As animals increases, the number of days decreases. This is in inverse proportion.

Let the required number of days be t

20 x 6 = 30 x t

t = 4

So, the food will last for 4 days.

Let the required number of machines be denoted by 'n' and number of days taken be 'd'.

As number of machines increases, the number of days required to make articles decreases, hence they are inversely proportional.
$n\times d$ = constant
$n_1\times d_1$ = $n_2\times d_2$

Using inverse proportion condition, 

 $\frac{42}{n}$ = $\frac{54}{63}$

$n=\frac{42\times 63}{54}$ 

n = 49

So, 49 machines are required.
Hence, the statement is false.

Given, the number of bottles can be filled by the machine in 6 hours = 480
Hence number of bottles it can fill in 1 hour = $\frac{480}{6}$= 80 bottles
Hence the number of bottles that can be filled in 2 days = 80 $\times$  2 $\times$ 24       ($\because$ Number of hours in a day = 24) 
= 3840 bottles

Let h represents the height of the tower that casts 300 m long shadow.

$\frac{14}{h}=\frac{10}{300}$
10 x h = 14 x 300

h = 420 metres

Given, the number of bottles that can be filled by the machine in 6 hours is 840.

Hence, the number of bottles it can fill in 1 hour
$=\frac{840}{6}$
= 140 bottles

Hence number of bottles that can be filled in 5 hours
$=140\times 5$
= 700 bottles 

$\frac{Lengthofsquare}{Perimeterofsquare}=\frac{a}{4a}=\frac{1}{4}$
which is a constant.

$\frac{Lengthofsquare}{Areaofsquare}=\frac{a}{a^2}=\frac{1}{a}$
which is not constant. 
$\frac{Lengthofsquare}{Lengthofdiagonal}=\frac{1}{\sqrt{2}}$
which is constant.

Hence, length of a square is in direct proportion with its perimeter and diagonal.

As Raghu goes at same speed , $\frac{Distance}{Time}$ remains constant.

Assume that the distance covered in 4 hours be x.

10x = 4 x 60 x 2

$x=48km$.

Hence Raghu can reach city A in given time.

When $x$ and $y$ are in direct proportion, we can write $\frac{x_1}{y_1}=\frac{x_2}{y_2}$

[$y_1,y_2$ are values of $y$ corresponding to the values $x_1,x_2$ of $x$ respectively] which gives $x_1{y}_2=x_2{y}_1$.

If x and y are in inverse proportion, and $y_1$, $y_2$are values of y corresponding to the values $x_1$, $x_2$ of x respectively, then

          $x_1$$y_1$=$x_2$$y_2$

As the number of children increases, the number of sweets for each decreases. It is an inverse proportion.

Let the required number of sweets be x.

AB = constant
$A_1\times B_1$ = $A_2\times B_2$
$5\times 24$ = $x\times 20$
$\frac{5}{x}$ = $\frac{20}{24}$

$x$ = $\frac{5\times 24}{20}$ = 6

So, each child gets 6 sweets.