Question 1

A room is 15 ft long and 12 ft broad. A rectangular mat is to be placed on the floor, leaving a uniform space of 1.5 ft from all the walls (on every side). What will be the cost of the mat at the rate of ₹3.50 per sq ft?

Accepted Answer

Introduction:This question tests area calculation after reducing dimensions due to margins. The mat does not cover the entire floor; it leaves a uniform gap from the walls. That means both length and breadth of the usable mat area reduce by twice the margin (once on each opposite side). After finding the mat’s area, multiplying by the cost per square foot gives the total cost. This is a direct, single-formula aptitude problem, but it checks attention to detail: many mistakes happen when people subtract the margin only once instead of twice. Given Data / Assumptions: Room length = 15 ft Room breadth = 12 ft Space from walls on each side = 1.5 ft Cost rate = ₹3.50 per sq ft Mat dimensions reduce on both sides, so subtract 2 * 1.5 from each dimension Concept / Approach:Mat length = 15 - 2*1.5 and mat breadth = 12 - 2*1.5. Then mat area = length * breadth. Finally cost = area * rate. The key idea is “uniform border on all sides” means the reduction happens twice per dimension. Step-by-Step Solution:Mat length = 15 - 2*1.5 = 15 - 3 = 12 ftMat breadth = 12 - 2*1.5 = 12 - 3 = 9 ftMat area = 12 * 9 = 108 sq ftCost = 108 * 3.50 = 378 Verification / Alternative Check:Check the border logic: leaving 1.5 ft on left and 1.5 ft on right consumes 3 ft of total length. Similarly, leaving 1.5 ft on top and bottom consumes 3 ft of total breadth. So 12 ft by 9 ft is consistent. Multiplying 108 by ₹3.50 is also straightforward: 108*3 = 324 and 108*0.5 = 54, total 378. Why Other Options Are Wrong:₹472.50 or ₹496: usually from using wrong mat dimensions or wrong multiplication.₹630: often from taking full room area 15*12 = 180 and multiplying by 3.5.₹405: comes from subtracting margin only once in one dimension. Common Pitfalls:Subtracting 1.5 only once instead of twice from length/breadth.Using room area instead of mat area.Mixing up “per sq ft” with “per ft”.Arithmetic error in 108 * 3.50. Final Answer:₹378

Question 2

Find the ratio of the areas of the incircle and the circumcircle of a square. (Assume the square has side length s. The incircle touches all 4 sides, and the circumcircle passes through all 4 vertices.)

Accepted Answer

Introduction:This question tests geometric relationships between a square and its associated circles. The incircle of a square is the largest circle that fits inside the square, touching all four sides. The circumcircle is the circle that passes through all four vertices of the square. The ratio of their areas depends only on their radii. Since circle area is proportional to r^2, we need to express both radii in terms of the square’s side length, then compare the squared radii. The problem checks whether you remember how to relate a square’s diagonal to the circumradius and how the incircle radius relates to the side length. Given Data / Assumptions: Let square side = s Incircle radius r_in = s/2 (diameter equals side) Square diagonal = s*sqrt(2) Circumcircle radius r_out = (diagonal)/2 = s*sqrt(2)/2 = s/sqrt(2) Circle area formula: A = pi * r^2 Concept / Approach:Compute r_in^2 and r_out^2 in terms of s, then take ratio A_in : A_out = (pi*r_in^2) : (pi*r_out^2). The pi cancels, so only squared radii matter. This makes the solution clean and independent of any numeric side length. Step-by-Step Solution:r_in = s/2 => r_in^2 = s^2/4r_out = s/sqrt(2) => r_out^2 = s^2/2Area ratio = r_in^2 : r_out^2 = (s^2/4) : (s^2/2)Cancel s^2: ratio = (1/4) : (1/2)Multiply both terms by 4: ratio = 1 : 2 Verification / Alternative Check:Pick an easy value, say s = 2. Then incircle radius = 1, area = pi*1^2 = pi. Diagonal = 2*sqrt(2), circumradius = sqrt(2), area = pi*(sqrt(2)^2) = 2pi. Ratio = pi : 2pi = 1 : 2. This numeric check confirms the algebraic result. Why Other Options Are Wrong:1:4 or 4:1: would require one radius to be double the other, which is not true here.2:1: reverses the correct ratio.3:2: does not match the squared-radius relationship of a square’s circles. Common Pitfalls:Using side s as the circumcircle diameter (incorrect; diagonal is the diameter).Forgetting area depends on r^2, not r.Mixing up inradius and circumradius formulas.Not cancelling pi and s^2 cleanly, leading to confusion. Final Answer:1 : 2

Question 3

The perimeter of a rectangle is 84 m. Its length is 6 m more than its breadth. What is the area of the rectangle in sq m?

