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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