Problem Solving

if 0<r<1<s<2 which of the following must be less than 1?

a. r/s
b. rs
c. s-r

A a only
B b only
C c only
D a and b only
E a and c only



Please explain

ANS : D

Its A

Hi Brent ,

In B) rs

IF i test the value like:

r=0.5 and s=1.5 then rs will less than 1(0.75)

and
r=0.7 and s=1.9 then rs will be greater than 1(1.33)

so we will remove this option right? because it's not giving unique information.

Pls correct me if i am wrong.

anksm22 wrote:if 0<r<1<s<2 which of the following must be less than 1?

a. r/s
b. rs
c. s-r

A a only
B b only
C c only
D a and b only
E a and c only

0 < r < 1.
1 < s < 2.

An alternate approach is to perform the 3 operations -- r/s, rs, and s-r -- using EVERY COMBINATION OF ENDPOINTS.
Here, there are four endpoints to consider:
lower limit for r = 0.
upper limit for r = 1.
lower limit for s = 1.
upper limit for s = 2.

a: r/s
(lower limit for r)/(lower limit for s) = 0/1 = 0.
(lower limit for r)/(upper limit for s) = 0/2 = 0.
(upper limit for r)/(lower limit for s) = 1/1 = 1
(upper limit for r)/(upper limit for s) = 1/2 = 1/2.

The smallest result -- 0 -- is the lower limit for r/s.
The greatest result -- 1 -- is the upper limit for r/s.
Thus:
0 < r/s < 1.

Since r/s must be less than 1, the correct answer choice must include a.
Eliminate B and C.

b: rs
(lower limit for r)/(lower limit for s) = 0*1 = 0.
(lower limit for r)/(upper limit for s) = 0*2 = 0.
(upper limit for r)/(lower limit for s) = 1*1 = 1
(upper limit for r)/(upper limit for s) = 1*2 = 2.

The smallest result -- 0 -- is the lower limit for rs.
The greatest result -- 2 -- is the upper limit for rs.
Thus:
0 < rs < 2.

Since rs does NOT have to be less than 1, the correct answer choice cannot include b.
Eliminate D.

c: s-r
(lower limit for s)/(lower limit for r) = 1-0 = 1.
(lower limit for s)/(upper limit for r) = 1-1 = 0.
(upper limit for s)/(lower limit for r) = 2-0 = 2.
(upper limit for s)/(upper limit for r) = 2-1 = 1.

The smallest result -- 0 -- is the lower limit for s-r.
The greatest result -- 2 -- is the upper limit for s-r.
Thus:
0 < s-r < 2.
Since s-r does NOT have to be less than 1, the correct answer choice cannot include c.
Eliminate E.

The correct answer is A.