An excellent question which tests you on the concepts of inequalities.
The topic of inequalities is a mystical one. It has taken many a people down a slippery slope.
One of the most important reasons for this, is that all those people tended to deal with inequalities in the same way they dealt with equations. They cancelled out terms on either side without thinking about the signs; they cross multiplies, again without thinking about the signs.
These are very common mistakes that a lot of students make on Inequalities questions.
Even in this question, there is a chance that, when you looked at statement I, some of you thought that you could cross multiply. You cannot do that because you do not know whether r is positive or negative.
Remember,
‘Multiplying both sides of an inequality will flip the inequality sign’.So, if r were to be positive, you could cross multiply with glee. But, if r were to be negative, cross multiplying without changing the sign would be wrong.
A better and safer approach in inequalities questions is always:
1. To make the RHS ZERO
2. To express the expression on the LHS as a product or a quotient
A product or a quotient on the left hand side makes analysis easier, than having a sum or a difference.
Let’s look at the question now. We know that none of r, s and t are ZERO. This is a necessary condition that the question has to specify, because, as you see in the statements, some of these variables are in the denominator.
Clearly, the denominator cannot be ZERO. This is actually a subtle clue that the question is giving you, from which you need to understand that some of r, s and t might come in the denominator.
The individual statements are insufficient, since neither of them provides data about all 3 variables.
Answer options A, B and D can be eliminated.
Combining the statements, we have the following:
From statement I, \(\frac{rs}{s}\) > \(\frac{rs}{r}\). This can be rewritten as,
\(\frac{rs}{s}– \frac{rs}{r}\) > 0. On simplification, this gives us (r-s) > 0. This means, r > s.
From statement II, \(\frac{st}{s}\) > \(\frac{ts}{t}\). This can be rewritten as,
\(\frac{st}{s} – \frac{ts}{t}\) > 0. On simplification, this yields (t-s) > 0. This means, t > s.
The main question is, ‘ Is r>s>t’? From the above analysis, the answer is a definite NO.
The correct answer is C.
Hope this helps!
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