Accepted Answer

Introduction:This question tests the standard perimeter and area formulas of a rectangle along with a simple linear relation between length and breadth. From the perimeter, we can determine the sum of length and breadth. Since the length is given as a fixed amount more than the breadth, we can solve the two-variable problem using substitution. Once length and breadth are found, the area is simply the product L * B. This is a classic aptitude question that checks whether you correctly interpret “length is 6 m more than breadth” and apply the perimeter formula without mistakes. Given Data / Assumptions: Perimeter P = 84 m Length L = B + 6 Perimeter of rectangle: P = 2(L + B) Area of rectangle: A = L * B Concept / Approach:Convert perimeter into L + B by dividing by 2. Then use substitution L = B + 6 to find B. Finally compute L and multiply L and B to get the area. This is a direct two-step algebra problem with no unit conversion needed. Step-by-Step Solution:2(L + B) = 84 => L + B = 42Given L = B + 6So (B + 6) + B = 422B + 6 = 42 => 2B = 36 => B = 18L = B + 6 = 18 + 6 = 24Area = L * B = 24 * 18 = 432 Verification / Alternative Check:Check perimeter: 2(24 + 18) = 2*42 = 84 m, correct. Since both constraints match, the dimensions are correct, so area must be 24*18 = 432 sq m. This also aligns with the idea that a rectangle with moderate side lengths should not produce an extremely large or small area relative to its perimeter. Why Other Options Are Wrong:330, 333, 360, 362: would correspond to different length-breadth pairs that do not satisfy both perimeter 84 and difference 6 simultaneously. Common Pitfalls:Forgetting to divide perimeter by 2 to get L + B.Treating “6 m more” as L = 6B (incorrect).Multiplying wrong values or making arithmetic error in 24*18.Mixing up perimeter and area formulas. Final Answer:432 sq m

Question 4

What is the least number of square tiles required to pave (cover completely) the floor of a room that is 15.17 m long and 9.02 m broad? Assume tiles are identical squares and there should be no cutting of tiles (use the largest possible square tile).

Accepted Answer

Introduction:This question tests the “least number of square tiles” concept, which is fundamentally a greatest common divisor (GCD) problem in disguise. To avoid cutting tiles, the side length of the square tile must exactly divide both the room’s length and breadth. To minimize the number of tiles, we must choose the largest possible tile side that still divides both dimensions. That largest possible tile side is the GCD of the two dimensions (expressed in the same unit). After finding the tile side, the number of tiles is (length/tile_side) * (breadth/tile_side). This is a multi-step aptitude question involving unit conversion and Euclidean algorithm logic. Given Data / Assumptions: Room length = 15.17 m Room breadth = 9.02 m Use identical square tiles, no cutting allowed Convert to cm to avoid decimals: 15.17 m = 1517 cm, 9.02 m = 902 cm Largest tile side = GCD(1517, 902) Concept / Approach:Find GCD using Euclidean algorithm:GCD(a, b) = GCD(b, a mod b) until remainder becomes 0. Then compute number of tiles along each side and multiply to get total tiles. This ensures the tiles fit perfectly and the count is minimal. Step-by-Step Solution:Convert: 15.17 m = 1517 cm, 9.02 m = 902 cmCompute GCD(1517, 902)1517 mod 902 = 615902 mod 615 = 287615 mod 287 = 41287 mod 41 = 0, so GCD = 41 cmTiles along length = 1517 / 41 = 37Tiles along breadth = 902 / 41 = 22Total tiles = 37 * 22 = 814 Verification / Alternative Check:If tile side is 41 cm, then 37 tiles make exactly 37*41 = 1517 cm (15.17 m) and 22 tiles make exactly 22*41 = 902 cm (9.02 m). So the flooring is covered with no cutting and no gaps. Any larger tile side would fail to divide at least one dimension, forcing cutting. Hence 814 is the least possible number of tiles. Why Other Options Are Wrong:714, 713, 744, 614: these would correspond to smaller tile sizes or incorrect division steps, meaning either tiles do not fit exactly or the chosen tile is not the largest possible, increasing the count unnecessarily. Common Pitfalls:Not converting metres to centimetres and making rounding errors with decimals.Using LCM instead of GCD.Choosing a tile side that divides one dimension but not the other.Mistake in Euclidean algorithm steps or final multiplication 37*22. Final Answer:814

Question 5

A rectangular plot measures 90 m by 50 m and is enclosed by a wire fence. Fence poles are placed 5 m apart along the boundary (including poles at the corners). How many poles are needed in total to complete the fencing?

