Area, General Mathematics

Question: A square and a rectangle have equal areas. If their perimeters are p1 and p2 respectively, than :
A
`p1 < p2`

B
`p1 = p2`

C
`p1 > p2`

D
None

Note: Not available
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Question: If the perimeters of a square and a rectangle are the same, than the areas A and B enclosed by them would satisfy the condition :
A
`A < B`

B
`A <= B`

C
`A > B`

D
`A >= B`

Note: Let side of a square = a. length of rect. = x & breadth of rect. = y. `4a = 2 (x + y) => (x + y)/2 = a` Area of square` = a^2 = ((x + y)^2/2)` So, A`= ((x + y)^2/2)` Area of rect. = xy. So, B = xy. `(x + y)/2 > x^1/2 y^1/2` [:. A.M. > G.M.] `=> ((x + y )^2/2)`> xy i. e.A > B.
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Question: The ratio of the area of a square to that of the square drawn on its diagonal is :
A
`1 : 1`

B
`1 : 2`

C
`1 : 3`

D
`1 : 4`

Note: Let side of a square = a. Then, its diagonal` = sqrt(2) a`. Ratio of areas of squares with sides a & `sqrt(2) a` `= a^2/(sqrt(2) a)^2` `= 1/2 or 1 : 2`
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Question: If the ratio of the areas of two squares is 9 : 1, the ratio of their perimeters is :
A
`9 : 1`

B
`3 : 1`

C
`1 : 3`

D
`3 : 4`

Note: Let the sides of two squares be a and b respectively. Then, `a^2/b^2 = 9/1` or `(a/b)^2` `= (3/1)^2` or `a/b = 3/1` :. Ratio of their perimeters `= (4a)/(4b) = a/b = 3/1` or ` 3 : 1`
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Question: If the diagonal of a rectangle and a square, each is equal to 80 cm and the difference of their areas is 100 sq. cm, the side of the rectangle are :
A
`35 cm, 15 cm`

B
`30 cm, 10 cm`

C
`28 cm, 12 cm`

D
` 25 cm, 15 cm`

Note: perimeter of square = 80 cm. So, side of square = 20 cm. Area of the square` = (20 xx 20) cm^2` `= 400 cm^2` With same perimeters, area of the square is larger. :. Area of rectangle`= (400 - 100) cm^2` `= 300 cm^2` Let length of rect. = x & its breadth = y. Then, xy = 300 and x + y = 40 `:. (x - y) = sqrt((x + y))^2- 4xy` `= sqrt(1600 - 1200) = 20` Solving `x + y = 40 & x - y = 20,` we get` x = 30, y = 10` :. Sides are` 30 cm, 10 cm.`
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Question: Of the two square fields, the area of one is 1 hectare, while the other one is broader by 1%. The difference in their areas is :
A
`100 m^2`

B
`101 m^2`

C
`200 m^2`

D
`201 m^2`

Note: Area of first square` = 10000 sq.` metres. Side of this square` = sqrt(10000) m = 201 m^2` Side of the new square` = 101 m` Difference in areas` = (101)^2 - (100)^2` ` = 201 m^2`
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Question: The length of a rectangle is twice its breadth. If its length is decreased by 5 cm and breadth is increased by 5 cm, the area of the rectangle is increased by 75 sq. cm. The length of the rectangle is
A
`20 cm`

B
`30 cm`

C
`40 cm`

D
`50 cm`

Note: Let breadth = x. Then, length = 2x `(2x - 5) (x + 5) - 2x xx x` `= 75 or x = 20`. `:. Length = 40 cm.`
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Question: The cost of cultivating a square field at the rate of Rs. 135 per hectare is Rs. 1215. The cost of putting a fence around it at the rate of 75 paise per metre would be :
A
`Rs. 360`

B
`Rs. 810`

C
`Rs. 900`

D
`Rs. 1800`

Note: Area `= (total cost)/Rate` `= (1215/135) hectares` `= (9 xx 10000) sq.m,` :. Side of the square` = sqrt(90000) = 300 m.` perimeter of the field` = (300 xx 4) m.` `= 1200 m.` cost of fencing` = Rs. (1200 xx 3/4)` `= Rs. 900`
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Question: Within a rectangular garden 10 m wide and 20 m long, we wish to pave a walk around the borders of uniform so as to leave an area of `96 m^2`for flowers.How wide should the walk be ?
A
1 m

B
2 m

C
2.1 m

D
2.5 m

Note: Let the width of walk be x metres. Then, `(20 - 2x) (10 - 2x) = 96` `or 4x^2 + 60x - 104` `= 0 or x^2 + 15x - 26 = 0` `or (x - 13) (x - 2) = 0` So, `x = 2 [:. x =/ 13]
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Question: The cost of carpeting a room 18 m long with a carpet 75 cm wide at 45 paise per metre is Rs. 81. The breadth of the room is :
A
7 m

B
7.5 m

C
8 m

D
8.5 m

Note: Length of the carpet` = (Total Cost/Rate/m)` ` = (8100/450) m` ` = 180 m` Area of the room` = (180 xx 75/100) m^2` ` = 135 m^2` :. Area of the room` = Area of the carpet = 135 m^2` :. Breadth of the room` = (Area/Length)` ` = (135/18) m^2` ` = 7.5 m.`
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Question: A square and a rectangle have equal areas. If their perimeters are p1 and p2 respectively, than :

Question: If the perimeters of a square and a rectangle are the same, than the areas A and B enclosed by them would satisfy the condition :

Question: The ratio of the area of a square to that of the square drawn on its diagonal is :

Question: If the ratio of the areas of two squares is 9 : 1, the ratio of their perimeters is :

Question: If the diagonal of a rectangle and a square, each is equal to 80 cm and the difference of their areas is 100 sq. cm, the side of the rectangle are :

Question: Of the two square fields, the area of one is 1 hectare, while the other one is broader by 1%. The difference in their areas is :

Question: The length of a rectangle is twice its breadth. If its length is decreased by 5 cm and breadth is increased by 5 cm, the area of the rectangle is increased by 75 sq. cm. The length of the rectangle is

Question: The cost of cultivating a square field at the rate of Rs. 135 per hectare is Rs. 1215. The cost of putting a fence around it at the rate of 75 paise per metre would be :

Question: Within a rectangular garden 10 m wide and 20 m long, we wish to pave a walk around the borders of uniform so as to leave an area of `96 m^2`for flowers.How wide should the walk be ?

Question: The cost of carpeting a room 18 m long with a carpet 75 cm wide at 45 paise per metre is Rs. 81. The breadth of the room is :