Direct & Inverse Relation - Part 2

Hope you would have got the answer as Rs. 7,500 being the total cost when 50 people go to the picnic.

The answer is not important, the approach is.

If you have used the relation, y = kx + k1 and then substituted (y, x) as (6250, 40) and (3750, 20) and solved the two equations simultaneously to arrive at k = 125 and k1 as 1250, you have done a lot of unnecessary work, found a lot of unnecessary details and yes, wasted quite a lot of time and energy.

The approach should have been as simple as:
Increase of 20 students (from 20 to 40) causes the cost to increase by Rs. 2500 (from Rs. 3750 to Rs. 6250).
An increase of 10 more students will cause cost to increase further by Rs. 1250.
Thus, total cost for 50 students will be Rs. 6,250 + Rs. 1,250 = Rs. 7500.

Theory: Whenever y is partly constant and partly varies directly with x, then the 'change in y' is directly proportional to the 'change in x'.
Thus, the increase/decrease in cost is directly proportional to increase/decrease in number of people.

If need be, from the above working, one can also find, k and k1, though to arrive at the answer one does not need it. But for someone who wants to find all these details …

Analogy: Say the picnic consists of hiring a bus (the fixed cost, independent of how many people are going to the picnic, the bus hiring cost remains the same) and ordering lunch-boxes for as many people as going to the picnic (the directly varying part).

When 20 additional people join the picnic, the cost goes up by Rs. 2500. Why does the cost increase?
Because 20 additional lunch-boxes have to be ordered. Thus, 1 lunch box will cost Rs. 2500/20 = Rs. 125. This is k of the equation. If you have used equations to solve, look carefully, we have mentally done exactly what you have done when you subtracted the two equations.

How does one find the fixed cost? 20 lunch boxes costs Rs. 2,500. Then, when 20 people are going to the picnic, why is the total cost Rs. 3,750? Because the additional Rs. 1,250 is the bus hiring cost. This is k1.

The above is easy to understand and does not require too much dwelling on it. But what is important, if you are using the short-cut, is not to use it wrongly e.g. can the following question also be solved in the same way?

Q. 2: When 20 people go to the picnic, the cost per head is Rs. 80. When 30 people go to the picnic, the cost per head is Rs. 60. What will be the cost per head if 50 people were to go to the picnic?

Can we do the following?
An increase of 10 people (20 to 30) caused the cost per head to reduce by 20 (Rs. 80 to Rs. 60)
Hence an increase of 20 people (30 to 50) will cause the cost per head to reduce by Rs. 40.
Thus, the required cost per head is 50 – 40 = Rs. 10

For those who do not see any problem, try finding the cost if instead of 50 people, 60 people were to go on picnic.

So how does one find the cost per head when 50 people go on the picnic?

One way is to translate this questions to that if ‘number of people’ and ‘total cost’ which is exactly same as Q. 1. The challenge is can we find a logical and oral way without taking this indirect route of total cost?

visit www.takshzila.com for online CAT prep courses
visit www.takshzilabeta.com for free CAT articles, videos, daily vocab gyan, etc.