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21. If increasing the length of a rectangular field by 5 metres, area also increases by 30 sq metres, then find the value of its width.
  A.  5 m
  B.  8 m
  C.  4 m
  D.  6 m
     
   
View Answer

Shortcut:
If the length of a rectangle is increased by 'm' units and due to this increase, area of the rectangle also increases by 'n' sq units, then
Width =
n / m
units.
Here, m = 5, n = 30
Using these values in the shortcut, we get:
Width =
30 / 5
= 6
Hence, the width is 6 metres.

22. There is a rectangular field of area 60 sq cm. Sum of its diagonal and length is 5 times of its breadth. Find the breadth of the rectangular field.
  A.  3 cm
  B.  4 cm
  C.  5 cm
  D.  9 cm
     
   
View Answer

Shortcut:
There is a rectangle of area 'A sq unit. If the sum of its diagonal and length is 'n' times of its breadth, then
Length of the rectangle =
√(
A(n2 − 1) / 2 x n
)

Breadth of the rectangle =
√(
2An / n2 − 1
)

Here, A = 60, n = 5
Using these values in the shortcut, we get:
Length =
√(
60(52 − 1) / 2 x 5
)
=
√(
60 x 24 / 10
)
= 12
Breadth =
√(
2 x 60 x 5 / 52 − 1
)
= 5
Hence, the breadth is 5cm.

23. Length of a rectangular blackboard is 8 cm more than that of its breadth. If its length is inicreased by 7 cm and its breadth is decreased by 4 cm, its area remains unchanged. Find the length and breadth of the rectangular blackboard.
  A.  Length: 28 cm; Breadth: 10 cm
  B.  Length: 28 cm; Breadth: 20 cm
  C.  Length: 18 cm; Breadth: 20 cm
  D.  Length: 38 cm; Breadth: 30 cm
     
   
View Answer

Shortcut:
There is a rectangle. Its length is 'x' units more than its breadth. If its length is increased by 'y' units and its breadth is decreased by 'z' units, the area of the rectangle is unchanged. Length and breadth of the rectangle are
(x + z)y / y − z
) and
(x + y)z / y − z
units respectively.
Here, x = 8, y = 7, z = 4
Using these values in the shortcut, we get:
Length =
(8 + 4)7 / 7 − 4
=
12 x 7 / 3
= 28
Breadth =
(8 + 7)4 / 7 − 4
= 20
Hence, the length is 28 cm and breadth is 20 cm.

24. Length of a rectangular field is increased by 7 m and breadth is decreased by 3 m, area of the field remains unchanged. If lenght be decreased by 7 m and breadth be increased by 5 m, again area remains unchanged. Find the length and breadth of the rectangular field.
  A.  Length: 28 cm; Breadth: 15 cm
  B.  Length: 18 cm; Breadth: 15 cm
  C.  Length: 28 cm; Breadth: 25 cm
  D.  Length: 18 cm; Breadth: 25 cm
     
   
View Answer

Shortcut:
Length of a rectangle is increased by 'a' units and breadth is decreased by 'b' units, area of the rectangle remains the same. If length be decreased by 'c' units and breadth by increased by 'd' units, in this case also area of the rectangle remains unchanged.
Length and breadth of the rectangle are given by c x
(
ad + ab / ad − bc
)
and d x
(
ba + bc / ad − bc
)
units respectively.
Here, a = 7, b = 3, c = 7, d = 5
Using these values in the shortcut, we get:
Length = 7 x
(
7 x 5 + 7 x 3 / 7 x 5 − 3 x 7
)
= 7 x
(
35 + 21 / 35 − 21
)
=
7 x 56 / 14
= 7 x 4 = 28
Breadth = d
(
ba + bc / ad − bc
)
= 5 x
3 x 7 + 3 x 7 / 7 x 5 − 3 x 7
= 5 x
(
21 + 21 / 35 − 21
)
=
42 / 14
= 3 x 5 = 15
Hence, the length is 28 cm and breadth is 15 cm.

25. The length of a rectangular hall is 16 m. If it can be partitioned into two equal square rooms, what is the length of the partition?
  A.  9 m
  B.  6 m
  C.  4 m
  D.  8 m
     
   
View Answer

Shortcut:
To find the area of a square if length of one of the sides is given.
Areaof a square = (side)2
Here, Let the length of partitiion be y m.
then, 16 x y = y2 + y2
or, 16 x y = 2y2
or, y = 8
Hence, the length is 8 m.

26. Find the sides of two squares, which contain together 12.25 hectares, the sides of the squares being in the ratio of 3:4.
  A.  220 m; 280 m
  B.  210 m; 280 m
  C.  210 m; 260 m
  D.  230 m; 290 m
     
   
View Answer

Shortcut:
To find the area of a square if length of one of the sides is given.
Area of a square = (side)2
Here, Let the sides are 3y and 4y metres.
then, (3y)2 + (4y)2 = 12.25 x 10000 = 122500
or, y = 70
∴ sides are 3 x 70 = 210 m amd 4 x 70 = 280 m

27. What would be the length of the diagonal of a square plot whose area is equal to the area of a rectangular plot of 45 m (Length) and 40 m (Width)
  A.  80 m
  B.  40 m
  C.  60 m
  D.  90 m
     
   
View Answer

Shortcut:
To find the area of a square if length of the diagonal is given.
Area of a square =
1 / 2
(diagonal)2
Here, Area of rectangular plot = 45 x 40 = 1800 sq m
Area of square plot = Area of rectangular plot (given)
So, Area of square plot = 1800 sq m
Using this value in the shortcut, we get:
Area of a square =
1 / 2
(diagonal)2
1800 =
1 / 2
(diagonal)2
then, (diagonal)2 = 2 x 1800 = 3600
or, Diagonal = 60
∴ Length of the diagonal of square = 60 m

28. Perimeter of a squae field is 16 m. Find the length of its sides.
  A.  4 m
  B.  6 m
  C.  8 m
  D.  9 m
     
   
View Answer

Shortcut:
To find the perimeter of a square if its length of side is given.
Perimeter of a square = 4 x side
Here, perimeter = 16, side = ?
Using these values in the shortcut, we get:
Perimeter of a square = 4 x side
16 = 4 x side
or, side =
16 / 4
= 4
Hence, the length of its sides is 4 m.

29. If the ratio of areas of two squares is 9 : 1, the ratio of their perimeters is:
  A.  3 : 2
  B.  3 : 1
  C.  5 : 2
  D.  6 : 1
     
   
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, Required ratio =
√
9 / 1
= 3 : 1 [∵ Perimeter ∝ √(Area)]

30. Perimeter and area of a rectangle are 82 cm and 400 sq cm. Find the difference in length and width.
  A.  8 cm
  B.  4 cm
  C.  9 cm
  D.  3 cm
     
   
View Answer

Shortcut:
To find the the difference in length and width of a rectangle when perimeter and area are given.
Difference in length and width of a rectangle =
√[(
perimeter / 2
)
2 − 4 x Area
]

Here, Perimeter = 82, area = 400
Using these values in the shortcut, we get:
Required value =
√[(
82 / 2
)
2 − 4 x 400
]

= √[(41)2 − 1600]
= √81 = 9
Hence, the difference is 9 cm.

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