Question 1

Engine consumption comparison: 9 engines of type A consume 24 tonnes of coal when each runs 8 h/day. If 3 engines of type A consume as much as 4 engines of type B, how much coal is required for 8 engines of type B each running 13 h/day?

Accepted Answer

Introduction / Context:This question involves two engine types with a stated consumption equivalence. By first finding consumption per engine-hour for type A and then using the given equivalence to infer type B’s consumption, we can compute the required coal for type B in the new schedule.Given Data / Assumptions:Type A: 9 engines * 8 h/day consume 24 tonnes ⇒ per engine-hour for A is constantEquivalence: 3 A-engines consume as much as 4 B-engines (same time)Target: 8 B-engines running 13 h/day; find total coalConcept / Approach:Let a be tonnes per engine-hour for A; then 9*8*a = 24 ⇒ a = 1/3. From 3A ≡ 4B, per engine-hour for B is b = (3/4)*a = 1/4. Then compute 8*13*b for the required total consumption.Step-by-Step Solution:Type A per engine-hour a = 24 / (9*8) = 24/72 = 1/3 tonneType B per engine-hour b = (3/4) * a = (3/4) * (1/3) = 1/4 tonneTotal for 8 B-engines, 13 h: 8 * 13 * (1/4) = 26 tonnesVerification / Alternative check:Equivalent A-engines for 8 B-engines: 4B ≡ 3A ⇒ 8B ≡ 6A. Consumption via A-rate: 6 * 13 * (1/3) = 26 tonnes—same result, confirming consistency.Why Other Options Are Wrong:20, 23: underestimate consumption given the derived 1/4 tonne per engine-hour for B.39: overestimates; requires higher per-hour usage than implied by the equivalence.24: still under the computed requirement of 26.Common Pitfalls:Applying 3A = 4B backwards (treating B as more consumptive), or forgetting to convert to per engine-hour before scaling.Final Answer:26 metric tonnes

Question 2

Scaling workforce to hit a target: If 40 men make 30 boxes in 20 days, how many more men are needed to make 60 boxes in 25 days (same efficiency)?

Accepted Answer

Introduction / Context:This is a direct work-rate proportionality problem. Boxes made are proportional to men * days (assuming identical hours and efficiency). We first compute output per man-day from the first scenario and then determine required men for the new target, finally reporting “how many more” than 40 are needed.Given Data / Assumptions:Base: 40 men make 30 boxes in 20 daysTarget: 60 boxes in 25 daysSame daily hours and same efficiencyConcept / Approach:Boxes per man-day = 30 / (40*20). For target, let M men be needed so that M * 25 * (boxes per man-day) = 60. Solve for M, then subtract 40 to get “additional men”.Step-by-Step Solution:Boxes per man-day = 30 / 800 = 0.0375Let M be required men: M * 25 * 0.0375 = 60M = 60 / (25 * 0.0375) = 60 / 0.9375 = 64Additional men = 64 − 40 = 24Verification / Alternative check:Proportion method: Men ∝ (Boxes)/(Days) with same productivity ⇒ M = 40 * (60/30) * (20/25) = 40 * 2 * 0.8 = 64. Additional = 24. Same result by a quicker ratio approach.Why Other Options Are Wrong:20, 27, 28, 18: Do not match the computed increment from 40 to the required 64 men.Common Pitfalls:Forgetting it asks for “more men” (M − 40), or inverting a ratio (using 25/20 instead of 20/25). Keep ratio direction consistent.Final Answer:24

Question 3

Combined men-and-boys work rates: 2 men and 7 boys finish a job in 14 days; 3 men and 8 boys finish it in 11 days. In how many days will 8 men and 6 boys finish three times this job?

