From the table itself we observe that as x increases y decreases
 \(\therefore\) x and y are inversely proportional
\(\therefore\) xy = 8 x 32 = 16 x 16 = 32 x 8 = 256 x 1 = 256

Time taken to cover 90 km = 2hrs 30mins
= 150 minutes [\(\begin{matrix} 1hour=60minutes \\ 2hour=120mnutes \end{matrix}\)]
Time taken to cover 1 km = \(\frac { 150 }{ 90 } \) x 210 minutes
= 350 minutes
= 5 hours 50 minutes
Thus, the time taken to travel 210 km is 5 hours 50 minutes.

Using unitary method we can solve this as follows:
The cost of 2 notebooks = Rs. 24
The cost of 1 notebook = \(\frac { 24 }{ 2 } \) = Rs.12
Therefore, the cost of 9 notebooks = 9 x Rs.12
= Rs.108
Hence, Anbu has to pay Rs.108 for 9 notebooks.

Let x be the required number of charts.
As weight increases, the number of charts also increases. So the quantities are in direct proportion.
Hence \(\frac { { x }_{ 1 } }{ { y }_{ 1 } } =\frac { { x }_{ 2 } }{ { y }_{ 2 } } \)

\(\frac { 15 }{ 50 } =\frac { x }{ 2500 } \)
15 x 2500 = x \(\times\) 50
x \(\times\) 50 = 15 x 2500
x = \(\frac { 15\times 2500 }{ 50 } \) = 750
Therefore, 750 charts will weigh 2\(\frac {1}{2}\) kilogram.

Let x be the number of pencils required for 18 children. As the number of children increases, number of pencils also increases.
In the case of direct proportion we take \(\frac { { x }_{ 1 } }{ { y }_{ 1 } } =\frac { { x }_{ 2 } }{ { y }_{ 2 } } \)

\(\frac { 6 }{ 24 } =\frac { 18 }{ x } \)
\(\frac { 6\times x }{ 24 } \) = 18
x = \(\frac { 18\times 24 }{ 6 } \) = 72
Hence, 72 pencils are required for 18 children.

To weave 25 baskets time taken = 35days
\(\therefore\) To weave 1 basket time taken = \(\cfrac { 35 }{ 25 } days\)

Length of the road = 600m
\(\cfrac { 2 }{ 3 } \) rd of the road = \(\cfrac { 2 }{ 3 } \times 600m=400m\)
In \(2\cfrac { 1 }{ 2 } \) hrs sumathi sweeps 600m
\(\therefore\) In 1 hr sumathi Sweeps \(\cfrac { 600 }{ 2\frac { 1 }{ 2 } } m=\cfrac { 600 }{ \frac { 5 }{ 2 } } m=120\times 2=240m\)
In \(1\frac { 1 }{ 2 } \) hrs Ramani sweeps 400 m.
\(\therefore\) In 1hr Ramani sweeps \(\cfrac { 400 }{ 1\frac { 1 }{ 2 } } m=\cfrac { 400 }{ 3 } m=400\times \cfrac { 2 }{ 3 } m=\cfrac { 800 }{ 3 } =266.66m\)
\(\therefore\) Ramanl sweeps more. in 1 hr
\(\therefore\) Ramani sweeps speedily.

The cost of one box = Rs. 200
Increased cost of one box = Rs. 260
Let x be the number of boxes bought by Vendan.
As cost of boxes increases the number of boxes decreases.
This is in inverse proportion. Therefore, x₁y₁ = x₂ y₂

13 x 200 = x \(\times\) 260
x x 260 = 13 x 200
x = \(\frac { 13\times 200 }{ 260 } \)
x = 10
Therefore, he can buy 10 boxes for the same amount.

Let x be the required number of days. The decrease in number of workers lead to the increase in number of days. (Therefore, both are in inverse proportion)
For inverse proportion x₁y₁ = x₂ y₂

Hence 60 x 7 = 42 \(\times\) x
42 \(\times\) x = 60 x 7
x = \(\frac { 60\times 7 }{ 42 } \)
x = 10
In 10 days 42 workers can spin a bale of cotton.