Question 1

A contractor agrees to complete a piece of work in 40 days. He initially engages 100 men. After 35 days, he adds 100 more men (so 200 men now) and just manages to complete the work in the originally stipulated 40 days. If he had not engaged the addi…

Accepted Answer

Introduction / Context: This is a practical work and manpower planning question. It examines how changes in workforce size at different stages affect total time, and asks you to work backwards to see what would happen without extra help. Such questions are common in aptitude tests under the topic of time and work or chain rule. Given Data / Assumptions: - Contract time: 40 days. - Initially, 100 men work for the first 35 days. - After 35 days, 100 more men are added, so 200 men work for the last 5 days. - With this pattern, the work is completed exactly in 40 days. - All men work with the same constant efficiency. - We must find the delay if only the initial 100 men worked until completion. Concept / Approach: We calculate the total work in man-days based on the actual schedule (with extra men). Then we compute how many days 100 men alone would need for the same amount of work. The difference between this time and the contracted 40 days gives the number of days the contractor would be behind schedule if he had not hired the additional 100 men. Step-by-Step Solution: Step 1: Compute work done by 100 men in the first 35 days: 100 * 35 = 3500 man-days.Step 2: In the last 5 days, 200 men work: 200 * 5 = 1000 man-days.Step 3: Total work W = 3500 + 1000 = 4500 man-days.Step 4: If only 100 men were employed throughout, let T be the number of days required.Step 5: Then 100 * T = 4500 → T = 4500 / 100 = 45 days.Step 6: The scheduled time was 40 days, so the delay would be 45 - 40 = 5 days. Verification / Alternative check: You can test the result by imagining the original plan: 100 men for 40 days provide only 100 * 40 = 4000 man-days, which is less than the needed 4500 man-days. The shortfall of 500 man-days would require an extra 500 / 100 = 5 more days with 100 men, confirming a total of 45 days and a 5-day delay. Why Other Options Are Wrong: Values like 3, 6, 7 or 9 days do not match the actual shortfall in man-days. They result from miscalculating total work (e.g., forgetting the last 5 days with 200 men) or incorrectly dividing by the number of men when converting man-days back to days. Common Pitfalls: Students often confuse the timeframe and might incorrectly multiply 200 men by the full 40 days, or forget to separate the work phases. Another common error is to assume work is proportional only to men or only to days. Always compute man-days for each phase separately and then add them to find the total work. Final Answer: If no additional men had been engaged, the contractor would have finished 5 days behind schedule.

Question 2

Eighteen binders can bind 900 books in 10 days. Assuming they all work at the same constant rate, how many binders will be required to bind 660 books in 12 days under similar working conditions?

Accepted Answer

Introduction / Context: This is a straightforward chain rule or work-rate question. It tests your ability to use the proportionality between number of workers, time, and amount of work. Given one scenario of binders binding books in a certain time, you must compute how many workers are needed in a different scenario. Given Data / Assumptions: - 18 binders bind 900 books in 10 days. - All binders work at the same constant rate. - We need the number of binders required to bind 660 books in 12 days. - Work (books bound) is directly proportional to (number of binders) * (number of days). Concept / Approach: We can compute the daily output per binder, or directly use proportional reasoning. First we find how many books are bound per binder per day. Then we use that rate to determine how many binders are needed to achieve the target (660 books) in the specified time (12 days). Step-by-Step Solution: Step 1: Total binder-days in the original scenario = 18 * 10 = 180 binder-days.Step 2: Total books bound = 900, so books per binder-day = 900 / 180 = 5 books.Step 3: For the new scenario, we need to bind 660 books in 12 days. Let the required number of binders be N.Step 4: Each binder in one day binds 5 books, so in 12 days one binder binds 5 * 12 = 60 books.Step 5: Therefore, N binders in 12 days will bind N * 60 books.Step 6: Set this equal to the required 660 books: N * 60 = 660 → N = 660 / 60 = 11. Verification / Alternative check: We can also use direct proportion with a chain rule: required binders = 18 * (900 / 660) * (10 / 12)⁻¹, simplifying to the same value. However, the unitary method using binder-days and books per binder-day is usually more intuitive and less error-prone. Why Other Options Are Wrong: Values slightly above or below 11, like 13 or 14, come from arithmetic mistakes such as miscomputing 900 / 180 or 660 / 60. A high value like 22 would imply too many binders relative to the work, and they would finish much earlier than 12 days at the same rate. Common Pitfalls: Errors include confusing proportional relationships, for example multiplying when you should divide, or mixing up the 10 and 12 days when scaling. Some students also forget that more time available means fewer binders are needed, not more. Keeping track of units (books, binder-days, days) helps avoid such mistakes. Final Answer: 11 binders are required to bind 660 books in 12 days.

Question 3

Thirty-six men can complete a certain piece of work in 18 days. Assuming all men work at the same constant rate, in how many days will 27 men complete the same work?