Accepted Answer

Introduction / Context:This problem involves two worker types (men and boys) with different efficiencies. By forming two equations from the given completion times, we can solve for the individual daily rates, combine them for a new team, and then scale the duration for three times the original amount of work.Given Data / Assumptions:(2m + 7b) completes 1 job in 14 days ⇒ daily rate = 1/14(3m + 8b) completes 1 job in 11 days ⇒ daily rate = 1/11Find time for 8m + 6b to complete 3 jobsConcept / Approach:Let m = work/day by one man and b = work/day by one boy. Solve the linear system: 2m + 7b = 1/14 and 3m + 8b = 1/11. Then compute the combined rate of 8m + 6b and invert to get time for 1 job; finally multiply by 3 for three jobs.Step-by-Step Solution:From 2m + 7b = 1/14 and 3m + 8b = 1/11Subtracting (after scaling): 5b = 3/14 − 2/11 = 5/154 ⇒ b = 1/154Then 2m + 7*(1/154) = 1/14 ⇒ 2m + 1/22 = 1/14 ⇒ 2m = 4/154 ⇒ m = 1/77Rate of 8m + 6b = 8/77 + 6/154 = 16/154 + 6/154 = 22/154 = 11/77 per dayTime for 1 job = 1 / (11/77) = 7 days ⇒ Time for 3 jobs = 3 * 7 = 21 daysVerification / Alternative check:Sanity: 8 men contribute 8/77 and 6 boys 6/154 per day; sum is 11/77, a clean fraction yielding 7 days per job. Scaling to triple work preserves linearity: 3 * 7 = 21 days.Why Other Options Are Wrong:18, 34, 39, 24: Do not match the computed rate and scaled duration for triple work.Common Pitfalls:Arithmetic slips in fraction subtraction or forgetting to take the reciprocal when converting rate to time. Keep fractions aligned over a common denominator.Final Answer:21

Question 4

Work–men–days (direct proportion): If 20 men can build a similar wall 56 meters long in 6 days, what length of such a wall can 35 men build in 3 days?

Accepted Answer

Introduction / Context:This is a classic unitary-method (work–time) question. For similar work, total output is directly proportional to the product men * days, assuming all men work at the same constant rate. We use this proportionality to scale from one scenario to another.Given Data / Assumptions:20 men in 6 days build 56 meters of wall.We need the length built by 35 men in 3 days (same wall type and productivity per man).Rates are constant and additive across workers.Concept / Approach:Total work (length of wall) ∝ men * days. Hence, length2 = length1 * (men2 * days2) / (men1 * days1). Compute once and select the matching option.Step-by-Step Solution:Known output: 56 m for (20 men * 6 days) = 120 man-days.Required scenario: (35 men * 3 days) = 105 man-days.Per man-day output = 56 / 120 = 0.4667 m.Length for 105 man-days = 105 * 0.4667 ≈ 49 m.Verification / Alternative check:Use direct ratio: length2 = 56 * (35*3)/(20*6) = 56 * 105 / 120 = 56 * 0.875 = 49 m. Same result via either method.Why Other Options Are Wrong:40, 43, 48: All underestimate the proportional increase from 120 to 105 man-days relative to the 56 m benchmark; only 49 fits the precise scaling.Common Pitfalls:Dividing by the wrong base (e.g., scaling by men only and forgetting days) or assuming inverse proportion. Here, output grows with both men and days.Final Answer:49

Question 5

Meshing gears (inverse revolutions): A 6-cog wheel meshes with a larger 14-cog wheel. When the smaller wheel makes 21 revolutions, how many revolutions does the larger wheel make?

