5 Marks Questions

Question 15 Marks

A square sheet of $2\ cm$ is cut from each corner of a rectangular piece of aluminium sheet of length $12\ cm$ and breadth $8\ cm.$ Find the area of the left over aluminium sheet.

Answer

We have, Side of each square sheet cut from each corner $= 2\ cm,$
Length of the rectangular piece of sheet $= 12\ cm$ and
Breadth of the rectangular piece of sheet $= 8\ cm$
As, the area of the square = (Side $\times$ Side)
$= 2 \times 2$
$= 4\ cm^2$
So, the area of four square sheets = (Side $\times$ Side)
$= 4 \times 4$
$= 16\ cm^2$
Also, the area of the rectangular sheet = (Length $\times$ Breadth)
$= 12 \times 8$
$= 96\ cm^2$
Now, the area of the sheet left over = (Area of the rectangular sheet - Area of the square sheet)
$= 96 - 16$
$= 80\ cm^2$
So, the area of the left over aluminium sheet is $80\ cm^2$.

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Question 25 Marks

Find the total cost of levelling the shaded path of uniform width $2$ metres, laid in the rectangular field shown below, if the rate per $m^2$ is $Rs. 100.$

Answer

The shaded path can be formed as two rectangles as shown below:

Now, the area of the shaded path $= (38 \times 2) + (50 \times 2)$
$= 76 + 100$
$ = 176m^2$
As, the rate of levelling the shaded path $= Rs. 100$ per $m^2$
So, the total cost of levelling the shaded path $= 176 \times 100$
$= Rs. 17,600$
Disclaimer: The answer given in the textbook is incorrect. The same has been corrected above.

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Question 35 Marks

A room is $8m$ long and $6m$ wide. Its floor is to be covered with rectangular tiles of size $25\ cm$ by $20\ cm$. How many tiles will be required? Find the cost of these tiles at $Rs. 25$ per tile.

Answer

We have,
Length of the room $= 8m = 800\ cm,$
Breadth of the room $= 6m = 600\ cm,$
Length of the tile $= 25\ cm$ and
Breadth of the tile $= 20\ cm$
Now,
The number of the tiles required $=\frac{\text{Area of the floor of the room}}{\text{Area of a tile}}$
$=\frac{\text{Length of the room}\times\text{Breadth of the room}}{\text{Length of the tile}\times\text{Breadth of the tile}}$
$=\frac{800\times600}{25\times20}$
$=960$
So, the number of the tiles required to cover the floor of the room is $960.$
Also,
Since, the cost of $1$ tile $= Rs. 25$
So, the cost of the $960$ tiles such tiles $= 25 \times 960 = Rs. 24,000.$

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Question 45 Marks

A square sheet of side $5\ cm$ is cut out from a rectangular piece of an aluminimum sheet of length $9\ cm$ and breadth $6\ cm$. What is the area of the aluminium sheet left over?

Answer

We have, Side of the square sheet cut out $= 5\ cm,$
Length of the rectangular sheet $= 9\ cm$ and
Breadth of the rectangular sheet $= 6\ cm$
Now, the area of the square sheet cut out = (Side $\times $ Side)
$= 5 \times 5$
$= 25\ cm^2$
Also, the area of the recatngular sheet = (Length $\times $ Breadth)
$= 9 \times 6$
$= 54\ cm^2$
So, the area of the sheet left over = (Area of the rectangular sheet - Area of the square sheet cut out)
$= 54 - 25$
$= 29\ cm^2$
Hence, the area of the aluminium sheet left over is $29\ cm^2$

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Question 55 Marks

The area of a rectangular ground is $120m^2$ and its length is $12m$. Find the cost of fencing the ground at the rate of $Rs.125$ per metre.

Answer

We have,
Area of the rectangular ground $= 120m^2$
Length of the ground $= 12m$
Breadth of the ground $=\frac{\text{Area of the ground}}{\text{Length of the ground}}$
$=\frac{120}{12}$
$10\text{m}$
Also,
Perimeter of the ground $= 2($Length + Breadth$)$
$= 2(12 + 10)$
$=2 \times 22$
$=44m$

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Question 65 Marks

The area of a square field is $36m^2$. A path of uniform width is laid around and outside of it. If the area of the path is $28m^2$, then find the width of the path.

Answer

As, the area of the square field $= 36m^2$
So, the side of the square field $=\sqrt{36}=6\text{m}$
Also, the area of the outer square = (Area of the square + Area of the path)
$= 36 + 28$
$= 64m^2$
So, the side of the outer square $=\sqrt{64}=8\text{m}$
Now, the width of the path $=\frac{\text{side of the outer square - side of the inner square}}{2}$
$=\frac{8-6}{2}$
$=\frac{2}{2}$
$=1\text{m}$

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Question 75 Marks

The length and breadth of a rectangular park are in the ratio $5 : 3$ and its perimeter is $128m$. Find the area of the park.

Answer

Let the length of the park be $5x\ m$ and its breadth be $3x\ m.$
As,
perimeter of the park $= 2($Length of the park + Breadth of the park$)$
$=2(5x + 3x)$
$=2 \times 8x$
$=16x\ m$
But the perimeter of the park $= 128m$
$\Rightarrow16\text{x}=128$
$\Rightarrow\text{x}=\frac{128}{16}$
$\Rightarrow\text{x}=8$
$\therefore$ The length of park $= 5x$
$= 5 \times 8$
$= 40m$
The breadth of the park $= 3x = 3 \times 8 = 24m$
Now, the area of the park = (Length $\times $ Breadth)
$= 40 \times 24$
$= 960m^2$
So, the area of the park is $960m^2$.

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