Chapter 3.7 - Graphs of Direct Proportional changes

In the previous sections we discussed the various features of direct proportional changes. In this section we will see it’s graphical representation.

The first example we saw is the change in quantity of turpentine with the change in quantity of paint. We saw table 3.1. which shows the quantities. It is shown here again for convenience:

Table 3.1

We saw that it is a case of direct proportion. In the table, we see that the quantities are denoted as ‘x’ and ‘y’. Each column form a pair. We can think of them as 'coordinates', and plot a graph with them. The points in the table 3.1 give the following graph:

Fig.3.15

We can see that all the points in the table falls in a line. This is a property of ‘direct proportion’. That is., if the change in two quantities is directly proportional, then, on a graph, all the sets of corresponding points will lie on a straight line. This property can be used to find y for any given value of x. Let us do an example in the above graph. Suppose, we want to know the quantity of turpentine for 4.5 litres of paint. This can be determined from the graph as shown below:

Fig.3.16

The following procedure is used in the making of the above graph:
• Quantity of paint is plotted along the X axis. So mark 4.5 on the X axis.
• From that mark, draw a perpendicular to the X axis
• This perpendicular line will meet the graph at a certain point.
• From this point of intersection, draw a parallel to the X axis towards the left.
• This parallel line will meet the Y axis at a certain point. This point is (0,562.5)
• So the required value of y is 562.5.
So the quantity of turpentine required for 4.5 litres of paint is 562.5 ml. The point (4.5,562.5) lies on the graph as shown in the fig.3.16 above.

It may be noted that 562.5 does not lie on main grid lines. So it may be difficult to read it from the graph. On a computer screen, a small portion can be enlarged to get the value. On a graph paper, there will be sub divisions. These sub divisions will help us to get the value. We have discussed about it in precision plotting.

Let us now look at the second example. The bill amount (y) is directly proportional to the quantity of rice (x). The table is shown here again for convenience.

Table 3.2

Let us plot the graph:

Fig.3.17

Again we find that all the points lie on a straight line. If we want to know the price of say, 9.5 kg of rice, we can draw lines perpendicular and parallel to X axis and read the price from Y axis. The procedure is same as that for the first example that we saw above. The modified graph is shown below:

Fig.3.18

We find that the price of 9.5 kg of rice is ₹ 361.

---

We will now see a solved example based on the above discussion.
Solved example 3.16
The owner of a car noted down the quantity of petrol he filled in the car, and the corresponding cost, on various occasions. The table below is taken from his notes.

Quantity (l)	Cost ₹
3	210
5	350
9	630
7	490
11	770

Plot the graph. Using the graph, answer the following:
(i) Does the cost change proportionately with the quantity?  
(ii) What will be the cost of 14 litres of petrol? 
(iii) How much petrol can be purchased for an amount of ₹ 600?

Solution:
(i) The plot of the points in the given table is shown below:

Fig.3.19

All the points lie on a straight line. So it is a case of direct proportion.

(ii) We have to mark a point at 14 l on the X axis and draw a perpendicular. (See fig.3.20 given below)
• But this perpendicular does not meet the graph. So we have to extend the graph along the same line, upwards. 
• Now the two will meet. From the point of intersection, draw a line towards the left and parallel to X-axis. 
This will meet the Y axis at 980. So the cost of 17 litres of petrol is ₹ 980

Fig.3.20

(iii) In part (ii), we calculated the cost corresponding of certain quantity of petrol. But here, we have to find the quantity of petrol corresponding to certain amount of money. So we have work in a sort of reverse order. The procedure is explained below. See fig.3.20.
• First mark 600 on the Y axis.
• Draw a line parallel to the X axis through this point.
• It will meet the graph at a certain point.
• From this point draw a perpendicular to the X axis It will meet the X axis at 8.57
So the quantity corresponding to ₹ 600 is 8.57 l

---

Check:We will try to get the above results using the usual method that we learned earlier. All the steps can be shown in a single table as shown below:

(i) We see that all the given values give a constant ratio of 70.00. So it is a case of direct proportion.
(ii) The cost of y1 and the corresponding quantity of 14 l  should give the same ratio 70.00. So we can write:
y1/14 = 70.00 ⇒ y1 = 14 × 70 = ₹ 980
(iii) The cost of ₹ 600 and the corresponding quantity x1 should give the same ratio 70.00. So we can write:
600/x1 = 70.00 ⇒ x1 = 600/70 = 8.57 l.

These are the same results that we obtained graphically.

---

We have completed the discussions on the 'Direct proportions and it's applications'. In the next section, we will see the 'Inverse proportions'.

PREVIOUS      CONTENTS       NEXT

                        Copyright©2016 High school Maths lessons. blogspot.in - All Rights Reserved