Accepted Answer

Introduction / Context:For meshed gears, revolutions are inversely proportional to the number of cogs (teeth). Fewer cogs imply more turns to pass the same number of teeth, and vice versa. We apply inverse proportion to map revolutions between the two wheels.Given Data / Assumptions:Smaller wheel: 6 cogs; Larger wheel: 14 cogs.Smaller wheel makes 21 revolutions.No slipping; perfect meshing; same tooth engagement count.Concept / Approach:rev_small * cogs_small = rev_large * cogs_large. Solve for rev_large using the product equality of engaged teeth counts.Step-by-Step Solution:rev_large = rev_small * cogs_small / cogs_large= 21 * 6 / 14 = 126 / 14 = 9Verification / Alternative check:Tooth counts: small engages 21 * 6 = 126 teeth; for the big gear to see 126 teeth with 14 per revolution, it must make 126/14 = 9 revolutions.Why Other Options Are Wrong:4 or 7: Undercount the necessary turns to match 126 engaged teeth.14: Would over-rotate the larger gear compared to meshing constraints.Common Pitfalls:Using direct proportion (wrong here) instead of inverse proportion. Always remember: more cogs ⇒ fewer revolutions for a fixed tooth transfer.Final Answer:9

Question 6

Manpower adjustment on a fixed deadline: A contractor plans to finish a job in 40 days. He employs 100 men initially and adds 100 more after 35 days, finishing on time. If he had not added extra men, how many days late would the work be?

Accepted Answer

Introduction / Context:This is a work–time problem converting varying workforce over time into total man-days. Once the total man-days for completion are known, we can compute the time if only the original crew worked throughout and compare with the target to find the delay.Given Data / Assumptions:Target duration = 40 days.First 35 days: 100 men.Last 5 days: 200 men (100 original + 100 additional).Uniform productivity; man-days add linearly.Concept / Approach:Total work W equals the sum of man-days used in the actual schedule. Then compute how long 100 men alone would take: time_alone = W / 100. The delay = time_alone − 40, if positive.Step-by-Step Solution:Total work W = (100 men * 35 days) + (200 men * 5 days)= 3500 + 1000 = 4500 man-days.If only 100 men worked: time = 4500 / 100 = 45 days.Delay relative to 40-day target = 45 − 40 = 5 days late.Verification / Alternative check:Check proportionality: replacing the final 5 days of 100 men with 200 men adds 500 extra man-days, exactly the shortfall that would have caused a 5-day delay at 100 man-days/day.Why Other Options Are Wrong:7, 12, 18: Do not match the precise calculation showing a 5-day delay.None: Correct here because 5 is not listed among the numerical choices.Common Pitfalls:Forgetting that “additional 100 men” means total 200 in the last 5 days, or miscomputing total man-days. Always aggregate man-days and then divide by the constant crew size for alternative schedules.Final Answer:None

Question 7

Inverse men–days relation: 36 men can complete a piece of work in 18 days. In how many days will 27 men complete the same work (same rate)?

Accepted Answer

Introduction / Context:With constant productivity, work done equals men * days. For a fixed job, men and days are inversely proportional. Reducing workers increases the required days proportionally.Given Data / Assumptions:36 men finish in 18 days.Find days for 27 men at same rate.Concept / Approach:Use men1 * days1 = men2 * days2. Solve for days2 with the given numbers.Step-by-Step Solution:36 * 18 = 27 * days2days2 = (36 * 18) / 27 = (36/27) * 18 = (4/3) * 18 = 24Verification / Alternative check:Total man-days = 36 * 18 = 648. With 27 men, days = 648 / 27 = 24. Same result.Why Other Options Are Wrong:23, 34, 39: Do not preserve the constant man-days 648 for the fixed job.Common Pitfalls:Using direct instead of inverse proportion between men and days for fixed work. Keep the product men * days constant for the given job.Final Answer:24

Question 8

Scaling workforce and scope: Some persons can complete a job in 12 days. If the number of persons is doubled, how long will it take to complete half of that job?

