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31. The cost of levellling and turfing a square cricket field at Rs 16000 per hectare is Rs 262440. Find the cost of surrounding it with a railing costing Rs 25 per metre.
  A.  Rs 41,500
  B.  Rs 40,500
  C.  Rs 42,300
  D.  Rs 42,300
     
   
View Answer

Shortcut:
To find the the diagonal and the perimeter of a square if its area is given.
(i) Length of diagonal of a square = √(2 x area)
(ii) Perimeter of a square = √(16 x area)
Here, Area =
262440 / 16000
= 16.4025 hectare = 164025 sq m
Perimeter = √16 x 164025 = 1620 m
∴ required cost = 1620 x 25 = Rs 40500

32. The ratio of the area of the two squares is 16:9. Find the ratio of their sides, ratio of their perimeters and the ratio of their diagonals.
  A.  Sides ratio= 4:3; Perimeters ratio=4:3; Diagonals ratio= 4:3
  B.  Sides ratio= 2:3; Perimeters ratio=5:3; Diagonals ratio= 4:3
  C.  Sides ratio= 2:3; Perimeters ratio=7:3; Diagonals ratio= 2:5
  D.  Sides ratio= 3:5; Perimeters ratio=2:9; Diagonals ratio= 5:4
     
   
View Answer

Shortcut:
If the ratio of the areas of square A and B is a : b, then
(i) the ratio of their sides = √a : √b
(ii) the ratio of their Perimeters = √a : √b
(iii) the ratio of their diagonals = √a : √b
Here, a = 16, b = 9
Required ratio = √16 : √9 = 4 : 3

33. There is a square of side 22 cm. Find the radius of the circle whose perimeter equals to the perimeter of the square.
  A.  10 cm
  B.  11 cm
  C.  8 cm
  D.  14 cm
     
   
View Answer

Shortcut:
If the perimeter of a square is equal to the perimeter of a circle, then the side of the square is π x
r / 2
and radius of the circle is
2y / π

Where y is the side of the square and r is the radius of the circle.
Here, y = 22
Using this value in the shortcut, we get:
Radius of the circle =
2 x 22 / 22/7
= 14
Hence, the radius of the circle is 14 cm.

34. There is a circle of radius 7 cm. Find the side of the square whose perimeter equals the perimeter of the circle.
  A.  9 cm
  B.  6 cm
  C.  11 cm
  D.  13 cm
     
   
View Answer

Shortcut:
If the perimeter of a square is equal to the perimeter of a circle, then the side of the square is π x
r / 2
and radius of the circle is
2y / π

Where y is the side of the square and r is the radius of the circle.
Here, r = 7
Using this value in the shortcut, we get:
The side of the square =
22 / 7
x
7 / 2
= 11
Hence, the side of the square is 11 cm.

35. Length of a square is increased by 8 cm. Its area becomes 208 sq cm. Find its perimeter.
  A.  108 cm
  B.  104 cm
  C.  106 cm
  D.  110 cm
     
   
View Answer

Shortcut:
If the side of a square is increased by 'm' units and its area becomes 'n' square units, the side of the square is given by
n / m
units, its area is given by
n2 / m2
sq units and its perimeter is 4 x
n / m
units.
[Note: If the side of a square is increased by 'm' units and its area increases by 'n' units then the side of the square is given by
1 / 2
[
n / m
− m
]
units.
Here, m = 8, n = 208
Using this value in the shortcut, we get:
The side of the square = 4 x
208 / 8
= 104
Hence, the perimeter is 104 cm.

36. A square room is surrounded by a verandah of width 2 metres. Area of the verandah is 64 sq meters. Find the area of the room.
  A.  39 sq m
  B.  32 sq m
  C.  36 sq m
  D.  30 sq m
     
   
View Answer

Shortcut:
A square room is surrounded by a verandah (on the outside of the square room) of width 'd' metres. If the area of the verandah is 'A' sq m, then the area of the room is
(
A − 4d2 / 4d
)
2 sq metres and obviously side of the square room is given by
A − 4d2 / 4d
metres.