Accepted Answer

Introduction:This question tests perimeter calculation and equal spacing logic. When poles are placed at a fixed distance along a closed boundary, the total number of poles depends on the total length of the boundary (perimeter) divided by the spacing. Because the fencing forms a closed loop, the last point coincides with the starting point. If the perimeter is exactly divisible by the spacing and poles are placed at corners naturally, then the number of poles required equals perimeter/spacing. The key is to compute the perimeter correctly and interpret “5 m apart” properly for a closed fence. Given Data / Assumptions: Length = 90 m Breadth = 50 m Spacing between poles = 5 m Rectangle perimeter P = 2(L + B) Fence is a closed loop around the rectangle Concept / Approach:Compute the perimeter first. Then divide by spacing to get the number of equal segments. In a closed loop with exact divisibility, number of poles equals number of segments because the final pole overlaps the starting position rather than adding an extra distinct pole. Step-by-Step Solution:Perimeter P = 2(90 + 50) = 2*140 = 280 mNumber of 5 m intervals around boundary = 280 / 5 = 56Therefore, total poles needed = 56 Verification / Alternative Check:Along the 90 m side, poles create 90/5 = 18 intervals, and along 50 m side, 50/5 = 10 intervals. Total intervals around rectangle = 2*18 + 2*10 = 36 + 20 = 56 intervals, matching the perimeter method. Since each interval corresponds to one step from a pole to the next in a closed loop, the count of unique poles equals 56. Why Other Options Are Wrong:60 or 65 poles: would imply smaller spacing than 5 m or incorrect perimeter.34 or 36 poles: typically result from dividing only one side or forgetting to double for opposite sides. Common Pitfalls:Using area instead of perimeter.Forgetting perimeter includes all four sides.Adding 1 extra pole unnecessarily even when the fence is a closed loop and spacing divides exactly.Arithmetic mistake in 280/5. Final Answer:56 poles

Question 6

The length and breadth of a rectangular field are in the ratio 8 : 7. The boundary of the field is painted at a cost of ₹10 per metre, and the total painting cost is ₹3000. Find the area of the rectangular field in sq m.

Accepted Answer

Introduction:This problem combines perimeter (boundary length) with ratio-based dimensions. Since painting cost is given per metre for the boundary, the total cost directly gives the perimeter. Once perimeter is known and the ratio of length to breadth is given, we can find the actual dimensions by introducing a common multiplier. After computing the length and breadth, area is found by multiplying them. This is a typical aptitude “ratio + perimeter + area” question that checks multi-step reasoning and correct translation from cost to perimeter. Given Data / Assumptions: Length : breadth = 8 : 7 Painting cost per metre = ₹10 Total painting cost = ₹3000 Perimeter P = total boundary length Perimeter formula: P = 2(L + B) Concept / Approach:First compute perimeter: P = 3000/10. Then express L = 8k and B = 7k. Substitute into perimeter equation to solve for k. Finally compute area = L * B. This keeps the process clean and prevents guesswork. Step-by-Step Solution:Perimeter P = 3000 / 10 = 300 mSo 2(L + B) = 300 => L + B = 150Let L = 8k and B = 7kThen L + B = 15k = 150k = 10L = 8*10 = 80 m, B = 7*10 = 70 mArea = 80 * 70 = 5600 sq m Verification / Alternative Check:Check perimeter: 2(80 + 70) = 2*150 = 300 m, matches boundary length from cost. Painting cost then is 300*10 = ₹3000, consistent. With correct dimensions, area must be 5600 sq m. The magnitude also makes sense for an 80 m by 70 m field. Why Other Options Are Wrong:1400, 3600, 4400, 4800: these correspond to incorrect k values or mixing up perimeter with area. They do not match L + B = 150 along with 8:7 ratio. Common Pitfalls:Using total cost as perimeter directly without dividing by ₹10 per metre.Forgetting that ratio parts must sum to 15k.Using L - B instead of L + B from perimeter relation.Arithmetic errors in 80*70. Final Answer:5600 sq m

Question 7

A rectangle has length L and width W. If the length is decreased by 4 cm and the width is increased by 3 cm, the new figure becomes a square. Also, this square has the same area as the original rectangle. Find the perimeter (in cm) of the original r…

Accepted Answer

Introduction:This is an algebraic geometry problem involving a rectangle transformed into a square under changes in dimensions. Two conditions are given: (1) after changing dimensions, the new shape is a square, meaning the new length equals the new width; and (2) the new square has the same area as the original rectangle. These two conditions create a solvable system of equations. The problem tests careful equation setup and handling of area equality correctly. The “same area” condition is crucial; without it, there would be many rectangles that could become a square after adjustments. Given Data / Assumptions: Original length = L cm Original width = W cm New length = L - 4 New width = W + 3 New figure is a square: L - 4 = W + 3 Area preserved: L * W = (L - 4) * (W + 3) Concept / Approach:Use the square condition to express L in terms of W. Then substitute into the area equation. Because L - 4 equals W + 3, the square side can be written as (W + 3), making the area equation simpler. Solve for W, compute L, then perimeter = 2(L + W). Step-by-Step Solution:Square condition: L - 4 = W + 3 => L = W + 7Square side = L - 4 = (W + 7) - 4 = W + 3Area equality: L * W = (L - 4) * (W + 3)Substitute L = W + 7: (W + 7)W = (W + 3)(W + 3)W^2 + 7W = W^2 + 6W + 97W - 6W = 9 => W = 9L = W + 7 = 16Perimeter = 2(L + W) = 2(16 + 9) = 50 Verification / Alternative Check:Original area = 16*9 = 144. New dimensions: L - 4 = 12 and W + 3 = 12, so it becomes a 12 by 12 square. New area = 12*12 = 144, same as original. Both conditions are satisfied, so the perimeter 50 cm is confirmed. Why Other Options Are Wrong:20, 30, 40, 60: these perimeters would correspond to different L and W values that cannot satisfy both “becomes a square” and “same area” simultaneously. Common Pitfalls:Using only the square condition and forgetting the same-area condition.Making the area equation L*W = (L-4) + (W+3) (wrong; area uses multiplication).Algebra mistakes when expanding (W+3)^2.Computing perimeter as L+W instead of 2(L+W). Final Answer:50 cm

Question 8

A wheel of diameter 40 cm travels a distance of 176 m. How many revolutions does the wheel make while covering this distance? (Assume no slipping.)

