The following is a question from the Common Admission Test which is an aptitude exam conducted in India for admission into postgraduate management programmes of premier management institutes of the country.

The salaries of Ram and Shyam are in the ratio 6:7 while their expenditures are in the ratio 7:8 respectively. Which of the following can be the ratio of their savings? A) 5:6 B) 6:7 C) 7:8 D) None of these



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HINT #1: Shyam makes $1/6$ more money but only spends $1/7$ more money, so what do you think the ratio of their savings can be?

HINT #2: Put Ram's money on the $x$-axis and Shyam's money on the $y$-axis. The salaries are represented by a vector $S=(6a, 7a)$ for some $a > 0$, and the expenditures are represent by a vector $E = (7b, 8b)$ for some $b > 0$. Draw the diagram, subtract the two vectors to arrive at the savings $V = S-E$. What can you conclude about the possible slope of the $V$?

Can you finish from here, or do you need more hint?

Expanding on Hint #1: If Shyam makes $1/6$ more money and also spends $1/6$ more money, then clearly he will also save $1/6$ more money. I hope that is obvious. If not, try a few numbers, e.g. they make $6000, 7000,$ and spend $600, 700,$ and therefore save $5400, 6300.$ All three pairs are $6:7.$

If Shyam makes $1/6$ more money, but spends less than $1/6$ more money, then he will end up saving more than $1/6$ more money. This is less obvious, but follows from the previous paragraph. Again, e.g. they make $6000, 7000,$ but spend $700, 800,$ and save $5300, 6200.$ The last pair is $1:1.170$ which is higher than $6:7 = 1:1.167$.

Now consider the specific case of Shyam spending $1/7$ more money. This is less than $1/6$ more money, so he will save more than $1/6$ more money. This immediately rules out answers (B) and (C). But is the answer (A) or (D)? I.e. can the savings ratio reach $5:6 = 1: 1.2$?

If the expenditures $\ll$ the salaries, the savings ratio will remain close to (but slightly higher than) $6:7$. (I hope this is obvious, e.g. imagine them spending $0.07, 0.08.$) If the expenditures are very big, e.g. in the limit when Ram spends all his salary while Shyam doesn't, the ratio will become infinite (division by zero). So as the expenditures increase, any ratio above the $6:7$ ratio is reachable. Thus the answer is (A), i.e. $5:6$ is a possible ratio.

If you need more concrete reasoning, imagine they make $6000, 7000,$ and spend $7x, 8x$ for some value of $x$. Thus you are trying to solve:

$${7000 - 8x \over 6000 - 7x} = {6 \over 5} \implies 35000 - 40x = 36000 - 42x \implies 2x = 1000 \implies x=500$$

I.e. they spend $3500, 4000$ and save $2500, 3000$ which is indeed a $5:6$ ratio. But IMHO more importantly, you should be able to realize $5:6$ is reachable without actually solving for $x$.

  • $\begingroup$ the question is from an entrance test for admmsion into management programmes of Indian universities. We are expected to arrive at the answer in 2-3 minutes. So using the 2nd approach would consume time I think. Can you please elaborate your 1st approach though? $\endgroup$ – Shahbaaz Sheikh Apr 11 at 1:54
  • 1
    $\begingroup$ hope my expanded reasoning is helpful! IMHO if you have only 2-3 minutes, you need the abstract reasoning without going to solve the equation listed. but since you have more than 2-3 minutes right now, i also suggest you actually draw the vector diagram. anyway, if this is good enough, please accept the answer (by clicking on the checkmark). $\endgroup$ – antkam Apr 11 at 2:30
  • $\begingroup$ this was great! Thanks! $\endgroup$ – Shahbaaz Sheikh Apr 11 at 2:41

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