rs.190.38 as shown below:
• Cost for 3 kg is Rs 114.00
• 5 kg is 1.67 times of 3 kg
• So we need to pay 1.67 times of 114 also
• 1.67 x 114 = 190.38

If we take lesser quantities, the bill will also decrease. For example, if we take 0.5 kg, the bill will be for Rs.19.00 as shown below:
• Cost for 3 kg is Rs 114.00
• 0.5 kg is 1/6 times of 3 kg (since 0.5/3 = 1/6)
• So we need to pay 1/6 times of 114 also
• 1/6 x 114 = 19.00
As in the case of paint and turpentine, we can make a table 3.2 given below:

Table 3.2

So, a change in the quantity of rice will cause a change in the bill amount. One more example is given below in Table 3.3. It shows the variation of the quantity of tea dust with the number of cups of tea.

Table 3.3

In this way, we come across many situations where we see a change in one quantity when there is change in another quantity. We say that, in all these situations, there is 'variation'. Now we will examine whether there is any definite ‘rule’ for these variations. First of all, what is the advantage of having a definite ‘rule’? 

Let us check the first example of Paint and turpentine. At the first instance when 250 ml of turpentine was used for 2 litres of paint, we got an excellent finish. When we want more than 2l of paint, obviously we must add more turpentine. But ‘how much more’ do we need to add? Adding ‘any larger’ quantity of turpentine will not give the same finish. If we add turpentine with out any rule, some portions of the fence will have a good finish, while other portions will have a poor finish. 

In the second example of the purchase of rice, if there is no rule, the owner of the super market may charge us large amounts for higher quantities. That is., he will fix price according to his convenience, and we will lose money.  

In the third example, if we add ‘any larger’ number of spoon of tea dust for larger number of cups of tea, OR, ‘any smaller’ number of spoon of tea dust for smaller number of cups of tea, the result will taste weird.


So a rule is essential. It should help us to calculate the value of 'y' for any value of 'x'.   Let us try to find it: Take the table 3.1. The values are arranged in 9 columns. Concentrate on the columns from (iii) to (ix). In each of these columns, the upper value is the x value and the lower value is the y value. 
• Take any column. Let us take column (v). The x value is 3 and the y value is 375. Take the ratio y/x. It is equal to 375/3 = 125. 
• Take any other column. Let us take column (vii). Take the ratio y/x. It is equal to 1125/9 = 125. 
• Let us try one more time. Take column (iv) Take the ratio y/x. It is equal to 250/2 = 125.

So we find that, which ever column we take, the ratio y/x remains the same. In other words, the ratio y/x = A constant


This can be put in another way: y = A constant × x. Recall that our ultimate aim is to formulate a rule for calculating 'y'. That is., we want a rule to calculate the value of y (quantity of turpentine) for the various values of x (quantity of paint). We have found a rule to do that. The rule is: 
y = 125x. So, we can determine the quantity of turpentine (in ml) just by multiplying the quantity of paint (in l) by 125.

In the next section we will try to find similar rules for the other two examples: Purchase of rice, and Making of tea.

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