Given:\(\cfrac { 9 }{ 20 } ,\cfrac { 3 }{ 4 } ,\cfrac { 7 }{ 12 } \)
LCM of 20, 4, 12 is 60
\(\cfrac { 9 }{ 30 } \times \cfrac { 3 }{ 3 } =\cfrac { 27 }{ 60 } ,\cfrac { 3 }{ 4 } \times \cfrac { 15 }{ 15 } =\cfrac { 45 }{ 60 } ,\cfrac { 7 }{ 12 } \times \cfrac { 5 }{ 5 } =\cfrac { 35 }{ 60 } \)
\(\Rightarrow \cfrac { 27 }{ 60 } <\cfrac { 35 }{ 60 } <\cfrac { 45 }{ 60 } \Rightarrow \cfrac { 45 }{ 60 } >\cfrac { 35 }{ 60 } >\cfrac { 27 }{ 60 } \)
\(\therefore\)  Descending order is \(\cfrac { 3 }{ 4 } ,\cfrac { 7 }{ 12 } ,\cfrac { 9 }{ 20 } \)
(i) LCM of 3 and 7 is 21
 \(\Rightarrow \cfrac { 2 }{ 3 } \times \cfrac { 7 }{ 7 } +\cfrac { 5 }{ 7 } \times \cfrac { 3 }{ 3 } \)
\(\cfrac { 14 }{ 21 } +\cfrac { 15 }{ 21 } =\cfrac { 14+15 }{ 21 } =\cfrac { 29 }{ 21 } \)
(ii) LCM of5 and 8 is 40
\(\Rightarrow \cfrac { 3 }{ 5 } \times \cfrac { 8 }{ 8 } -\cfrac { 3 }{ 8 } \times \cfrac { 5 }{ 5 } \)
= \(\cfrac { 24 }{ 60 } -\cfrac { 15 }{ 40 } =\cfrac { 24-15 }{ 40 } =\cfrac { 9 }{ 40 } \)

Given: \(\cfrac { 3 }{ 5 } ,\cfrac { 9 }{ 10 } ,\cfrac { 11 }{ 15 } \)
LCM of 5; 10 and 15 is 30.
\(\cfrac { 3 }{ 5 } \cfrac { 6 }{ 6 } \times \cfrac { 18 }{ 30 } ,\cfrac { 9 }{ 10 } \times \cfrac { 3 }{ 3 } =\cfrac { 27 }{ 30 } ,\cfrac { 11 }{ 15 } \times \cfrac { 2 }{ 2 } =\cfrac { 22 }{ 30 } \)
\(\Rightarrow \cfrac { 18 }{ 30 } <\cfrac { 22 }{ 30 } <\cfrac { 27 }{ 30 } \)
\(\therefore\) Ascending order :\(\cfrac { 3 }{ 5 } ,\cfrac { 11 }{ 15 } ,\cfrac { 9 }{ 10 } \)

Given:\(\cfrac { 3 }{ 8 } ,\cfrac { 3 }{ 5 } \)
L. C. M of 8 and 5 is 40
\(\cfrac { 3 }{ 8 } \times \cfrac { 5 }{ 5 } =\cfrac { 15 }{ 40 } ,\cfrac { 3 }{ 5 } \times \cfrac { 8 }{ 8 } =\cfrac { 24 }{ 40 } \)
\(\therefore \ \cfrac { 24 }{ 40 } >\cfrac { 15 }{ 40 } \Rightarrow \cfrac { 3 }{ 5 } >\cfrac { 3 }{ 8 } \)
\(>\cfrac { 3 }{ 8 } \\ \) is greater

(i) \(\cfrac { 1 }{ 3 } \) is greater than \(\cfrac { 1 }{ 5 } \)
That is \(\cfrac { 1 }{ 3 } \ge \cfrac { 1 }{ 5 } \)
(ii) \(\cfrac { 2 }{ 5 } \) is smaller \(\cfrac { 5 }{ 8 } \)
That is \(\cfrac { 2 }{ 5 } \le \cfrac { 1 }{ 5 } \)

Length of the staircase = \(5\cfrac { 1 }{ 2 } \) m
To one step = \(\cfrac { 1 }{ 4 } \) m
No. of the steps will be there = \(\cfrac { 5\frac { 1 }{ 2 } }{ \frac { 1 }{ 4 } } \)
= \(\cfrac { \frac { 11 }{ 2 } }{ \frac { 1 }{ 4 } } \Rightarrow \cfrac { 11 }{ 2 } \times \cfrac { 4 }{ 1 } =22\)
22 steps will be there in the staircase.