Accepted Answer

Introduction:This question tests the relationship between circular motion and linear distance. Each full revolution of a wheel covers a distance equal to the wheel’s circumference. Therefore, the number of revolutions equals total distance traveled divided by the circumference. The main challenge is unit consistency: diameter is given in cm while distance is given in m, so we must convert to the same unit before dividing. This is a standard aptitude question that checks circumference formula usage and careful unit conversion. Given Data / Assumptions: Wheel diameter = 40 cm Distance traveled = 176 m Convert 176 m to cm: 176 m = 17600 cm Circumference C = pi * d Number of revolutions = distance / circumference Concept / Approach:Compute circumference using C = pi*d. Convert distance to cm to match. Then revolutions = 17600 / (40*pi). Use pi = 22/7 to get an exact integer result, which is typical in aptitude problems. Step-by-Step Solution:Distance = 176 m = 17600 cmCircumference = pi * d = pi * 40 = 40pi cmRevolutions = 17600 / (40pi) = 440 / piUsing pi = 22/7: revolutions = 440 * (7/22) = 20 * 7 = 140 Verification / Alternative Check:If revolutions are 140, then distance covered = 140 * circumference = 140 * 40pi cm = 5600pi cm. Using pi = 22/7, distance = 5600 * 22/7 = 800 * 22 = 17600 cm = 176 m. This matches the given distance exactly, confirming correctness. Why Other Options Are Wrong:240 or 340: too high, would require either a much smaller wheel or longer distance.40 or 120: too low, typically from forgetting unit conversion or using radius instead of diameter in circumference. Common Pitfalls:Not converting metres to centimetres.Using circumference as 2*pi*d instead of pi*d.Using radius 40 cm instead of diameter 40 cm.Rounding pi too early and losing exactness. Final Answer:140

Question 9

A wheel makes 1000 revolutions while covering a distance of 88 km. Find the radius of the wheel. (Assume no slipping and use pi = 22/7 if needed.)

Accepted Answer

Introduction:This question tests the connection between revolutions and distance traveled by a wheel. In one revolution, a wheel covers a distance equal to its circumference. Therefore, total distance = (number of revolutions) * (circumference). When total distance and revolutions are given, we can find circumference per revolution, and then compute radius using circumference formula C = 2*pi*r. Unit conversion is important because the distance is given in km and radius will come out in metres if we convert properly. Given Data / Assumptions: Number of revolutions = 1000 Total distance = 88 km = 88000 m Distance per revolution (circumference) = total distance / revolutions Circumference formula: C = 2*pi*r Use pi = 22/7 for exact result Concept / Approach:First compute circumference of the wheel by dividing total distance by revolutions. Then solve r = C / (2*pi). Because aptitude problems often use pi = 22/7, we will compute with that to get a clean answer. Step-by-Step Solution:Total distance = 88 km = 88000 mCircumference C = 88000 / 1000 = 88 mC = 2*pi*r => r = C / (2*pi) = 88 / (2*pi) = 44 / piUsing pi = 22/7: r = 44 * (7/22) = 2 * 7 = 14 m Verification / Alternative Check:If radius is 14 m, circumference = 2*pi*14 = 28pi. With pi = 22/7, circumference = 28 * 22/7 = 88 m. Over 1000 revolutions, distance = 1000*88 = 88000 m = 88 km, exactly matching the given distance. So the radius is confirmed. Why Other Options Are Wrong:13, 12, 11, 10 m: would produce circumferences smaller than 88 m, meaning 1000 revolutions would cover less than 88 km. Common Pitfalls:Forgetting to convert 88 km into metres before dividing.Using C = pi*r or C = pi*d incorrectly.Dividing by pi instead of 2*pi when finding radius.Rounding pi and missing the exact integer radius. Final Answer:14 m

Question 10

A circular swimming pool is surrounded by a concrete wall that is 4 ft wide (uniform width). The area of the concrete wall (the ring between the outer circle and inner pool) is 11/25 of the area of the pool. Find the radius of the pool in feet.

