There are three different hoses used to fill a pool: hose...

LUANDATO wrote:There are three different hoses used to fill a pool: hose x, hose y, and hose z. Hose x can fill the pool in a days, hose y in b days, and hose z in c days, where a > b > c. When all three hoses are used together to fill a pool, it takes d days to fill the pool. Which of the following must be true?

I) d < c
II) d > b
III) c/3 < d < a/3

A) I only
B) III only
C) I and III only
D) II only
E) I, II and III

The OA is C.

Two of the three statements are fairly straightforward. Clearly the hoses together will fill the pool more quickly than the hoses individually, and thus the time together, or d, will be smaller than any of the individual times. Thus I is true and II is false.

We also know that if each of the three hoses took c days to complete the job, then together they'd have taken c/3 days. But, in actuality, two of the three hoses took more than c days, so together they'd take more than c/3 days, and thus d > c/3.

Similarly, if each of the three hoses took a days to complete the job, together, they'd have taken a/3 days. But two of the three hoses took fewer than a days, so together they'd take less than a/3 days and d < a/3.

Taken together, we get statement III - c/3 < d < a/3. Thus, I and III are true. The answer is C

LUANDATO wrote:There are three different hoses used to fill a pool: hose x, hose y, and hose z. Hose x can fill the pool in a days, hose y in b days, and hose z in c days, where a > b > c. When all three hoses are used together to fill a pool, it takes d days to fill the pool. Which of the following must be true?

I. d<c
II. d>b
III. c/3<d<a/3

A) I only
B) III only
C) I and III only
D) II only
E) I, II and III

Roman numeral I is true. When all three hoses work together, the number of days it takes must be less than the number of days it takes for any individual hose to complete the job by itself. Using the same argument, we see that Roman numeral II can’t be true.

If each hose works as fast as hose z, it will take exactly c/3 days. However, since they do not, and since hose z is the fastest, they must take more than c/3 days. That is, d > c/3. Similarly, if each hose works as fast as hose x, it will take exactly a/3 days. However, since they do not, and since hose x is the slowest, they must take less than a/3 days. That is, d < a/3. We see that c/3 < d < a/3, which means Roman numeral III is true also.

Answer: C