Quantity $\mathrm{I} :$$15$ men can complete $1$ work in $16$ $\mathrm{days}$ working $4$ hours a day. How many men are required to complete same amount of work in $15$ $\mathrm{days}$ working $8$$\mathrm{hours}$ a day?

Quantity $\mathrm{II} :$ A tank is filled with water using $2$ pipes  A, and B. Pipe A and B can fill the swimming pool completely in $5$ $\mathrm{hours}$ and $3 \mathrm{hours}$ respectively. If a pipe C empties the tank in $6 \mathrm{hours}$, find the total time taken to fill the tank completely if Pipes A, B and C are opened alternatively for $1$ $\mathrm{hour}$ each :

Quantity $\mathrm{III} :$X takes $\frac{5}{2}$ $\mathrm{days}$ to complete ${\frac{1}{7}}^{\mathrm{th}}$ of a work whereas, Y takes $\frac{7}{3}$ $\mathrm{days}$ to complete ${\frac{1}{5}}^{\mathrm{th}}$ of the same work. How much time do they take to complete the work together?

$\mathrm{Quantity} \mathrm{I}=18 {\mathrm{x}}^3{\mathrm{y}}^3, \mathrm{Quantity} \mathrm{II}=12 {\mathrm{x}}^4{\mathrm{y}}^4, \mathrm{if} \mathrm{x}>0 \& \mathrm{y}<0$

Find the relation between these two quantities.

The speed of a boat in still water and the speed of the current is in ratio $5:1$. If the difference between distances covered by the boat in going $2 \mathrm{hours}$ upstream and $2 \mathrm{hours}$  downstream is $8 \mathrm{km}$.

I. Quantity I - Speed of boat in still water

II. Quantity II -A cyclist goes $30 \mathrm{km}$ in $3 \mathrm{hours}$ 

Find the relation between these two quantities.

Consider the following statements:
(i) When $0<\mathrm{x}<1$, then $\frac{1}{1+\mathrm{x}}<1-\mathrm{x}+{\mathrm{x}}^2$.

(ii) When $0<\mathrm{x}<1$, then $\frac{1}{1+\mathrm{x}}>1-\mathrm{x}+{\mathrm{x}}^2$.

(iii) When $-1<\mathrm{x}<0$, then $\frac{1}{1+\mathrm{x}}<1-\mathrm{x}+{\mathrm{x}}^2$.

(iv) When $-1<\mathrm{x}<0$, then $\frac{1}{1+\mathrm{x}}>1-\mathrm{x}+{\mathrm{x}}^2$.

Then the correct statements are

Consider the following inequalities:

$$\begin{gathered} 1. \frac{a^2-b^2}{a^2+b^2}>\frac{a-b}{a+b}, \text{where} a>b>0 \\ 2. \frac{a^3+b^3}{a^2+b^2}>\frac{a^2+b^2}{a+b}, \text{where} a>b>0 \end{gathered}$$

Which of the above is/ are correct?

Two equations are given, you have to solve the both the equations:

$\left(i\right) X^2=196$

$\left(ii\right) Y^2+2y-48=0$

I.$(7{)}^{450}>(9{)}^{300}$

II .${\left(2\right)}^{900}>{\left(7\right)}^{450}$