Accepted Answer

Introduction:This question tests area relationships in concentric circles (a ring or annulus). The pool is the inner circle with radius r. The concrete wall creates an outer circle with radius (r + 4). The wall area is the difference between outer area and inner area. The problem gives a fractional relationship between wall area and pool area, which leads to a quadratic equation in r. Solving that quadratic carefully yields the pool’s radius. This is a multi-step geometry and algebra integration question, making it harder than direct formula substitution. Given Data / Assumptions: Pool radius = r ft Wall width = 4 ft Outer radius = r + 4 Pool area = pi * r^2 Wall area = pi * (r + 4)^2 - pi * r^2 Given: wall area = (11/25) * pool area Concept / Approach:Set up: pi((r+4)^2 - r^2) = (11/25)*pi*r^2. Cancel pi to simplify. Expand (r+4)^2 - r^2 to get a linear expression in r, then solve the resulting quadratic equation for r. Choose the positive root because radius must be positive. Step-by-Step Solution:Wall area: (r+4)^2 - r^2 = (r^2 + 8r + 16) - r^2 = 8r + 16Equation: 8r + 16 = (11/25) * r^2Multiply by 25: 200r + 400 = 11r^2Rearrange: 11r^2 - 200r - 400 = 0Discriminant = 200^2 + 4*11*400 = 40000 + 17600 = 57600sqrt(57600) = 240r = (200 + 240) / (2*11) = 440/22 = 20 Verification / Alternative Check:With r = 20, outer radius = 24. Pool area = pi*400. Wall area = pi*(576 - 400) = pi*176. Ratio wall/pool = 176/400 = 11/25, exactly matches the condition. This confirms r = 20 ft is correct and also shows why the other root (negative) is invalid for a radius. Why Other Options Are Wrong:10, 30, 40, 16 ft: these do not satisfy the exact fractional area condition 11/25 when substituted into the ring-area ratio. Common Pitfalls:Forgetting to subtract inner area from outer area for wall area.Not cancelling pi, leading to messy calculations.Expanding (r+4)^2 incorrectly.Choosing the negative quadratic root, which is not physically meaningful. Final Answer:20 ft

Question 11

The breadth of a rectangular field is 60% of its length. If the perimeter of the field is 800 m, what is the area of the field in sq m?

Accepted Answer

Introduction:This question tests perimeter-to-area conversion when a proportional relationship between breadth and length is given. If breadth is 60% of length, we can write breadth as 0.6 times length. The perimeter gives an equation involving length and breadth. Solving that equation provides the actual dimensions, and then area is computed as length * breadth. This is a standard aptitude problem that checks whether you correctly convert a percentage relationship into an algebraic equation and avoid common mistakes like treating 60% as 60 metres or confusing perimeter with area. Given Data / Assumptions: Perimeter P = 800 m Breadth B = 60% of length L => B = 0.60L Perimeter formula: P = 2(L + B) Area formula: A = L * B Concept / Approach:Use perimeter to find L + B. Substitute B = 0.6L to solve for L. Then compute B, and finally compute area. This is a multi-step but straightforward algebraic substitution problem. Step-by-Step Solution:2(L + B) = 800 => L + B = 400B = 0.60LSo L + 0.60L = 4001.60L = 400 => L = 400 / 1.60 = 250B = 0.60 * 250 = 150Area = L * B = 250 * 150 = 37500 Verification / Alternative Check:Check perimeter: 2(250 + 150) = 2*400 = 800 m, matches. Also breadth is 150, which is 60% of 250, matches. Therefore area must be 250*150 = 37500 sq m. The value is also reasonable for a field with sides 250 m and 150 m. Why Other Options Are Wrong:47500, 57500, 77500: come from incorrect length calculation or wrong multiplication.35000: typically due to rounding or using 0.6 on the wrong dimension. Common Pitfalls:Forgetting perimeter is 2(L + B).Treating 60% as 60 instead of 0.60.Mixing up which dimension is 60% of the other.Computing area using perimeter values directly. Final Answer:37500 sq m

Question 12

A typist uses a sheet measuring 20 cm by 30 cm. Margins are left as follows: 2 cm on each side (left and right) and 3 cm on top and bottom. What percentage of the page area is used for typing?

Accepted Answer

Introduction:This question tests area calculation with margins and percentage conversion. A margin reduces the usable typing region. Since margins are on both sides, we subtract twice the margin value from each dimension. After finding the usable (typing) area, we divide it by the total page area and multiply by 100 to get the percentage used. This is a straightforward problem, but it checks attention to “each side” and “top and bottom,” which are common sources of errors. Given Data / Assumptions: Sheet size = 20 cm by 30 cm Side margins: 2 cm on left and 2 cm on right Top and bottom margins: 3 cm each Total area = 20 * 30 Typing area = (20 - 2*2) * (30 - 2*3) Concept / Approach:Compute total area. Compute effective width and effective height after margins. Compute typing area. Then percentage used = (typing area / total area) * 100. This uses basic rectangle area and percent formula. Step-by-Step Solution:Total area = 20 * 30 = 600 sq cmEffective width = 20 - 2*2 = 20 - 4 = 16 cmEffective height = 30 - 2*3 = 30 - 6 = 24 cmTyping area = 16 * 24 = 384 sq cmPercentage used = (384 / 600) * 100384/600 = 64/100 = 0.64Percentage used = 64% Verification / Alternative Check:Reduce the fraction: 384/600 simplifies by dividing by 6 to 64/100. That directly equals 64%. This confirms the arithmetic without a calculator. The result also makes sense: margins reduce both dimensions noticeably, so used area should be significantly less than 100%. Why Other Options Are Wrong:74%, 84%, 94%: too high; would imply very small margins or incorrect subtraction (only subtracting once).70%: usually from rounding errors or using wrong effective dimensions. Common Pitfalls:Subtracting 2 cm only once instead of left + right.Subtracting 3 cm only once instead of top + bottom.Computing percentage on the remaining blank margin area instead of typing area.Arithmetic mistake in 16*24 or fraction conversion. Final Answer:64%

Question 13

The difference between the circumference of a circle and its semi-diameter is 37 cm. (Here, semi-diameter means the radius.) What is the diameter of the circle in cm? (Use pi = 22/7.)

