Let the required number of bottles to be filled in 5 hours be x
Since, more number of bottles, more number of hours would be required.
\(\therefore\) The given quantities very directly
\(\therefore \ \frac { 540 }{ x } =\frac { 6 }{ 5 } \Rightarrow 6\times x=5\times 540\)
\(\Rightarrow \ x=\frac { 5\times 540 }{ 6 } =5\times 90=450\)
Thus, the required number of bottles = 450

It is given that, a water tank casts a shadow 21 m long and a tree of height 9.5 m casts a shadow 8 m long at the same time. Here the lengths of the shadows are directly proportional to their heights.
Let height = h and the length of shadow = l
Here, h1 = 9.5 m, l₁= 8 m, h2 = X, l₂ = 21 m
For direct proportion,
\(\frac{h_1}{l_1}=\frac{h_2}{l_2} \Rightarrow \frac{9.5}{8}=\frac{x}{21}\)
∴ x = \(\frac{9.5\times 21}{8}=24.9\)m
∴ The height of the tank is 24.9 m.

When x = 20,y= 40, then
\(\frac { x }{ y } =\frac { 20 }{ 40 } =\frac { 20\div 20 }{ 40\div 20 } \) [.: HCF of 20 and 40 = 20]
\(\Rightarrow \frac { x }{ y } =\frac { 1 }{ 2 } \)
When x = 17,y = 34, then
\(\frac { x }{ y } =\frac { 17 }{ 34 } =\frac { 17\div 17 }{ 34\div 17 } =\frac { 1 }{ 2 } \) [HCF of17 and 34 = 17]
When x = 14,y = 28, then
\(\frac { x }{ y } =\frac { 14 }{ 28 } =\frac { 14\div 14 }{ 28\div 14 } =\frac { 1 }{ 2 } \) [HCF of 14 and 28 = 14​​​​​​​]
When x = 11,y = 22, then​​​​​​​
\(\frac { x }{ y } =\frac { 11 }{ 22 } =\frac { 11\div 11 }{ 22\div 11 } =\frac { 1 }{ 2 } \) [HCF of 11 and 22=11]​​​​​​​
When x = 8,y = 16,then​​​​​​​
\(\frac { x }{ y } =\frac { 8 }{ 16 } =\frac { 8\div 8 }{ 16\div 8 } =\frac { 1 }{ 2 } \) [HCF of 8 and 16=8]
When x = 5,y = 10,then​​​​​​​
\(\frac { x }{ y } =\frac { 5 }{ 10 } =\frac { 5\div 5 }{ 10\div 5 } =\frac { 1 }{ 2 } \) [HCFof 5and10=5]
When x = 2,y = 4, then​​​​​​​
\(\frac { x }{ y } =\frac { 2 }{ 4 } =\frac { 2\div 2 }{ 4\div 2 } =\frac { 1 }{ 2 } \) [HCF of 2 and 4 = 2]
From above,we can say that the values of​​​​​​​ \(\frac { x }{ y } \) is same​​​​​​​,in everycaseit is​​​​​​​ \(\frac { 1 }{ 2 } \) So, these values of x and y are directly proportional​​​​​.

Few more examples for direct proportion are as follows:
(i) Length of the cloth purchased and its total cost.
(ii) Number of months and total salary.
(iii) Number of hours of production and the amount of the commodity produced

More is the number of persons, less is the time to complete the task.
\(\therefore\) It is a case of inverse variation,
Now, we have:

\(\therefore \ 5\times 4=4\times x\Rightarrow x=\frac { 5\times 4 }{ 4 } =5\)
Thus, the required number of days = 5.

448 persons

As number of persons increases, number of days will decreases for finishing the same amount of flour.
This is a case of inverse proportion.
Let number of persons = n
Number of days = d
\(n\propto \frac { 1 }{ d } \)
Here, n₁ = 300, d₁ = 42 days.
n₂ = 300 + 20 = 320, d₂ = ?
300 \(\times\) 42 = 320 \(\times\) d₂
\({ d }_{ 2 }=\frac { 300\times 42 }{ 320 } =\frac { 315 }{ 8 } =39\frac { 3 }{ 8 } days\)

Let the number of parts of red pigment be x and the amount of base = y mL
Here, x₁ = 1,y₁ = 75 and y₂ =  1800, x₂ = ?
\(\Rightarrow \frac { 1 }{ 75 } =\frac { { x }_{ 2 } }{ 1800 } \Rightarrow { x }_{ 2 }\times 75=1\times 1800\)
\({ x }_{ 2 }=\frac { 1800 }{ 75 } =24\Rightarrow { x }_{ 2 }=24\)
Hence, 24 parts of red pigment should be mixed with 1800 mL of base.

45 minutes

2 hours

4 days

14 km

Let the desired time to fill the tank be x minutes. Thus, we have the following table.

Lesser the number of pipes, more will be the time required by it to fill the tank. So, this is a case of inverse proportion.
Hence, 80 × 6 = x × 5 [x₁ y₁ = x₂ y₂ ]
or \(\frac{80 \times 6}{5}=x\) 
or x = 96
Thus, time taken to fill the tank by 5 pipes is 96 minutes or 1 hour 36 minutes.

x = 6

Direct

24 sweets

Change in radius = 10cm

No

40 x 16 = ? x 10 ⇒ ? = 64.