Each of the questions given below consists of two statements numbered I and II given below it. Please read the questions carefully and decide whether the data provided in the statement(s) is / are sufficient to answer the given question.

1)

Two taps $A$ and $B$ , when opened together, can fill a tank in $6$ hours. How long will it take for the pipe $A$ alone to fill the tank ?

I. $B$ alone take $5$ hours more than $A$ to fill the tank.

II. The ratio of the time taken by $A$ to that taken by $B$ to fill the tank is $2:3$

A) I alone sufficient while II alone not sufficient to answer

B) II alone sufficient while I alone not sufficient to answer

C) Either I or II alone sufficient to answer

D) Both I and II are not sufficient to answer

E) Both I and II are necessary to answer

Option C

### Explanation:

$(A+B)$’s $1$ hour filling work $=\frac{1}{6}$

I. Suppose $A$ takes $x$ hours to fill the tank.

Then, $B$ takes $(x+5)$ hours to fill the tank.

$\therefore$  ($A$’s $1$ hour work) + ($B$’s $1$ hour work) = $(A+B)$’s $1$ hour work

$\Leftrightarrow\frac{1}{x}+\frac{1}{(x+5)}$  $=\frac{1}{6}$

$\Leftrightarrow\frac{(x+5)+x}{x(x+5)}$ $=\frac{1}{6}$

$\Leftrightarrow x^{2}-5x$ $=12x+30$

$\Leftrightarrow x^{2}-7x-30=0$

$\Leftrightarrow x^{2}-10x+3x-30$ $=0$

$\Leftrightarrow x(x-10)+3(x-10)$ $=0$

$\Leftrightarrow (x-10)(x+3)$ $=0$

$\Leftrightarrow x=10$.

So, $A$ alone takes 10 hours to fill the tank.

II. Suppose $A$ takes $2x$ hours and $B$ takes $3x$ hours to fill the tank. Then,

$\frac{1}{2x}+\frac{1}{3x}=\frac{1}{6}$

$\Leftrightarrow\left(\frac{1}{2}+\frac{1}{3}\right).\frac{1}{x}$ $=\frac{1}{6}$

$\Leftrightarrow\frac{5}{6x}$ $=\frac{1}{6}$

$\Leftrightarrow x=5$.

So, $A$ alone takes $(2\times 5)=10$ hours to fill the tank.

Thus, each one of I and II gives the answer.

$\therefore$ the correct answer is (C).