Accepted Answer

Introduction:This question tests understanding of circle measures and forming an equation from a word statement. The circumference of a circle is 2*pi*r. The term “semi-diameter” is another name for radius. The question says the difference between circumference and radius is 37 cm. That means (2*pi*r - r) = 37. Solving this equation gives r, and then diameter = 2r. The problem checks whether you correctly interpret “semi-diameter” and handle pi = 22/7 to get an exact integer value. Given Data / Assumptions: Circumference C = 2*pi*r Semi-diameter = radius r Given: C - r = 37 Use pi = 22/7 Diameter d = 2r Concept / Approach:Translate statement into equation: 2*pi*r - r = 37. Factor r: r(2*pi - 1) = 37. Substitute pi = 22/7, solve for r, then compute diameter = 2r. This is a direct algebraic approach and avoids confusion. Step-by-Step Solution:2*pi*r - r = 37r(2*pi - 1) = 37Using pi = 22/7: 2*pi - 1 = 44/7 - 1 = (44 - 7)/7 = 37/7So r * (37/7) = 37r = 37 * (7/37) = 7 cmDiameter = 2r = 14 cm Verification / Alternative Check:With r = 7, circumference = 2*pi*r = 2*(22/7)*7 = 44 cm. Difference C - r = 44 - 7 = 37 cm, matching the condition. Therefore the diameter is 14 cm. The clean cancellation of 37 is a strong confirmation of correctness. Why Other Options Are Wrong:12, 16, 18, 20 cm: would imply different radii. Substituting those radii into C - r does not produce 37 cm when using pi = 22/7. Common Pitfalls:Interpreting semi-diameter as half of radius (incorrect).Using circumference formula pi*d but mixing d and r incorrectly.Forgetting to use pi = 22/7 consistently.Computing diameter as r instead of 2r. Final Answer:14 cm

Question 14

The sides of a triangle are 26 cm, 24 cm, and 10 cm. Find the area of this triangle in sq cm.

Accepted Answer

Introduction:This question tests triangle area computation from side lengths using Heron’s formula. When height is not directly given and the triangle is not obviously right-angled (though it could be checked), Heron’s formula provides a reliable method: area = sqrt(s*(s-a)*(s-b)*(s-c)), where s is the semi-perimeter. The sides here (26, 24, 10) produce neat factors, resulting in an exact area. The problem checks whether you can compute semi-perimeter correctly and handle the multiplication under the square root without mistakes. Given Data / Assumptions: Sides: a = 26 cm, b = 24 cm, c = 10 cm Semi-perimeter s = (a + b + c) / 2 Heron’s formula: Area = sqrt(s*(s-a)*(s-b)*(s-c)) Concept / Approach:Compute s first. Then compute (s-a), (s-b), (s-c). Multiply the four values and take the square root. Because the numbers factor nicely, we simplify before taking the square root to avoid large arithmetic errors. Step-by-Step Solution:s = (26 + 24 + 10) / 2 = 60 / 2 = 30s - a = 30 - 26 = 4s - b = 30 - 24 = 6s - c = 30 - 10 = 20Area = sqrt(30 * 4 * 6 * 20)Group terms: (30*20) * (4*6) = 600 * 24 = 14400Area = sqrt(14400) = 120 Verification / Alternative Check:Since 10, 24, 26 is actually a right triangle set (10^2 + 24^2 = 100 + 576 = 676 = 26^2), the triangle is right-angled with legs 10 and 24. So area can also be (1/2)*10*24 = 120 sq cm. This confirms the Heron’s formula result perfectly and provides a strong consistency check. Why Other Options Are Wrong:108, 112, 116: usually come from wrong semi-perimeter or wrong multiplication under the square root.144: often comes from forgetting the 1/2 in the right-triangle area method. Common Pitfalls:Forgetting to divide perimeter by 2 for s.Making subtraction mistakes in s-a, s-b, s-c.Not recognizing simplification opportunities before taking square root.Arithmetic errors when multiplying 30*4*6*20. Final Answer:120 sq cm

Question 15

Find the area of the square whose side is equal to the diagonal of a rectangle of length 3 cm and breadth 4 cm. Give the answer in sq cm.

