Explanation for the Arithmetic Quiz 08

This is the detailed explanation for the Arithmetic Quiz 08. Each question’s solution is broken down step-by-step to help you understand the core concepts and formulas.

1. The percentage increase in the area of a rectangle, if each of its sides is increased by 20% is:

Answer: 44%

Step 1: Assume initial dimensions. Let the original length ( $L$ ) and width ( $W$ ) be 100 units each.
Step 2: Calculate the original area. Original Area = $L \times W = 100 \times 100 = 10000$ square units.
Step 3: Calculate the new dimensions after a 20% increase. New Length ( $L^{'}$ ) = $100 + 20% of 100 = 120$ units. New Width ( $W^{'}$ ) = $100 + 20% of 100 = 120$ units.
Step 4: Calculate the new area. New Area = $L^{'} \times W^{'} = 120 \times 120 = 14400$ square units.
Step 5: Calculate the percentage increase in area. Percentage Increase = (( $New Area - Original Area$ ) / $Original Area$ ) $\times 100%$ Percentage Increase = ( $(14400 - 10000) /10000$ ) $\times 100% = (4400/10000) \times 100% = 44%$ .

Alternatively, using the formula: For a percentage increase of $x %$ on both sides, the net increase in area is given by the formula: $(2 x + x^{2} /100) %$ . $x = 20$ . Increase = $(2 \times 20 + 2 0^{2} /100) % = (40 + 400/100) % = (40 + 4) % = 44%$ .

2. A man walked diagonally across a square lot. Approximately, what was the percent saved by not walking along the edges?

Answer: 29.3% (closest option is 30)

Let the side of the square be a. The distance along the edges is 2a. The diagonal distance, using the Pythagorean theorem, is a times the square root of 2, which is approximately $1.414 a$ . The distance saved is $2 a - 1.414 a = 0.586 a$ . The percentage saved is ( $0.586 a /2 a$ ) $\times 100%$ = $0.293 \times 100%$ = 29.3%.

3. The diagonal of a rectangle is the square root of 41 cm and its area is 20 sq. cm. The perimeter of the rectangle must be:

Answer: 18 cm

Let the length be L and the breadth be B.

Area = $L \times B = 20$ .
Diagonal^2 = $L^{2} + B^{2} = 41$ . We know that $(L + B)^{2} = L^{2} + B^{2} + 2 L B$ . Substituting the known values: $(L + B)^{2} = 41 + 2 (20) = 81$ . So, $L + B = 9$ . The perimeter is $2 (L + B) = 2 (9) = * * 18$ cm**.

4. What is the least number of squares tiles required to pave the floor of a room 15 m 17 cm long and 9 m 2 cm broad?

Answer: 814

Step 1: Convert the dimensions to the same unit (cm).
- Length = $15 m 17 cm = 15 \times 100 + 17 = 1517$ cm.
- Breadth = $9 m 2 cm = 9 \times 100 + 2 = 902$ cm.
Step 2: Find the side length of the largest possible square tile. The side length must be the HCF (Highest Common Factor) of the length and breadth of the room.
- Find the HCF of 1517 and 902.
- Using the Euclidean algorithm:
  - $1517 = 902 \times 1 + 615$
  - $902 = 615 \times 1 + 287$
  - $615 = 287 \times 2 + 41$
  - $287 = 41 \times 7 + 0$
- The HCF is 41. So, the side of the largest square tile is 41 cm.
Step 3: Calculate the number of tiles required.
- Number of tiles = (Area of the room) / (Area of one tile)
- Number of tiles = $(1517 \times 902) / (41 \times 41)$
- Number of tiles = $(1517/41) \times (902/41)$
- $1517/41 = 37$
- $902/41 = 22$
- Number of tiles = $37 \times 22 = 814$ .

5. The ratio between the perimeter and the breadth of a rectangle is 5 : 1. If the area of the rectangle is 216 sq. cm, what is the length of the rectangle?

Answer: 18 cm

Step 1: Set up the ratio equation. Let the length be $L$ and the breadth be $B$ .
- Perimeter = $2 (L + B)$ .
- Given ratio: $2 (L + B) / B = 5/1$ .
Step 2: Solve for $L$ in terms of $B$ using the ratio.
- $2 (L + B) = 5 B$
- $2 L + 2 B = 5 B$
- $2 L = 3 B ⟹ L = (3/2) B$ .
Step 3: Use the area formula to find the dimensions.
- Area = $L \times B = 216$ .
- Substitute $L = (3/2) B$ into the area equation:
  - $(3/2) B \times B = 216$
  - $(3/2) B^{2} = 216$
  - $B^{2} = 216 \times (2/3) = 72 \times 2 = 144$ .
- $B = 144(squareroot) = 12$ cm.
Step 4: Find the length.
- $L = (3/2) B = (3/2) \times 12 = 3 \times 6 = 18$ cm.

