Each of these questions followed by three statements. Please study the questions and decide which of the statement(s) is/are necessary to answer the question.

1)

A tank is fitted with two taps $A$ and $B$. In how much time will the tank be full if both the taps are opened together ?

I. $A$ is $50\%$ more efficient than $B$.

II. $A$ alone takes $16$ hours to fill the tank.

III. $B$ alone takes $24$ hours to fill the tank.

A) II and III only

B) All I, II and III

C) I and II only

D) I and III only

E) Any two of the three

Option E

### Explanation:

II. $A$’s $1$ hour work $=\frac{1}{16}$.

Suppose $B$ fills the tank in $x$ hours. Then, $B$’s $1$ hour work $=\frac{1}{x}$.

I. Work done by $A$ in $1$ hour $=150\%$ of $\frac{1}{x}$ $=\left(\frac{1}{x}\times\frac{150}{100}\right)$  $=\frac{3}{2x}$.

$\therefore\frac{3}{2x}=\frac{1}{16}$

$\Leftrightarrow x=24$.

So, $B$ can fill the tank in 24 hours.

$(A+B)$’s $1$ hour work $=\left(\frac{1}{16}+\frac{1}{24}\right)$ $=\frac{5}{48}$.

$\therefore$ $(A+B)$ can fill the tank in $\frac{48}{5}$ hrs.

Thus I & II give the answer.

III. Work done by $B$ in $1$ hour $=\frac{1}{24}$.

From II & III, we get the same answer.

From III & I, we get :

$A$’s $1$ hour work $=150\%$ of $\frac{1}{24}$ $=\left(\frac{1}{24}\times\frac{150}{100}\right)$ $=\frac{1}{16}$.

Thus, III & I, we get the answer.

$\therefore$  Correct answer is (E).