Accepted Answer

Introduction:This question tests the use of Pythagoras theorem to find the diagonal of a rectangle and then using that diagonal as the side of a square. The diagonal of a rectangle forms the hypotenuse of a right triangle whose legs are the rectangle’s length and breadth. Once the diagonal is found, the square’s side is known directly, and the area is the square of that side. This is a short and direct question, but it checks whether you correctly identify the diagonal formula and avoid confusing diagonal with perimeter or area. Given Data / Assumptions: Rectangle length = 3 cm Rectangle breadth = 4 cm Rectangle diagonal d = sqrt(L^2 + B^2) Square side = d Square area = side^2 Concept / Approach:Use Pythagoras theorem: d = sqrt(3^2 + 4^2). Then square area = d^2. Notably, area becomes (3^2 + 4^2) directly because (sqrt(x))^2 = x. This gives a very clean computation. Step-by-Step Solution:Diagonal d = sqrt(3^2 + 4^2)d = sqrt(9 + 16) = sqrt(25) = 5 cmSquare side = 5 cmSquare area = 5^2 = 25 sq cm Verification / Alternative Check:The 3-4-5 right triangle is a standard Pythagoras triple, so the diagonal being 5 cm is immediately consistent. If the square side is 5 cm, its area must be 25 sq cm. Also note the shortcut: square area = d^2 = (3^2 + 4^2) = 25 directly, confirming the same result without computing the square root explicitly. Why Other Options Are Wrong:16 or 9: correspond to using 4 or 3 as the square side (ignoring diagonal).4: usually comes from confusing breadth with diagonal.20: comes from using 3*4+? or misapplying formula. Common Pitfalls:Adding 3 and 4 to get diagonal (incorrect; diagonal uses squares).Using rectangle area (12) as diagonal or square side (nonsense).Forgetting the square side equals diagonal, not half diagonal.Arithmetic error in 3^2 + 4^2. Final Answer:25 sq cm

Question 16

A rectangular field is to be fenced on three sides only, leaving one side of 20 ft completely uncovered. If the area of the field is 680 sq ft, how many feet of fencing will be required?

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Introduction:This question tests combining rectangle area with partial perimeter (fencing on only three sides). The field has one side left uncovered, and that uncovered side is given as 20 ft. Using area = length * breadth, we can find the other dimension. Then, because only three sides are fenced, we do not take the full perimeter. Instead, we add the uncovered side’s opposite side (which is equal in length) and the two adjacent sides. This type of question checks whether you correctly interpret “three sides fenced” and do not accidentally compute full perimeter. Given Data / Assumptions: Area of rectangle = 680 sq ft One side uncovered = 20 ft Let the uncovered side be the length L = 20 ft Breadth B = area / L Fencing required on three sides = L + 2B (since the opposite length side is fenced, and both breadth sides are fenced) Concept / Approach:Use area to find the missing dimension: B = 680/20. Then compute fencing as three sides. If the uncovered side is 20, the fenced sides are: the other 20 plus both breadths. Total fencing = 20 + B + B = 20 + 2B. Step-by-Step Solution:Given L (uncovered side) = 20 ftArea = L * B => 680 = 20 * BB = 680 / 20 = 34 ftFencing on three sides = 20 + 34 + 34Total fencing = 20 + 68 = 88 ft Verification / Alternative Check:Check area: 20*34 = 680 sq ft, correct. If we fenced all four sides, perimeter would be 2(20 + 34) = 108 ft. Since one 20 ft side is left open, fencing should be 108 - 20 = 88 ft, matching our direct three-side sum. This alternate method confirms the answer cleanly. Why Other Options Are Wrong:108 ft: is the full perimeter (fencing on all sides), not three sides.44 ft or 22 ft: ignores one or both breadth sides.11 ft: not consistent with the given area and side length. Common Pitfalls:Computing full perimeter even though only three sides are fenced.Using 20 as breadth instead of the uncovered side length.Forgetting opposite sides of a rectangle are equal.Arithmetic mistake in 680/20. Final Answer:88 ft

Question 17

The breadth of a rectangular field is 75% of its length. If the perimeter of the field is 1050 m, what is the area of the field (in sq m)?

Accepted Answer

Introduction / Context:This problem tests basic mensuration for rectangles using perimeter and a relationship between length and breadth. The key idea is to convert the percentage statement into an algebraic relationship, use the perimeter formula to find the actual dimensions, and then compute area. Because a rectangle's perimeter depends only on the sum of length and breadth, once we express breadth in terms of length (or vice versa), we can solve in one clean equation. After the dimensions are known, area is simply length * breadth. This is a standard one-equation substitution problem that rewards careful unit handling (metres and square metres). Given Data / Assumptions: Breadth is 75% of length Perimeter = 1050 m Rectangle perimeter formula: P = 2*(L + B) Concept / Approach:Convert 75% into a multiplier: B = 0.75*L. Use the perimeter to solve for L, then compute B, then area = L*B. Step-by-Step Solution: Let length = L m, breadth = B m B = 75% of L = 0.75*L Perimeter: 2*(L + B) = 1050 => L + B = 525 Substitute B: L + 0.75L = 525 => 1.75L = 525 L = 525 / 1.75 = 300 m B = 0.75*300 = 225 m Area = L*B = 300*225 = 67500 sq m Verification / Alternative check:Check perimeter using found dimensions: 2*(300+225)=2*525=1050 m, which matches. So the dimensions are correct, and the area computed from them is reliable. Why Other Options Are Wrong: 65700 and 70500 come from arithmetic slips in 300*225. 54500 and 78700 result from incorrect L and B due to wrong handling of 75% or perimeter. Common Pitfalls:Using 75% as 75 (instead of 0.75), forgetting to divide perimeter by 2, or mixing up which dimension is 75% of the other. Final Answer:The area of the field is 67500 sq m.

