GSEB Class 7 Maths Solutions Chapter 11 પરિમિતિ અને ક્ષેત્રફળ Exercise 11.1

Question 1. A rectangular piece of land has a length of 500 meters and a width of 300 meters respectively.
(i) Find its area.
(ii) If 1 \( m^2 \) of land costs Rs 10,000, find its total cost.
Answer:
(i) Here, the length of the rectangular land plot \( (l) \) is 500 meters and its width \( (b) \) is 300 meters.
The area of the rectangular land plot is calculated as length multiplied by width \( (l \times b) \). \[ \text{Area} = 500 \times 300 \, m^2 = 1,50,000 \, m^2 \] Thus, the area of the rectangular land plot is 1,50,000 \( m^2 \).
(ii) The cost of 1 \( m^2 \) of land is given as Rs 10,000.
\( \implies \) Therefore, the total cost for 1,50,000 \( m^2 \) of land will be \( \text{Rs } (10,000 \times 1,50,000) \). \[ \text{Total Cost} = \text{Rs } 1,50,00,00,000 \] The total value of the land becomes Rs 1,50,00,00,000.
In simple words: To find the area, you just multiply the length by the width. The result tells you how much space the land covers. To get the total cost, take the price for one square meter and multiply it by the total area.

Exam Tip: Remember to include the correct units (\( m \) for length/width, \( m^2 \) for area, and Rs for cost) in your calculations and final answer for full marks.

Question 2. Find the area of a square garden whose perimeter is 320 meters.
Answer: The perimeter of a square garden is calculated as 4 times the length of its side.
\( \implies \) So, 4 multiplied by the side's length equals the square garden's perimeter.
\( \implies \) This means \( 4 \times \text{side length} = 320 \) meters.
\( \implies \) Therefore, the side length of the garden is \( \frac{320}{4} = 80 \) meters. The side length of the square garden measures 80 meters. Now, the area of the square garden is found by multiplying its length by its width (which are the same for a square), or side by side. \[ \text{Area} = 80 \times 80 \, m^2 = 6400 \, m^2 \] Hence, the square garden's area is 6400 \( m^2 \).
In simple words: First, divide the perimeter by 4 to find one side of the square. Then, multiply that side by itself to get the area.

Exam Tip: Always remember that all sides of a square are equal. The perimeter is the sum of all sides, and the area is the product of two sides.

Question 3. Find the width of a rectangular plot of land whose area is 440 \( m^2 \) and length is 22 meters. Also find its perimeter.
Answer: Let's assume the width of the rectangular land plot is \( b \) meters.
The length of this rectangular plot \( (l) \) is provided as 22 meters, and its area is 440 \( m^2 \).
The formula for the area of a rectangular plot is \( \text{Area} = l \times b \).
Substituting the given values, we get \( 22 \times b = 440 \).
\( \implies \) Therefore, the width \( b = \frac{440}{22} = 20 \) meters. Thus, the width of the rectangular plot is 20 meters. Next, the perimeter of the rectangular plot is calculated using the formula \( \text{Perimeter} = 2 \times (\text{length} + \text{width}) \). \[ \text{Perimeter} = 2 \times (22 + 20) \, m = 2 \times (42) \, m = 84 \, m \] Hence, the perimeter of the rectangular plot is 84 meters.
In simple words: To get the width, divide the area by the given length. Then, use both the length and the width in the perimeter formula, which is two times (length plus width).

Exam Tip: Make sure to distinguish between area and perimeter formulas. Area is a measure of surface, while perimeter is a measure of the boundary.

Question 4. The perimeter of a rectangle is 100 cm. If its length is 35 cm, find its width. Also find its area.
Answer: The perimeter of the rectangle is 100 cm, and its length \( (l) \) is 35 cm.
Let's suppose the width of the rectangle is \( b \) cm.
The perimeter of a rectangle is calculated as \( 2 \times (\text{length} + \text{width}) \).
\( \implies \) So, \( 100 = 2 \times (35 + b) \).
\( \implies \) Dividing by 2, we get \( 50 = 35 + b \).
\( \implies \) From this, \( b = 50 - 35 \).
\( \implies \) Thus, \( b = 15 \). The width of the rectangle is 15 cm. Now, the area of this rectangle is given by \( \text{length} \times \text{width} \). \[ \text{Area} = 35 \times 15 \, cm^2 = 525 \, cm^2 \] Consequently, the area of the rectangle is 525 \( cm^2 \).
In simple words: To find the width, first halve the perimeter, then subtract the length from that result. Once you have the width, multiply it by the length to get the area.

Exam Tip: For perimeter problems, always use the formula \( P = 2(l+b) \). For area, use \( A = l \times b \). Be careful with the units: cm for length/width, \( cm^2 \) for area.

Question 5. The areas of a square garden and a rectangular garden are equal. If the side of the square garden is 60 meters and the length of the rectangular garden is 90 meters, find the width of the rectangular garden.
Answer: The side length of the square garden \( (l) \) is 60 meters.
\( \implies \) Therefore, the area of the square garden is calculated as \( \text{length} \times \text{length} = 60 \times 60 \, m^2 = 3600 \, m^2 \). It is stated that the areas of the square garden and the rectangular garden are equal.
\( \implies \) So, the area of the rectangular garden is also 3600 \( m^2 \). Furthermore, the length of the rectangular garden is 90 meters. The formula for the area of a rectangular garden is \( \text{length} \times \text{width} \).
\( \implies \) This means \( 3600 = 90 \times \text{width} \).
\( \implies \) Consequently, the width is \( \frac{3600}{90} = 40 \) meters. Thus, the width of the rectangular garden is 40 meters.
In simple words: First, figure out the area of the square by multiplying its side by itself. Since the areas are equal, use that area with the rectangular garden's length to find its width (area divided by length).