Here, A = 64, d = 2
Using this value in the shortcut, we get:
The area of the room =
(
64 − 4 x 22 / 4 x 2
)
2
=
(
64 − 16 / 8
)
2 = (6)2 = 36
Hence, the area of the room is 36 sq m.

37. A square room has a verandah of area 64 sq m and width 2 m all round it on its inside. Find the area of the room.
  A.  95 sq m
  B.  105 sq m
  C.  85 sq m
  D.  100 sq m
     
   
View Answer

Shortcut:
If a square room has a verandah of area 'A' sq metres and width 'd' metres all round it on its inside, then the area of the room is
(
A + 4d2 / 4d
)
2 sq metres and obviously side of the square room is given by
A + 4d2 / 4d
metres

Here, A = 64, d = 2
Using this value in the shortcut, we get:
The area of the room =
(
64 + 4 x 22 / 4 x 2
)
2
=
(
64 + 16 / 8
)
2
=
(
80 / 8
)
2
= (10)2 = 100
Hence, the area of the room is 100 sq m.

38. A square field contains 2.89 hectares. It has to be fenced all-round and a path 10 m wide has to be laid out all round close to the fence inside. If the cost of fending is Rs 50 per m and the cost of preparing the path is Rs 10 per sq metre. Find the total expenses.
  A.  Rs 96,000
  B.  Rs 98,000
  C.  Rs 99,200
  D.  Rs 97,000
     
   
View Answer

Shortcut:
If a square room has a verandah of area 'A' sq metres and width 'd' metres all round it on its inside, then the area of the room is
(
A + 4d2 / 4d
)
2 sq metres and obviously side of the square room is given by
A + 4d2 / 4d
metres
Here, Area of the square = 2.89 hectares = 28900 sq m
Perimeter = √(16 x 28900) = 680 m
∴ Cost of fencing the square field = 680 x 50 = Rs 34000
Now, we use the shortcut
A = ?, d = 10
28900 =
(
A + 4 x 102 / 4 x 10
)
2
√(28900) =
(
A + 4 x 102 / 4 x 10
)

170 =
(
A + 400 / 40
)

170 x 40 = A + 400
A = 6800 − 400 = 6400 (Area of the path)
∴ Cost in preparing the path = 6400 x 10 = Rs 64000
∴ total expenses = Rs 34000 + Rs 64000 = Rs 98000

39. A room is 8 m long, 6 m broad and 3 m high. Find the area of the four walls of the room.
  A.  84 sq m
  B.  75 sq m
  C.  90 sq m
  D.  82 sq m
     
   
View Answer

Shortcut:
(i) To find the area of the four walls of a room, if its length, breadth and height are given.
Area of the four walls of a room = 2 x (Length + Breadth) x Height.
(ii) To find the height of a room, if area of four walls of the room and its length and breadth are given.
Height =
Area of four walls of the room / 2(Length + Breadth)
metres
Here, Length = 8, Breadth = 6, Height = 3
Using these values in the shortcut, we get:
Area of the four walls of a room = 2 x (8 + 6) x 3 = 84
∴ Area of four walls = 84 sq m

40. The cost of papering the walls of a room 12 m long at the rate of 45 paise per sq metre is Rs 113.4 and the cost of matting the floor at the rate of 35 paise per sq m is Rs 37.80. Find the height of the room.
  A.  5 m
  B.  4 m
  C.  6 m
  D.  8 m
     
   
View Answer

Shortcut:
(i) To find the area of the four walls of a room, if its length, breadth and height are given.
Area of the four walls of a room = 2 x (Length + Breadth) x Height.
(ii) To find the height of a room, if area of four walls of the room and its length and breadth are given.
Height =
Area of four walls of the room / 2(Length + Breadth)
metres
Here, Area of floor =
Total cost / Rate per sq m
=
3780 / 35
= 108 sq m.
∴ Breadth of the room =
Area of floor / Length of the room
=
108 / 12
= 9 m.
Now, area of four walls =
Total cost of papering / Rate per sq metre
=
11340 / 35
= 252 sq m.
Let the height of room be 'h' metres.
Then, 2 x (12 + 9) x h = 252
∴ h =
252 / 2 x 21
= 6 m

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