If r > 0 and S > 0, Is r/s < s/r?

Is \(\frac{r}{s}<\frac{s}{r}\)?

(1) \(\frac{r}{3s}=\frac{1}{4}\):

Re-write as: \(\frac{r}{s}=\frac{3}{4}\);

The above, on the other hand, can also be written as: \(\frac{s}{r}=\frac{4}{3}\);

So, we have that \((\frac{r}{s}=\frac{3}{4})<(\frac{4}{3}=\frac{s}{r})\). Thus the answer to the question is YES. Sufficient.

(2) \(s=r+4\);

This implies that \(s>r\). Since given that \(r>0\), then \(s>r>0\). And finally, \(s>r>0\) means that \(\frac{s}{r}>1>\frac{r}{s}\). Thus the answer to the question is YES. Sufficient.

Answer: D.
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Since r and s are both positive, we can simplify the inequality to \(r^2 < s^2\), and by taking the square root of both sides, \(r < s\).

(1) \(4r = 3s\)
\(r = \frac{3}{4}s\)
So r < s. Sufficient.

(2) Since \(r = s - 4\), \(r < s\). Sufficient.

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.
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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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Since r>0 and s>0 that means we can safely multiply or divide across an inequality and know that the direction will not change. So let's start with rephrasing the question stem to is R^2<S^2, and now we can take the square root and don't need to consider both positive and negative scenarios in taking the sqrt because we are given that both r and s > 0, so now the question is if R<S?

(1) If r/3s = 1/4 then that means that 4r = 3s. So let's again rephrase the stem to be is 4R<4s?. Now we can compare 3s to 4s and of course then if S and R are positive 4s>3s, so suffecient.
(2) Simply plug s into our rephrased stem so now we have is R < R+4, and again since r and s are greater than 0 this is sufficient.

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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Correct Option D

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

(1) r/(3s) = 1/4
(r/s)=(3/4)= 0.75 and reverse of it (s/r)=(4/3)=1.33
clearly 0.75<1.33, thus (r/s)<(s/r) - Sufficient

(2) s = r + 4
Statement says, s>r by 4 value and s and r both are integers
now (r/s) = Numerator<Denominator, value will be in decimals
and (s/r) = Numerator>Denominator, value will be more than 1
hence (r/s)<(s/r) - Sufficient

learning: understand relationship between both the variables, as in this case with (r) and (s),
while they are in Numerator and Denominator

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

(2) s = r + 4
Since s > 0 , r > 0
Say s = 5, r becomes 9
We can get r/s < s/r given both r or s.
Sufficient

Answer (D)

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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).
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How would the answer change if r and s could also be negative?
Would someone show me a step-by-step solution?

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