them, due to which, if the value of one of them changes, the value of the other also changes. We look into two types of variations here: (i) Direct variation (ii) Indirect variation. (i) Direct variation: When you go to the market, to buy more apples, you’ll have to spend more amount of money . If the cost of one kg of apples is ` , you pay as follows: Weight (Kg) Cost ( ` ) ( , ) ( , ) ( , ) ( , ) ( , ) Scale x axis cm = kg y axis cm = ` Fig.
. Weight (Kg) Cost ( ` ) y = x You find that ... This kind of proportionate variation is known as Direct variation . Here to find the cost, the weight is multiplied by the constant .
If we denote the variable weight as x and the variable cost as y we can express this algebraically as y = x. The multiplying constant here is . If k where k is a positive number (a constant), then x and y are said to vary directly. Here, k is known as the constant of proportionality.
Mathematics in real life: This figure shows that doubling the force doubles the displacement. This is a consequence of what is known as Hooke’s law . It states F = kx where F is the force needed to produce a displacement of x in the position of a spring. To double the displacement, you double the force on the spring; the constant of proportionality k depends on the stiffness of the spring.
So this is an example of a direct proportionality. Visualising Direct variation: To identify direct variation is to look at the equation and determine if it is of the form y = kx, where k is the constant of proportionality. Thus, an equation like y = x will