Mangai bought = \(6\cfrac { 3 }{ 4 } kg\) of apple 
Kalai bought = \(1\cfrac { 1 }{ 2 } \) times as Mangai bought
= \(\left[ 1\cfrac { 1 }{ 2 } \times 6\cfrac { 3 }{ 4 } \right] kg\) of apples

= \(\cfrac { 3 }{ 2 } \times \cfrac { 27 }{ 4 } \)
= \(\cfrac { 81 }{ 8 } \)
\(\therefore \) Kalai bought = \(10\cfrac { 1 }{ 8 } \) kg of apples.

Difference between \(2\cfrac { 1 }{ 2 }\ and\ 3\cfrac { 1 }{ 2 } \)
Difference = \(3\cfrac { 2 }{ 3 } -2\cfrac { 1 }{ 2 } \)
= \(\cfrac { 11 }{ 3 } -\cfrac { 5 }{ 2 } \)
= \(\cfrac { \left( 11\times 2 \right) -\left( 5\times 3 \right) }{ 6 } =\cfrac { 22-15 }{ 6 } \)
= \(\cfrac { 7 }{ 6 } \) ...(1)
Also given
Sum of \(1\cfrac { 1 }{ 2 }\ and\ 2\cfrac { 1 }{ 4 } \)
Sum =\(1\cfrac { 1 }{ 2 } +2\cfrac { 1 }{ 4 } \)
= \(\cfrac { 3 }{ 2 } +\cfrac { 9 }{ 4 } \)
= \(\cfrac { \left( 3\times 2 \right) +9 }{ 4 } =\cfrac { 6+9 }{ 4 } =\cfrac { 15 }{ 4 } \)
From (1) and (2),
Difference = \(\cfrac { 7 }{ 6 } \), Sum = \(\cfrac { 15 }{ 4 } \)
\(\cfrac { 7 }{ 6 } ,\cfrac { 15 }{ 4 } \)
\(\Rightarrow\) Lcm of 6 and 4 is 12
\(\cfrac { 7 }{ 6 } \times \cfrac { 2 }{ 2 } =\cfrac { 14 }{ 12 } ,\cfrac { 15 }{ 4 } \times \cfrac { 3 }{ 3 } =\cfrac { 45 }{ 12 } \)
\(\cfrac { 14 }{ 12 } <\cfrac { 45 }{ 12 } \Rightarrow \cfrac { 7 }{ 6 } <\cfrac { 15 }{ 4 } \)
\(\therefore \) The difference between \(2\cfrac { 1 }{ 2 }\ and\ 3\cfrac { 2 }{ 3 } \)

Total distance = \(5\cfrac { 3 }{ 4 } km=\cfrac { 23 }{ 4 } km\)
He had walked = \(2\cfrac { 1 }{ 2 } km=\cfrac { 5 }{ 2 } km\)
Difference = \(\left( \cfrac { 23 }{ 4 } -\cfrac { 5 }{ 2 } \right) km\)
= \(\cfrac { 23-\left( 5\times 2 \right) }{ 4 } =\cfrac { 23-10 }{ 4 } =\cfrac { 13 }{ 4 } km\)
\(\therefore 3\cfrac { 1 }{ 4 } km\) distance still he has to walk to reach his house

Sankari purchased
To stitch a long skirt = \(2\cfrac { 1 }{ 2 } \ m\ cloth=\cfrac { 5 }{ 2 } m\)
To Stitch blouse = \(1\cfrac { 3 }{ 4 } m=\cfrac { 7 }{ 4 } m\)
Total length of clothe = \(\cfrac { 5 }{ 2 } m+\cfrac { 7 }{ 4 } m|\)
= \(\cfrac { (5\times 2)+7 }{ 4 } =\cfrac { 10+7 }{ 4 } \)
= \(\cfrac { 17 }{ 4 } m\)
Cost of 1 m cloth = Rs. 120
\(\therefore \cfrac { 17 }{ 4 } m\ cloth=120\times \cfrac { 17 }{ 4 } \)
= Rs. 510
Cost of cloth purchased by Sankari =  Rs. 510