yvonne12 wrote:In a certain bathtub, both the cold water and the hot water fixtures leak. The cold water leaks alone and would fill an empty bucket in c hours, and the hot water leaks alone would fill the same bucket in H hours, where c<h. If both fixtures began to leak at the same time in the empty bucket at their respective constant rate and consequently, it took T hours to fill the bucket, which of the following must be true?

1. 0<T<H

2. C<T<H

3. C/2< T<H/2

Using rate formula,

1/C + 1/H = 1/T

1 - T has to be > 0 and < H. Combined time cannot exceed H or C

2 - Not true. Just take C=1 and H=2.
We get 1/1 + 1/2 = 1/(2/3), T=2/3
Combined rate cannot be greater than individual rates

3 - True/ Using the above rate-formula,
T = CH/(C+H)

T = C/2 = H/2 when C=H

Since we know C < H, T > C/2 and T < H/2

Hence 1 and 3 are correct.

In 1 hour the 2 fixtures will fill 1/H + 1/C of the bathtub = C + H / HC

They will fill the bathtub in HC / C + H hours

So T = C+H/HC

Since T represents the total hours taken by the 2 fixtures to fill the bathtub together; for sure T<H and T<C. Hence eliminate Choice 2. Choice 1 for sure is correct. T will always be less than H and always greater than 0 (for filling the tub some time would be required, it can't be 0).

For Statement III just try and substitute 2 pair of random figures for C and H (where C is not equal to H and C<H; say 3,4 or 5,7). It will hold true in all cases. Hence Statement I and III will hold true.

thanks