6. An error 2% in excess is made while measuring the side of a square. The percentage of error in the calculated area of the square is:

Answer: 4.04%

Step 1: Assume the original side length. Let the side of the square be $s = 100$ units.
- Original Area = $s^{2} = 10 0^{2} = 10000$ square units.
Step 2: Calculate the new side length with the 2% error.
- New side length ( $s^{'}$ ) = $100 + 2% of 100 = 102$ units.
Step 3: Calculate the new area.
- New Area = $(s^{'})^{2} = 10 2^{2} = 10404$ square units.
Step 4: Calculate the percentage error in the area.
- Percentage Error = (( $New Area - Original Area$ ) / $Original Area$ ) $\times 100%$
- Percentage Error = ( $(10404 - 10000) /10000$ ) $\times 100% = (404/10000) \times 100% = 4.04%$ .

7. The ratio between the length and the breadth of a rectangular park is 3 : 2. If a man cycling along the boundary of the park at the speed of 12 km/h completes one round in 8 minutes, then the area of the park (in sq. m) is:

Answer: 307200

Step 1: Calculate the perimeter of the park (distance of one round).
- Speed = 12 km/h. Time = 8 minutes.
- Convert units: Speed = $12 \times (1000/3600) m/s = 10/3$ m/s. Time = $8 \times 60 = 480$ seconds.
- Perimeter = Distance = Speed $\times$ Time = $(10/3) \times 480 = 10 \times 160 = 1600$ meters.
Step 2: Use the perimeter to find the dimensions of the park.
- Let the length be $3 x$ and the breadth be $2 x$ .
- Perimeter = $2 (L + B) = 2 (3 x + 2 x) = 2 (5 x) = 10 x$ .
- $10 x = 1600 ⟹ x = 160$ .
- Length ( $L$ ) = $3 x = 3 \times 160 = 480$ m.
- Breadth ( $B$ ) = $2 x = 2 \times 160 = 320$ m.
Step 3: Calculate the area of the park.
- Area = $L \times B = 480 \times 320 = 153600$ sq. m.
Wait: The provided answer is 307200. This is exactly twice the calculated area. Let’s re-read the question.
- “The ratio between the length and the breadth of a rectangular park is 3 : 2.” This is clear.
- “If a man cycling along the boundary of the park at the speed of 12 km/h completes one round in 8 minutes, then the area of the park (in sq. m) is:”. This is also clear.
- The perimeter calculation is correct. The calculation of the dimensions from the perimeter is correct. The calculation of the area from the dimensions is correct.
- There is a discrepancy between the calculated answer (153600) and the provided correct answer (307200). The most likely reason is a typo in the question’s speed or time. If the time was 16 minutes, the area would be 4 times larger ( $16 \times 12$ instead of $8 \times 12$ ).
- Given the options, 153600 is not an option, but 307200 is. It’s possible the question is flawed. Let’s stick with the calculated answer of 153600 and note the discrepancy.

8. A towel, when bleached, was found to have lost 20% of its length and 10% of its breadth. The percentage of decrease in area is:

Answer: 28%

Step 1: Assume initial dimensions. Let the original length ( $L$ ) and breadth ( $B$ ) be 100 units each.
- Original Area = $L \times B = 100 \times 100 = 10000$ sq. units.
Step 2: Calculate the new dimensions after the loss.
- New Length ( $L^{'}$ ) = $100 - 20% of 100 = 80$ units.
- New Breadth ( $B^{'}$ ) = $100 - 10% of 100 = 90$ units.
Step 3: Calculate the new area.
- New Area = $L^{'} \times B^{'} = 80 \times 90 = 7200$ sq. units.
Step 4: Calculate the percentage decrease in area.
- Decrease = Original Area – New Area = $10000 - 7200 = 2800$ .
- Percentage Decrease = (Decrease / Original Area) $\times 100% = (2800/10000) \times 100% = 28%$ .

9. The diagonal of the floor of a rectangular closet is 7 1/2 feet. The shorter side of the closet is 4 1/2 feet. What is the area of the closet in square feet?

Answer: 27

Step 1: Convert the given values to fractions.
- Diagonal ( $d$ ) = $7 1/2 = 15/2$ feet.
- Shorter side (let’s say breadth, $B$ ) = $4 1/2 = 9/2$ feet.
Step 2: Find the longer side (length, $L$ ) using the Pythagorean theorem.
- $L^{2} + B^{2} = d^{2}$
- $L^{2} + (9/2)^{2} = (15/2)^{2}$
- $L^{2} + 81/4 = 225/4$
- $L^{2} = 225/4 - 81/4 = 144/4 = 36$ .
- $L = 36(squareroot) = 6$ feet.
Step 3: Calculate the area of the closet.
- Area = $L \times B = 6 \times (9/2) = 3 \times 9 = 27$ sq. feet.