Exam Tip: When two shapes have equal areas, first calculate the known area, then use that value to find missing dimensions in the other shape.

Question 6. A wire is bent in the shape of a rectangle with length 40 cm and width 22 cm. If it is straightened and then re-bent into the shape of a square, what will be the measure of each of its sides? Also determine which shape encloses more area.
Answer: When the wire is shaped as a rectangle, its length \( (l) \) is 40 cm and its width \( (b) \) is 22 cm.
\( \implies \) The perimeter of the rectangle is calculated as \( 2 \times (l + b) \). \[ \text{Perimeter}_{\text{rectangle}} = 2 \times (40 + 22) \, cm = 2 \times (62) \, cm = 124 \, cm \] When this wire is reshaped into a square, its perimeter will remain 124 cm, because the length of the wire hasn't changed.
\( \implies \) The perimeter of a square is given by \( 4 \times \text{side length} \).
\( \implies \) So, \( 4 \times l = 124 \).
\( \implies \) Thus, \( l = \frac{124}{4} = 31 \) cm. The side length of the wire when it's in a square shape is 31 cm. Now, let's determine the area of the rectangular shape: \( \text{Area} = l \times b \). \[ \text{Area}_{\text{rectangle}} = 40 \times 22 \, cm^2 = 880 \, cm^2 \] And the area of the square shape: \( \text{Area} = l \times l \). \[ \text{Area}_{\text{square}} = 31 \times 31 \, cm^2 = 961 \, cm^2 \] Therefore, the square shape of the wire covers more area than the rectangular shape.
In simple words: First, find the perimeter of the rectangle. That perimeter is also the perimeter of the square. Divide the perimeter by 4 to get the side of the square. Then, calculate the area for both shapes (length times width for rectangle, side times side for square) and compare them to see which is bigger.

Exam Tip: Remember that reshaping a wire changes its shape but not its total length, which corresponds to the perimeter. For a fixed perimeter, a square will always enclose the maximum area compared to any rectangle.

Question 7. The perimeter of a rectangle is 130 cm. If its width is 30 cm, find its length. Also find its area.
Answer: The perimeter of the rectangle is 130 cm, and its width \( (b) \) is 30 cm.
Let's suppose the length of the rectangle is \( l \).
The perimeter of a rectangle is determined by the formula \( 2 \times (\text{length} + \text{width}) \).
\( \implies \) So, \( 130 = 2 \times (l + 30) \).
\( \implies \) Dividing both sides by 2 gives \( \frac{130}{2} = l + 30 \).
\( \implies \) This simplifies to \( 65 = l + 30 \).
\( \implies \) Therefore, \( l = 65 - 30 \).
\( \implies \) So, \( l = 35 \) cm. The length of the rectangle is 35 cm. Now, the area of the rectangle is calculated as \( \text{length} \times \text{width} \). \[ \text{Area} = 35 \times 30 \, cm^2 = 1050 \, cm^2 \] Consequently, the rectangle's length is 35 cm, and its area is 1050 \( cm^2 \).
In simple words: To find the length, first divide the perimeter by two, then subtract the width. After that, multiply the length by the width to get the area.

Exam Tip: Always double-check your calculations, especially when working with subtraction and multiplication to avoid small errors that can affect the final answer.

Question 8. A door with length 2 meters and width 1 meter is fitted in a wall. The length of the wall is 4.5 meters and width is 3.6 meters. If the cost of whitewashing the wall is Rs 20 per square meter, find the cost of whitewashing the wall.
Answer: Here, the length of the wall \( (l) \) is 4.5 meters, and its width \( (b) \) is 3.6 meters.
\( \implies \) The area of the wall is calculated as \( l \times b \). \[ \text{Area}_{\text{wall}} = 4.5 \times 3.6 \, m^2 = 16.2 \, m^2 \] The length of the door \( (l) \) is 2 meters, and its width \( (b) \) is 1 meter.
\( \implies \) The area of the door is calculated as \( l \times b \). \[ \text{Area}_{\text{door}} = 2 \times 1 \, m^2 = 2 \, m^2 \] The wall needs to be whitewashed, excluding the area occupied by the door.
\( \implies \) Therefore, the area to be whitewashed is \( \text{Area}_{\text{wall}} - \text{Area}_{\text{door}} \). \[ \text{Area}_{\text{whitewash}} = 16.2 \, m^2 - 2 \, m^2 = 14.2 \, m^2 \] The cost to whitewash 1 \( m^2 \) of space is given as Rs 20.
\( \implies \) So, the cost to whitewash 14.2 \( m^2 \) of space will be \( \text{Rs } (20 \times 14.2) \). \[ \text{Total Cost} = \text{Rs } 284 \] Thus, the total cost for whitewashing the wall is Rs 284.
In simple words: First, find the total area of the wall and the area of the door. Subtract the door's area from the wall's area to know how much surface to paint. Then, multiply that painted area by the cost per square meter.

Exam Tip: When calculating costs for painting or whitewashing, always subtract the areas of any openings (like doors or windows) from the total area of the surface to be covered.