Accepted Answer

Introduction / Context:For fixed productivity, time varies directly with work and inversely with manpower. Here both the team size and the amount of work change, so we combine both effects carefully.Given Data / Assumptions:Original: N persons finish the full job in 12 days.New plan: 2N persons, and only half of the original job.Concept / Approach:Total work W = N * 12 (person-days). Half work is W/2. With 2N persons, time t = (W/2) / (2N) = W / (4N). Substitute W = 12N to solve.Step-by-Step Solution:W = N * 12.t = (W/2) / (2N) = (12N/2) / (2N) = 6 / 2 = 3 days.Verification / Alternative check:Check proportionality: doubling persons halves the time; halving the work halves the time again. Starting from 12 days: 12/2/2 = 3 days.Why Other Options Are Wrong:16, 12, 8: Ignore the compounded effect of both changes (work halved and manpower doubled).Common Pitfalls:Applying only one of the two changes or adding/dividing in the wrong order. Always express in person-days to avoid confusion.Final Answer:3

Question 9

Same job, different crew size: If 36 men can do a piece of work in 25 hours, in how many hours will 15 men do the same work (same rate)?

Accepted Answer

Introduction / Context:Work done is men * hours for a fixed job. With fewer men, the required hours increase so that the product remains constant. We compute total man-hours and then divide by the new crew size.Given Data / Assumptions:36 men complete in 25 hours.New crew: 15 men.Concept / Approach:Total man-hours = men * hours = constant for the job. So hours2 = (36 * 25) / 15.Step-by-Step Solution:Total man-hours = 36 * 25 = 900.Hours for 15 men = 900 / 15 = 60 hours.Verification / Alternative check:Proportionality: reducing men by a factor of 36/15 = 2.4 increases time by 2.4; 25 * 2.4 = 60.Why Other Options Are Wrong:45, 55, 72: Do not preserve the constant 900 man-hours for completion.Common Pitfalls:Multiplying instead of dividing by the new crew size or mixing units. Keep the men * hours product equal to the known job total.Final Answer:60

Question 10

On a scale of map, 0.6 cm represents 6.6 km. If the distance between the points on the map is 80.5 cm, the actual distance between these points is

Accepted Answer

Concept & Scaling in Maps: Map scales represent a direct ratio between the distance on the map and the actual distance on the ground. Actual Distance = Map Distance × ( \frac{Actual Scale Reference}{Map Scale Reference} ) Step-by-Step Solution: • Given: Map scale is 0.6 cm = 6.6 km. • Calculate the actual distance represented by 1 cm on the map: (6.6/0.6) = 11 km. • Given map distance between points = 80.5 cm. • Actual distance = 80.5 × 11 = 885.5 km. Exam Strategy & Shortcut: Multiply 80.5 × 11 quickly using the 11-multiplication trick. Split the numbers: 805 × 11. 8 | 8+0 | 0+5 | 5 \rightarrow 8855. Add the one decimal place back to get 885.5. Common Pitfall: Setting up the ratio upside down or incorrectly dealing with the decimals in 0.6 and 6.6, which could lead to answers off by a factor of 10. Final Answer: Therefore, the correct answer is 885.5 km.

Question 11

An industrial loom weaves 0.128 metres of cloth every second. Approximately, how many seconds will it take for the loom to weave 25 metres of cloth?  (M.B.A., 2003)

Accepted Answer

Concept & Time and Work: When a constant rate of work is given, the total time required is the total work divided by the rate of work. Time = \frac{Total Work}{Rate of Work} Step-by-Step Solution: • Given rate: 0.128 metres per second. • Total required cloth (work): 25 metres. • Total time = (25/0.128) seconds. • Convert decimal to fraction to simplify: 0.128 = (128/1000). • Time = (25/\frac{128){1000}} = (25000/128). • 25000 \div 128 = 195.3125 seconds. • The approximate time is 195 seconds. Exam Strategy & Shortcut: To approximate (25000/128), recognize that 128 × 2 = 256. So 128 × 200 = 25600. Since 25000 is just slightly less than 25600, the answer must be slightly less than 200. Looking at the options, 195 is the only logical fit. Common Pitfall: Getting bogged down in exact long division of a 5-digit number by a 3-digit number. When asked for an approximation, use rounding and bounding to save time. Final Answer: Therefore, the correct answer is 195.