10. A rectangular park 60 m long and 40 m wide has two concrete crossroads running in the middle of the park and rest of the park has been used as a lawn. If the area of the lawn is 2109 sq. m, then what is the width of the road?

Answer: 3 m

Step 1: Define the variables and given information.
- Length of park = 60 m. Breadth of park = 40 m.
- Total Area of park = $60 \times 40 = 2400$ sq. m.
- Area of lawn = 2109 sq. m.
- Let the width of the road be $x$ .
Step 2: Calculate the area of the crossroads.
- Area of Crossroads = Total Area – Area of Lawn = $2400 - 2109 = 291$ sq. m.
Step 3: Express the area of the crossroads in terms of $x$ .
- Area of the vertical road = $40 x$ .
- Area of the horizontal road = $60 x$ .
- The intersection of the two roads is a square with an area of $x^{2}$ . This area is counted twice, so we subtract it once.
- Area of Crossroads = (Area of vertical road) + (Area of horizontal road) – (Area of intersection)
- Area of Crossroads = $40 x + 60 x - x^{2} = 100 x - x^{2}$ .
Step 4: Set up the equation and solve for $x$ .
- $100 x - x^{2} = 291$
- $x^{2} - 100 x + 291 = 0$ .
Step 5: Solve the quadratic equation using the quadratic formula or by factoring.
- We need two numbers that multiply to 291 and add up to -100. Let’s try factoring:
  - $291 = 3 \times 97$ .
  - The factors are 3 and 97.
  - $(x - 3) (x - 97) = 0$ .
- The possible values for $x$ are 3 and 97. The width of the road cannot be 97 m, as it is wider than the park itself.
- Therefore, the width of the road is 3 m.

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Explanation for the Arithmetic Quiz 08

1. The percentage increase in the area of a rectangle, if each of its sides is increased by 20% is:

2. A man walked diagonally across a square lot. Approximately, what was the percent saved by not walking along the edges?

3. The diagonal of a rectangle is the square root of 41 cm and its area is 20 sq. cm. The perimeter of the rectangle must be:

4. What is the least number of squares tiles required to pave the floor of a room 15 m 17 cm long and 9 m 2 cm broad?

5. The ratio between the perimeter and the breadth of a rectangle is 5 : 1. If the area of the rectangle is 216 sq. cm, what is the length of the rectangle?

6. An error 2% in excess is made while measuring the side of a square. The percentage of error in the calculated area of the square is:

7. The ratio between the length and the breadth of a rectangular park is 3 : 2. If a man cycling along the boundary of the park at the speed of 12 km/h completes one round in 8 minutes, then the area of the park (in sq. m) is:

8. A towel, when bleached, was found to have lost 20% of its length and 10% of its breadth. The percentage of decrease in area is:

9. The diagonal of the floor of a rectangular closet is 7 1/2 feet. The shorter side of the closet is 4 1/2 feet. What is the area of the closet in square feet?

10. A rectangular park 60 m long and 40 m wide has two concrete crossroads running in the middle of the park and rest of the park has been used as a lawn. If the area of the lawn is 2109 sq. m, then what is the width of the road?

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Muhammad Arslan

1. The percentage increase in the area of a rectangle, if each of its sides is increased by 20% is:

2. A man walked diagonally across a square lot. Approximately, what was the percent saved by not walking along the edges?

3. The diagonal of a rectangle is the square root of 41 cm and its area is 20 sq. cm. The perimeter of the rectangle must be:

4. What is the least number of squares tiles required to pave the floor of a room 15 m 17 cm long and 9 m 2 cm broad?

5. The ratio between the perimeter and the breadth of a rectangle is 5 : 1. If the area of the rectangle is 216 sq. cm, what is the length of the rectangle?

6. An error 2% in excess is made while measuring the side of a square. The percentage of error in the calculated area of the square is:

7. The ratio between the length and the breadth of a rectangular park is 3 : 2. If a man cycling along the boundary of the park at the speed of 12 km/h completes one round in 8 minutes, then the area of the park (in sq. m) is:

8. A towel, when bleached, was found to have lost 20% of its length and 10% of its breadth. The percentage of decrease in area is:

9. The diagonal of the floor of a rectangular closet is 7 1/2 feet. The shorter side of the closet is 4 1/2 feet. What is the area of the closet in square feet?

10. A rectangular park 60 m long and 40 m wide has two concrete crossroads running in the middle of the park and rest of the park has been used as a lawn. If the area of the lawn is 2109 sq. m, then what is the width of the road?

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About the author

Muhammad Arslan