Who saves more? Part 1

Ratios is a very easy topic (though very useful). And one of the easy type of questions on basic use of ratios is

Q. 1: If the incomes of A and B are in the ratio 6 : 5 and their expenses are in the ratio 4 : 3, what is the ratio of their savings?

Q. 2: If the incomes of A and B are in the ratio 6 : 5 and their expenses are in the ratio 6 : 5, what is the ratio of their savings?

(You have to assume that A and B save some amount and do not borrow any money)

Try the above before reading ahead.

In Q. 1, the uninitiated student, occasionally in hurry, assumes the incomes of A and B as Rs. 600 and Rs. 500 and the expenses as Rs. 400 and Rs. 300 and find the savings as Rs. 200 in each case, thus the needed ratio is found as 1 : 1

Needless to say this is the ratio only for the chosen values and this ratio will keep changing depending on the chosen values. Theoretically, if x and y are the constants of proportionality of the two ratios, the required ratio is , the value of which is dependent on x and y. Thus, the correct answer to Q. 1 is ‘cannot be determined’ or ‘no unique value’.

Q. 2 is also simple and using , the ratio is found as 6 : 5, always, irrespective of the values of x and y (not being equal when savings is 0 for both).

Thus, if the ratio of income and expenses are different, the ratio of savings cannot be determined. But if the ratio of income and expenses is the same, ratio of the saving will also be the same value.

The above is very easy and should have been known to most who have some preparation behind them. What is interesting is the following.

In each of the following, the ratio of incomes and expenses are different and hence ratio of savings cannot be found. But the question is “Who saves more?”.

Even if the ratio is not known, can we comment on who will save more and who will save less?

Q. 3: If the incomes of A and B are in the ratio 6 : 5 and their expenses are in the ratio 3 : 4, who saves more?

Q. 4: If the incomes of A and B are in the ratio 6 : 5 and their expenses are in the ratio 4 : 3, who saves more?

Q. 5: If the incomes of A and B are in the ratio 6 : 5 and their expenses are in the ratio 9 : 8, who saves more?

And each of the above is a different logic.

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