Question 12

A snapshot $1\frac{7}{8}" 	imes 2\frac{1}{2}"$ is to be enlarged so that the longer dimension is $4"$. What will be the dimension of the shorter side?  (DMRC, 2003)

Accepted Answer

Concept & Proportional Scaling: When an image is enlarged, the ratio of its dimensions remains constant. Therefore, the ratio of the new sides must equal the ratio of the original sides. \frac{New Shorter Side}{Original Shorter Side} = \frac{New Longer Side}{Original Longer Side} Step-by-Step Solution: • Identify original dimensions and convert them to improper fractions: • Original shorter side = 1(7/8) = (15/8)". • Original longer side = 2(1/2) = (5/2)". • New longer side = 4". Let the new shorter side be x. • Set up the proportion: (x/15/8) = (4/5/2). • Simplify the right side: (4/5/2) = 4 × (2/5) = (8/5). • Solve for x: x = (8/5) × (15/8). • The 8s cancel out, and 15 \div 5 = 3. • x = 3". Exam Strategy & Shortcut: Find the enlargement multiplier directly: Multiplier = \frac{New Longer}{Old Longer} = (4/5/2) = (8/5). Multiply the old shorter side by this multiplier: (15/8) × (8/5) = 3. Common Pitfall: Incorrectly identifying which side is the "longer" or "shorter" side initially, or making arithmetic errors when dividing by a fraction. Final Answer: Therefore, the correct answer is 3".

Question 13

On a certain map of India the actual distance of 1450 km between two cities Delhi and Kolkata is shown as 5 cm. What scale is used to draw the map?  (A.T.M.A., 2004)

Accepted Answer

Concept & Map Scales: A map scale is a ratio of the distance on the map to the corresponding actual distance on the ground, expressed in the same units. Scale = Map Distance : Actual Distance Step-by-Step Solution: • Given map distance = 5 cm. • Given actual distance = 1450 km. • Convert actual distance to centimeters to match the map units: • 1 km = 1000 meters = 100,000 cm = 10^5 cm. • Actual distance in cm = 1450 × 10^5 cm. • Formulate the scale ratio: 5 : 1450 × 10^5. • Simplify by dividing both sides by 5: 1 : 290 × 10^5. • Adjust to standard scientific notation format shown in options: 290 × 10^5 = 29 × 10^6. • The scale is 1 : 29 × 10^6. Exam Strategy & Shortcut: Focus on the significant digits first. You need the ratio of 5 to 1450. 1450 \div 5 = 290. Look at the options; only one contains the digits "29". You can safely select option (d) without perfectly calculating all the zeroes. Common Pitfall: Failing to convert kilometers to centimeters correctly, often missing a zero (10^4 instead of 10^5), leading to an incorrect power of 10 in the final ratio. Final Answer: Therefore, the correct answer is 1 : 29 × 10^6.

Question 14

A TV tower 36 metres high casts a shadow of 24 metres at a particular time of a day. What is the height of a minar with a three metre high flagstaff atop it, if both of these together cast a shadow of 50 metres at the same time of the day?

Accepted Answer

Concept & Direct Proportion: The ratio of the height of an object to its shadow length is constant at a specific time of day. Find this constant ratio to determine unknown heights. Ratio = \frac{Height}{Shadow Length} Step-by-Step Solution: • Calculate the constant height-to-shadow ratio from the TV tower: (36/24) = 1.5. • Let the height of the minar be h metres. The total height of the minar and the flagstaff together is (h + 3) metres. • Using the constant ratio for the minar and flagstaff combination: (h + 3/50) = 1.5. • Multiply to solve for the total height: h + 3 = 1.5 × 50 = 75 metres. • Subtract the height of the flagstaff to find the minar's height: h = 75 - 3 = 72 metres. Exam Strategy & Shortcut: Work purely in multiples. The height is 1.5 × the shadow. So, total height for a 50 m shadow is 50 × 1.5 = 75 m. Since the flagstaff is 3 m, the minar is 75 - 3 = 72 m. Common Pitfall: Forgetting to subtract the 3-metre flagstaff from the total calculated height of 75 metres, which leads directly to the distractor option (c). Final Answer: Therefore, the correct answer is 72 m.

