If r > 0 and s > 0, is r/s < s/r? (1) r/(3s) = 1/4 (2) s = r + 4

First I restated the problem since we are given r and s are greater than 0 --> therefore, the question can be solved by answering whether or not r^2<s^2.

(1) r = (3/4)s [r<s, so r^2 < s^2] Sufficient AD/BCE - elminate BCE
(3) s = r + 4 [r<s, so r^2 < s^2] Sufficent - elminate A ... answered D

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If r > 0 and s > 0, is r/s < s/r?

(1) r/(3s) = 1/4
(2) s = r + 4

In inequalities, the sign does not change when a positive integer is multiplied on both sides.
If we modify the question, r/x<s/r, or is r^2<s^2, of is r^2-s^2<0?, or (r-s)(r+s)<0? and r>0 and s>0, so we want to know whether
r-s>0?
For condition 1, in r/s=3/4, r and s are positive, so s>r, which answers the question 'yes' and is sufficient.
For condition 2, s-r=4>0. s>r, so this also answers the question 'yes' and is sufficient.
The answer becomes (D).

Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.

Solution:

Question Stem Analysis:

We need to determine whether r/s < s/r given that both r and s are positive. Notice that r/s < s/r if r < s when both r and s are positive. Therefore, we really need to determine whether r < s.

Statement One Alone:

r/(3s) = 1/4

4r = 3s

r = 3s/4

We see that r is ¾ of s. Given that both r and s are positive, we see that r < s. Therefore, r/s < s/r. Statement one alone is sufficient.

Statement Two Alone:

Since s = r + 4, r < s. Therefore, r/s < s/r. Statement two alone is sufficient.

Answer: D

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Answer: Option D

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If r > 0 and s > 0, is r/s < s/r?

Solution: Statement I
r/(3s) = ¼
r/s = 3/4 also s/r =4/3
This is sufficient to answer the Question, is r/s < s/r?
YES, r/s< s/r
3/4 < 4/3


From statement 2:
S= R+4 Also its given S>0 and R >0
So, we can conclude that S>R
S/R >1 and R/S <1
That's sufficient to answer the Question is r/s < s/r?

Option D

Thanks,
Clifin J Francis,
GMAT SME

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For statement 1, what if r=s? This condition is not mentioned in the question so we can assume this too, right?
Hence, marked B.

From (1) we get that \(\frac{r}{s}=\frac{3}{4}\), so r ≠ s (if r = s were true than r/s would be 1).

could we bring both to one side and do the following

s/r - r/s > 0?

is (s^2-r^2) / (rs) > 0 ?

for this fraction to be greater than 0, we need to have the numerator as a positive value, therefore, is s^2 > r^2? and since we know both are greater than 0 we can ask is s>r?

is this a valid way of doing this question?

is this a valid way of this doing this?

Yes, the approach you've mentioned is indeed valid. You could also apply the same reasoning and do the following:

If r > 0 and s > 0, is r/s < s/r?

Since both r and s are positive, we can safely cross multiply to obtain: is r^2 < s^2. Taking the square root results in: is |r| < |s|. Since r and s are both positive, this simplifies to: is r < s?

(1) r/(3s) = 1/4

From this equation, we can deduce that r/s = 3/4. Given that r and s are both positive, it follows that r < s. This means the answer to the question is YES. Sufficient.

(2) s = r + 4.

This equation directly tells us that s > r. As a result, the answer to the question is YES, making this statement sufficient as well.

Answer: D.

Answer in photo below: