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81. A rectangular hall 12 m long and 10 m broad, is surrounded by a verandah 2 m wide. Find the area of the verandah.
  A.  110 sq m
  B.  105 sq m
  C.  104 sq m
  D.  101 sq m
     
   
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Shortcut:
If a rectangular hall 'y' metre long and 'z' metre broad, is surrounded by a verandah (on the outside to the rectangular hall) 'd' metre wide, then the area of the verandah is given by 2d[(y + z) + 2d] sq metre.
or Area of verandah = 2(width of verandah ) x [length + breadth of room + 2(width of verandah)]

Here, y = 12, z = 10, d = 2
Using these values in the shortcut, we get:
Area of the verandah = 2 x 2[(12 + 10) + 2 x 2] = 2 x 2[22 + 4]
= 2 x 2(26) = 2 x 52 = 104
Hence, the area of the verandah is 104 sq metres.

82. A rectangular garden has 5 m wide road outside around all the four sides. The area of the road is 600 sq m. What is the ratio between the length and the breadth of that plot?
  A.  2:3
  B.  3:5
  C.  1:2
  D.  Data inadequate
     
   
View Answer

Shortcut:
If a rectangular hall 'y' metre long and 'z' metre broad, is surrounded by a verandah (on the outside to the rectangular hall) 'd' metre wide, then the area of the verandah is given by 2d[(y + z) + 2d] sq metre.
or Area of verandah = 2(width of verandah ) x [length + breadth of room + 2(width of verandah)]
Here, y = ?, z = ?, d = 5
Using these values in the shortcut, we get:
Area of the verandah = 2 x 5[(y + z) + 2 x 5] 600 = 2 x 5[(y + z) + 2 x 5]
600 = 2 x 5[(y + z) + 10]
60 = (y + z) + 10 y + z = 60 − 10 = 50
Hence, the data is inadequate to find the ratio.

83. A 5m wide lawn is cultivated all along the outside of a rectangular plot measuring 90m x 40m. The total area of the lawn is:
  A.  1400 sq m
  B.  13400 sq m
  C.  1580 sq m
  D.  1200 sq m
     
   
View Answer

Shortcut:
If a rectangular hall 'y' metre long and 'z' metre broad, is surrounded by a verandah (on the outside to the rectangular hall) 'd' metre wide, then the area of the verandah is given by 2d[(y + z) + 2d] sq metre.
or Area of verandah = 2(width of verandah ) x [length + breadth of room + 2(width of verandah)]
Here, y = 90, z = 40, d = 5
Using these values in the shortcut, we get:
Area of the lawn = 2 x 5[(90 + 40) + 2 x 5] = 2 x 5[(130) + 10]
= 2 x 5[(140] = 1400
Hence, the area of lawn is 1400 sq metre.

84. A rectangular grassy plot is 112 m by 78m. It has a gravel path 2.5m wide all round it on the inside. Find the area of the path and the cost of constructing it at Rs 2 per sq m?
  A.  Rs 1620
  B.  Rs 1850
  C.  Rs 1750
  D.  Rs 1910
     
   
View Answer

Shortcut:
If a rectangular plot 'y' metre long and 'z' metre broad, has a gravel path 'd' metre wide all round it on the inside, then the area of the path is given by 2d[(y + z) − 2d] sq metre.
or Area of path = 2(width of gravel path) x [length + breadth of plot − 2(width of gravel path)]
Here, y = 112, z = 78, d = 2.5
Using these values in the shortcut, we get:
Area of the path = 2 x 2.5[(112 + 78) − 2 x 2.5] = 5[(190) − 5]
= 5 x 185 = 925
∴ cost of construction = rate x area = 2 x 925 = Rs 1850

85. A path all around the inside of a rectangular park 37m by 30m occupies 570 sq m. Find the width of the path.
  A.  2 m
  B.  1 m
  C.  6 m
  D.  5 m
     
   
View Answer

Shortcut:
If a path all around the inside of a rectangular park 'y'm x 'z' metre occupies 'A' sq m, then the width of the path is given by
(y + z) − √[(y + z)2 − 4A] / 4
metre.
Here, y = 37, z = 30, A = 570
Using these values in the shortcut, we get:
Width of the path =
(37 + 30) − √[(37 + 30)2 − 4 x 570] / 4

=
67 − √[(67)2 − 2280] / 4

=
67 − √[(67 x 67) − 2280] / 4

=
67 − √[4489 − 2280] / 4

=
67 − √(2209) / 4

=
67 − 47 / 4

=
20 / 4
= 5
∴ width of the path 5 metre.

86. A rectangular garden is 15 m long and 10 m broad. It has 3 m wide pavements all around it both on its inside and outside. Find the total area of the pavements.
  A.  300 sq m
  B.  260 sq m
  C.  330 sq m
  D.  315 sq m
     
   
View Answer

Shortcut:
A rectangular garden is 'y' metres long and 'z' metres broad. It is to be provided with pavements 'd' metres wide all round it both on its outside as well as inside. Then the total area of the pavement is given by 4d(y + z) sq metre.
Here, y = 15, z = 10, d = 3
Using these values in the shortcut, we get:
Total area of the pavement = 4 x 3(15 + 10)
= 12 x 25 = 300
Hence, the total area of the pavement is 300 sq metre.

87. A square garden is 10 m long. It has 3 m wide pavements all round it both on its inside and outside. Find the total area of the pavements.
  A.  250 sq m
  B.  260 sq m
  C.  240 sq m
  D.  200 sq m
     
   
View Answer

Shortcut:
A square garden is 'y' metres long. It is to be provided with pavements 'd' metres wide all round it both on its outside as well as inside. Then the total area of the pavement is = 8 x d x y sq metres.
Here, y = 10, d = 3
Using these values in the shortcut, we get:
Total area of the pavement = 8 x 3 x 10 = 240
Hence, the total area of the pavement is 240 sq metre.

88. An oblong piece of ground measures 19m 2.5 dm by 12 m 5 dm. From the centre of each side a path 2 m wide goes across to the centre of the opposite side. What is the area of the path? Find the cost of paving these paths at the rate of Rs 1.32 per sq m.:
  A.  Rs 75.24
  B.  Rs 78.54
  C.  Rs 80.52
  D.  Rs 82.25
     
   
View Answer

Shortcut:
An oblong piece of ground measures y metres x 'z' metres. from the centre of each side a path 'd' metre wide goes across to the centre of the opposite side. (i) Area of the path = d x (y + z − d)
or Area of the path = (width of path)(length + breadth of park − width of path)
(ii) Area of the park minus the path = (y − d)(z − d)
or = (length of park − width of path ) x (breadth of park − width of the path)

Here, y = 19.25, z = 12.5, d = 2
Using these values in the shortcut, we get:
Area of the path = 2 x (19.25 + 12.5 − 2)
= 2 x 29.75 = 59.5 sq m. ∴ cost = rate x area = 1.32 x 59.5 = Rs 78.54

89. A field is 100 m long and 70 m wide. It has two roads each of same breadth. One road is parallel to the length and other parallel to breadth. If the cost of gravelling them at Rs 2 per sq m is Rs 3200, find the breadth of the roads.
  A.  15 m
  B.  12 m
  C.  8 m
  D.  10 m
     
   
View Answer

Shortcut:
An oblong piece of ground measures 'y' metres x 'z' metres. from the centre of each side a path 'd' metre wide goes across to the centre of the opposite side. (i) Area of the path = d x (y + z − d)
or Area of the path = (width of path)(length + breadth of park − width of path)
(ii) Area of the park minus the path = (y − d)(z − d)
or = (length of park − width of path ) x (breadth of park − width of the path)
Here, Area =
3200 / 2
= 1600 sq m
y = 100, z = 70, d = ?
Using these values in the shortcut, we get:
Area of the path = d x (100 + 70 − d)
1600 = 170d − d2
or, d2 − 170d = 1600
By solving the above equation, we get:
d = 160, 10
Hence breadth of the roads is 10 m.

90. There is a square field of side 5 m. A path 1 m wide runs through the centre of the field, one each across its opposite sides. Thwat is the total area of the path and the area of the remaining portion of the fiedl?
  A.  11 sq m
  B.  20 sq m
  C.  16 sq m
  D.  15 sq m
     
   
View Answer

Shortcut:
There is a square garden of side 'y' metres. From the centre of each side a path 'd' metres wide goes across to the centre of opposite side. (i) Area of the path = d x (2y − d) sq m
(ii) Area of the garden − the path = (y − d)2
Here, y = 5, d = 1
Using these values in the shortcut, we get:
(i) Area of the path = 1 x (2 x 5 − 1) = 9 sq m
(ii) Area of the garden − the path = (5 − 1)2 = (4)2 = 16 sq m

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