Question 15

In a camp, there is a meal for $120$ men or $200$ children. If $150$ children have taken the meal, how many men will be catered to with the remaining meal?

Accepted Answer

Concept & Unitary Method: When a resource can support either quantity A or quantity B, we can establish an equivalence ratio to convert the remaining portion from one demographic to the other. Step-by-Step Solution: • Given: Total meal = 120 men OR 200 children. • Calculation: Establish the equivalent ratio: 120 men = 200 children 1 child = (120/200) men = (3/5) men The meal is designed for 200 children, but 150 children have already eaten. Remaining meal capacity = 200 - 150 = 50 children. Convert the remaining capacity to men: Men = 50 × (3/5) = 30 men Exam Strategy & Shortcut: Find the fraction of the meal remaining. 150 out of 200 children ate, meaning 75% of the food is gone. 25% of the food is left. 25% of 120 men is simply (120/4) = 30 men. Common Pitfall: Trying to subtract children from men directly without applying the equivalency ratio. Final Answer: Therefore, the correct answer is 30.

Question 16

The cost of 4 dozen papers is ₹ 24. The cost of 1 score of papers (in rupees) is ....... [APTET, 2011]

Accepted Answer

Concept & Unit Conversions: To solve this, you must know standard numeric quantities. 1 dozen = 12 items 1 score = 20 items Step-by-Step Solution: • Identify Quantities: • 4 dozen = 4 × 12 = 48 papers. • 1 score = 20 papers. • Find Unit Price: • Cost of 48 papers = ₹ 24. • Cost of 1 paper = (24/48) = ₹ (1/2) or ₹ 0.50. • Calculate Required Cost: • Cost of 20 papers = 20 × (1/2) • Cost = ₹ 10. Exam Strategy & Shortcut: Use ratios directly: \frac{Cost_1}{Quantity_1} = \frac{Cost_2}{Quantity_2} (24/48) = (x/20) \implies (1/2) = (x/20) \implies x = 10. Common Pitfall: The most frequent mistake is not knowing what a "score" represents and guessing incorrectly, or confusing dozen (12) with a score (20). Final Answer: Therefore, the correct answer is 10.

Question 17

12 buckets of water fill a tank when the capacity of each bucket is 13.5 litres. How many buckets will be needed to fill the same tank, if the capacity of each bucket is 9 litres?

Accepted Answer

Concept & Constant Volume: The total volume of the tank remains constant regardless of the size of the bucket used to fill it. We equate the total volume calculated with the first bucket size to the total volume calculated with the second bucket size. N_1 × V_1 = N_2 × V_2 Step-by-Step Solution: 1. Calculate Total Capacity: • Volume of one large bucket = 13.5 litres. • Number of large buckets used = 12. • Total capacity of the tank = 12 × 13.5 litres. 2. Calculate Number of Smaller Buckets: • Volume of one small bucket = 9 litres. • Let the number of small buckets needed be x. • x × 9 = 12 × 13.5 3. Solve for x: • x = (12 × 13.5/9) • Simplify the fraction: (12/9) = (4/3). • x = (4/3) × 13.5 = 4 × 4.5 • x = 18 Exam Strategy & Shortcut: Use inverse proportionality. The capacity of the bucket dropped from 13.5 to 9. Ratio of bucket capacities = 13.5 : 9 = 135 : 90 = 3 : 2. Since capacity and number of buckets are inversely proportional, the ratio of the number of buckets will be 2 : 3. If 2 parts = 12 buckets, then 1 part = 6 buckets. So, 3 parts = 3 × 6 = 18 buckets. Common Pitfall: Performing the full multiplication (12 × 13.5 = 162) before dividing wastes time. Always keep the numbers in a fraction and cancel out common terms first. Final Answer: Therefore, the correct answer is 18.

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