Question 18

The length, breadth, and height of a room are in the ratio 7:3:1. If the breadth and height are doubled while the length is halved, by what percent does the total area of the four walls increase?

Accepted Answer

Introduction / Context:This question tests surface area of the four walls of a cuboid-shaped room (lateral surface area). The area of the four walls depends on height and the perimeter of the base: Area = 2*h*(l + b). When dimensions change, you should compute the new lateral area in terms of the old using ratios, not absolute measurements. Since only ratios are given, we represent dimensions using a common factor and compare the expressions. Finally, percent increase is computed as ((new - old) / old) * 100. Given Data / Assumptions: Length:breadth:height = 7:3:1 New length = half of old length New breadth = 2 times old breadth New height = 2 times old height Four walls area formula: 2*h*(l + b) Concept / Approach:Let l=7x, b=3x, h=x. Compute original lateral area A0. Update dimensions to l'=3.5x, b'=6x, h'=2x and compute new area A1. Then percent increase = ((A1-A0)/A0)*100. Step-by-Step Solution: Original: l = 7x, b = 3x, h = x A0 = 2*h*(l + b) = 2*x*(7x + 3x) = 2*x*(10x) = 20x^2 New: l' = (1/2)*7x = 3.5x, b' = 2*3x = 6x, h' = 2*x = 2x A1 = 2*h'*(l' + b') = 2*(2x)*(3.5x + 6x) = 4x*(9.5x) = 38x^2 Increase = 38x^2 - 20x^2 = 18x^2 Percent increase = (18x^2 / 20x^2)*100 = 90% Verification / Alternative check:The ratio A1/A0 = 38/20 = 1.9, meaning the new area is 190% of the old, so the increase is 90%. This matches the computed result. Why Other Options Are Wrong: 88% and 84% come from adding changes separately instead of using the full formula. 85% and 80% occur when the halving/doubling is applied to the wrong dimension or missed. Common Pitfalls:Using total surface area instead of four walls area, forgetting the factor 2 in 2*h*(l+b), or mishandling 7/2 as 3 (instead of 3.5). Final Answer:The total area of the four walls increases by 90%.

Question 19

A rectangle has dimensions 13 units by 9 units. How many square units are in its area?

Accepted Answer

Introduction / Context:This question tests the most basic area formula for a rectangle. Area measures how many unit squares cover the rectangle without gaps or overlaps. When a rectangle has side lengths in units, its area in square units is simply the product of length and breadth. The wording “13 by 9” indicates two perpendicular sides of 13 units and 9 units. Multiplying these gives the number of square units inside the rectangle. This is a direct substitution problem with no conversions, so accuracy comes from using the correct formula and correct multiplication. Given Data / Assumptions: Length = 13 units Breadth = 9 units Area of rectangle = length * breadth Concept / Approach:Use the rectangle area formula: A = L*B. Substitute 13 and 9 and compute the product to get the area in square units. Step-by-Step Solution: A = L * B A = 13 * 9 A = 117 square units Verification / Alternative check:Think of 13 columns of unit squares and 9 rows of unit squares. Total unit squares = 13*9, which confirms 117. Why Other Options Are Wrong: 13 and 9 are side lengths, not area. 13/9 is a ratio, not area. 22 is unrelated and could come from adding sides (13+9). Common Pitfalls:Adding side lengths instead of multiplying, confusing perimeter with area, or forgetting that area is measured in square units, not units. Final Answer:The area is 117 square units.

Question 20

Find the area of a square if one of its diagonals is 3.8 m long. What is the area of the square (in sq m)?

Accepted Answer

Introduction / Context:This question tests the relationship between the diagonal of a square and its area. In a square, the diagonal forms a right triangle with two equal sides. By the Pythagoras relationship, diagonal = side*sqrt(2). Once the side length is expressed in terms of the diagonal, area is side^2. A very efficient identity is: area of a square = (diagonal^2) / 2. Using this avoids calculating sqrt(2) explicitly and reduces rounding errors. Units must be handled carefully: diagonal is in metres, so area will be in square metres. Given Data / Assumptions: Diagonal d = 3.8 m For a square: d = s*sqrt(2) Area A = s^2 Concept / Approach:Use A = d^2 / 2. Compute 3.8^2, divide by 2, and keep the result in sq m. Step-by-Step Solution: A = d^2 / 2 d^2 = 3.8 * 3.8 = 14.44 A = 14.44 / 2 = 7.22 Verification / Alternative check:Compute side explicitly: s = 3.8 / sqrt(2). Then area s^2 = (3.8^2)/(2) = 7.22, which matches the direct method and confirms correctness. Why Other Options Are Wrong: 6.22 and 8.22 come from incorrect squaring or division. 4.22 and 3.22 result from using d/2 as side or other wrong diagonal-side relations. Common Pitfalls:Treating diagonal as a side, forgetting to square, or dividing by sqrt(2) incorrectly and introducing rounding errors too early. Final Answer:The area of the square is 7.22 sq m.

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