Circular Arrangement Part 2

Circular Arrangement Part 1

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?

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Direct & Inverse Relation - Part 1 (Intro)

We all know direct/inverse proportion very well

If x varies directly with y …
…. when x increases, y also increases and when x decreases, y also decreases.

This is a very layman-ish understanding but will serve our purpose. However in the above, a very important aspect is missed out! That of a proportional increase or decrease. So if x increases, y increases proportionally i.e. if x doubles, y will also double and if x becomes one-third, y will also become one-third.

Now consider the following question:

Q. 1: When 20 people go on a picnic, the cost is Rs. 3750 and when 40 people go on a picnic, the cost is Rs. 6250. What is the cost of the picnic, if 50 people go on the picnic?

The number of people has increased from 20 to 40 i.e. doubled.
Has the cost increased? Yes it has.
Has the cost increased proportionally i.e. has it doubled? No.

Thus, there is a ‘direct relation’ but not ‘direct proportion’.

This happens because: The cost of the picnic is partly constant and partly varies directly with the number of people.

Thus, the part that varies directly with number of people would have increased, in fact would have doubled. But the part that is constant would remain the same. Hence the total cost does increase, but not proportionally.

Now your task is to solve the question. That is not a big deal, if you are applying pencil to paper and solving it through equations. You have to devise a very logical and oral way to solve such situations of direct relation. Solution and further interesting twist tomo.

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Who saves more - Part 2

First with the answers …
Q. 3: A saves more
Q. 4: Cannot be determined
Q. 5: A saves more
 
Now for the logic
 
Q. 3 is an easy one, A earns more than B and spends less than B. So A will definitely save more.
 
There are numerous ways in which Q. 4 and 5 can be approached, of which three are listed below. What one must realise is that the expenses could have an entire range, from being very less to being as high as possible, RELATIVE to the income. The word ‘relative’ to the income is the key here.
 
Approach 1: Trying values
 
Assuming the incomes as Rs. 600 and Rs. 500 respectively, we construct two cases, once when the expenses are very less and once when the expenses are as high as income.
 
Q. 4:        Case 1: Less Expenses    Case 2: High Expenses
                          A       B                          A        B
Income              600    500                       600    500
Expenses             4        3                        600    450
Saving               596    497                          0      50
 
Thus, we have two cases, of which in one A saves more and in other B saves more. Thus, the answer can be nothing but ‘Cannot be determined’
 
Q. 5:         Case 1: Less Expenses    Case 2: High Expenses
                          A       B                         A        B
Income              600    500                       600    500
Expenses             9        8                        558    496
Saving               591    492                         42      4
 
Even if B spends entire 500, A would have some saving.
 
Thus, in both the extreme cases – expenses being very less or being as high as possible, A saves more. And hence for expenses being any value between these extremes, A must cave more.
 
Approach 2: Algebraically
 
Let’s use variables here.
 
Q. 4: Assume the incomes as 6x and 5x and the expenses as 4y and 3y respectively. Since neither guy can spend more than their incomes, 4y < 6x i.e. y < 1.5x AND 3y < 5x i.e. y < 1.66x. Thus, the value of y that satisfies both is y < 1.5x (It could also be equal to, but using the symbol of less than or equal to is very cumbersome in HTML)
 
Comparing the savings, 6x – 4y  ?  5x – 3y.
Transposing, x  ?  y
 
Now it is quite possible that x can be larger than y. And it is also possible that y can be larger than x (can be upto 1.5x). Thus, either the LHS or the RHS could be larger and thus we ‘cannot determine’ who saves more, could be either depending on the values of x and y.
 
Q. 5: Assume the incomes as 6x and 5x and the expenses as 9y and 8y respectively. Since neither guy can spend more than their incomes, 9y < 6x i.e. y < 0.66x AND 8y < 5x i.e. y < 0.625x. Thus, the value of y that satisfies both is y < 0.625x
 
Comparing the savings, 6x – 9y  ?  5x – 8y.
Transposing, x  ?  y
 
In this case we can be sure that x > y because y has to be less than 0.625x.
Thus, the LHS will always be greater than RHS and hence A will always save more than B
 
Approach 3: Logically (the suggested approach)
 
Q. 4: A earns 20% more than B. Let’s call this the ‘extra income’ of A. Also A spends 33.33% more than B. Let’s call this the ‘extra expense’ of A.
 
Now we cannot be sure whether ‘20% of B’s income’ i.e. ‘the extra income’ is greater or less than ‘33.33% of B’s expense’ i.e. ‘the extra expense’. If the expenses are very less as compared to the incomes, the ‘extra income’ is more than the ‘extra expense’ and so A will save more. But if the expenses are very close to the income levels, the ‘extra income’ will be less than the ‘extra expense’ and hence even after earning more, A will save less.
And so for this question, the answer to who saves more is ‘cannot be determined’
 
Q. 5: A earns 20% more than B and A spends 12.5% more than B
 
And 20% of B’s income will surely be more than 12.5% of B’s expense. Thus, the ‘extra income’ of A will be sufficient for his saving to be more than that of B, even after spending more than B, for any expense levels. So the answer for this question is that A will always save more.
 
Further Work
 
Now for Q. 3 and Q. 5, we found that A will always save more than B. Can we find savings of A will be what percentage more than saving of B? Obviously it will be a range of percentages (or else we could have found the ratio of saving uniquely, which is not the case, as part 1 of this series started with)  

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