If 0<r<1<s<2, which of the following must be less than 1?
I. \frac{r}{s}
II. rs
III. sr
A. I only
B. II only
C. III only
D. I and II
E. I and III
Problem Solving Question
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I) r/s
r and s are both POSITIVE, and r is less than s.
So, it MUST be the case that r/s < 1
II) rs
It could be the case that r = 0.9 and s = 1.8, in which case rs = 1.62
So, it NEED NOT be the case that rs < 1
III) s  r
It could be the case that r = 0.1 and s = 1.8, in which case s  r = 1.7
So, it NEED NOT be the case that s  r < 1
Answer: A
Cheers,
Brent
 gmatknight
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The best way to solve this problem is to pick a set of variables that meet the conditions of R and S and play around with them. The question asks what MUST be less than 1. Use that statement as your anchor. Ask yourself if, with the variables you have chosen, if it is even possible to make an answer greater than 1.
I'd also recommend you use the answer choices to shorten your time, but it would probably be best to understand the concept at play here first for those unfamiliar with this type of question. (For example, if you know for certain that statement (1) and (3) must be less than 1, you don't need to worry about figuring out (2): the answer choices don't have all three choices as an option!)
I'd also recommend you use the answer choices to shorten your time, but it would probably be best to understand the concept at play here first for those unfamiliar with this type of question. (For example, if you know for certain that statement (1) and (3) must be less than 1, you don't need to worry about figuring out (2): the answer choices don't have all three choices as an option!)
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The best approach for a lot of algebra questions like these is to definitely to pick numbers and think logically.
I r/s we know 0< numerator < 1 as well as, 1 < denominator < 2 so the answer definitely be less than 1.
II rs choose r = 0.2 s = 1.1 rs = 0.22 < 1 so not the case
III s  r choose r = 0.9 s = 1.1 1.1  0.9 = 0.2 so < 1
Therefore, A is correct answer.
I r/s we know 0< numerator < 1 as well as, 1 < denominator < 2 so the answer definitely be less than 1.
II rs choose r = 0.2 s = 1.1 rs = 0.22 < 1 so not the case
III s  r choose r = 0.9 s = 1.1 1.1  0.9 = 0.2 so < 1
Therefore, A is correct answer.
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We are given that 0 < r < 1 < s < 2 and need to determine which Roman Numeral expressions must be less than 1. Let’s test each expression.
I. r/s
Since r is less than s and both r and s are positive, r/s will always be less than 1. Roman numeral I is correct.
II. rs
The product of r and s does not have to be less than 1. For example, if r = 0.8 and s = 1.5 the product of r and s will be (0.8)(1.5) = 1.2, which is greater than 1. Roman numeral II is not correct.
III. s – r
s – r does not have to be less than 1. For example, if s = 1.9 and r = 0.1, s – r = 1.8. Roman number III is not correct.
Answer: A