Accepted Answer

Introduction / Context: This is another basic time and work problem illustrating inverse variation between the number of workers and the time required to finish a fixed amount of work. It reinforces the idea of total work measured in man-days and how changes in workforce size affect completion time. Given Data / Assumptions: - 36 men complete the work in 18 days. - We need the number of days required if 27 men do the same work. - Total work remains unchanged. - All men have the same constant efficiency. Concept / Approach: Total work can be expressed in man-days. If W is the total work, then: - W = men * days. Once we compute W using the first scenario, we divide by the new number of workers (27) to find the new time. Because work is fixed, the time and the number of workers are inversely proportional. Step-by-Step Solution: Step 1: Calculate total man-days in the first case: W = 36 * 18.Step 2: Compute: 36 * 18 = 648 man-days.Step 3: Let T be the number of days required for 27 men.Step 4: Then 27 * T must equal the same total work: 27 * T = 648.Step 5: Solve for T: T = 648 / 27.Step 6: Calculate 648 / 27: 27 * 24 = 648, so T = 24 days. Verification / Alternative check: Use inverse proportion: T2 / T1 = M1 / M2 = 36 / 27 = 4 / 3. So T2 = T1 * (4 / 3) = 18 * (4 / 3) = 18 * 4 / 3 = 24 days. This agrees perfectly with the man-day method and confirms the answer. Why Other Options Are Wrong: 28, 34 or 35 days are larger than necessary and would imply more man-days than required to complete the same work. 20 days is too small and would give fewer man-days (27 * 20 = 540) than the original 648 man-days, so the work would remain incomplete. Common Pitfalls: Some learners mistakenly use direct proportion (assuming more men require more time) or perform division incorrectly, such as mixing up 36 / 27 with 27 / 36. Another common mistake is miscalculating 36 * 18. Writing out the intermediate products and checking the proportion logic helps avoid these issues. Final Answer: 27 men will complete the work in 24 days.

Question 4

If a quarter kilogram (250 g) of carrots costs 60 paise, then, at the same rate, how many paise will 200 grams of carrots cost?

Accepted Answer

Introduction / Context: This is a simple unitary-method question involving direct proportion between quantity and cost. It checks whether you can scale prices correctly when the weight changes, a very common topic in basic arithmetic and aptitude tests. Given Data / Assumptions: - 1 quarter kilogram = 250 grams of carrots costs 60 paise. - We need the cost of 200 grams of carrots at the same rate per gram. - The cost is directly proportional to the weight when the rate per gram is constant. Concept / Approach: Because price ∝ quantity for a fixed unit rate, we can either compute the cost per gram and multiply by 200, or directly apply a fraction relating 200 g to 250 g. Both methods should lead to the same answer, demonstrating proportional reasoning. Step-by-Step Solution: Step 1: Recognise that 250 g of carrots cost 60 paise.Step 2: Cost per gram = total cost / total grams = 60 / 250 paise per gram.Step 3: For 200 g, cost = (60 / 250) * 200 paise.Step 4: Simplify: (60 * 200) / 250 = 12,000 / 250.Step 5: 12,000 / 250 = 48 (since 250 * 48 = 12,000).Step 6: Therefore, 200 g of carrots cost 48 paise. Verification / Alternative check: Use simple fraction scaling: 200 g is (200 / 250) = 4 / 5 of 250 g. Therefore, the cost for 200 g is 4 / 5 of 60 paise. Compute 4 / 5 * 60 = 4 * 12 = 48 paise. This confirms the earlier result obtained by the unitary method. Why Other Options Are Wrong: 54 or 62 paise correspond to using an incorrect fraction or miscomputing the division. 78 paise is more than the cost of 250 g and thus clearly impossible for a smaller quantity like 200 g. 40 paise would correspond to a different per-gram rate and does not match the given 60 paise for 250 g relationship. Common Pitfalls: Common mistakes include treating 200 g as 3/4 or 2/3 of 250 g instead of 4/5, or simplifying 12,000 / 250 incorrectly. Another issue is confusing paise and rupees, but here everything is in paise. Carefully setting up the proportional fraction and doing the arithmetic stepwise will avoid these errors. Final Answer: The cost of 200 grams of carrots is 48 paise.

Question 5

In an arithmetic progression, the sum of the 3rd and 15th terms is equal to the sum of the 6th, 11th and 13th terms. Under this condition, which term of the series must necessarily be equal to zero?

Accepted Answer

Introduction / Context: This question is about arithmetic progressions (A.P.) and tests your ability to express terms algebraically and use given conditions to deduce properties of the progression. Specifically, you must determine which term becomes zero when a certain relationship between sums of terms is given. Given Data / Assumptions: - We have an arithmetic progression with first term a and common difference d. - The sum of the 3rd and 15th terms equals the sum of the 6th, 11th and 13th terms. - We must find which term of this A.P. is necessarily equal to 0. Concept / Approach: In an A.P., the nth term is given by: a_n = a + (n - 1) * d. We will write expressions for the 3rd, 15th, 6th, 11th and 13th terms in terms of a and d. Then we impose the given equality, simplify to find a relationship between a and d, and finally determine for which n the term a_n equals zero. Step-by-Step Solution: Step 1: Write the required terms:a_3 = a + 2d, a_15 = a + 14d.Step 2: Sum of 3rd and 15th terms: a_3 + a_15 = (a + 2d) + (a + 14d) = 2a + 16d.Step 3: Write the other three terms: a_6 = a + 5d, a_11 = a + 10d, a_13 = a + 12d.Step 4: Sum of 6th, 11th and 13th terms: a_6 + a_11 + a_13 = (a + 5d) + (a + 10d) + (a + 12d) = 3a + 27d.Step 5: Given condition: 2a + 16d = 3a + 27d.Step 6: Rearrange: 2a + 16d - 3a - 27d = 0 → -a - 11d = 0 → a = -11d.Step 7: General term a_n = a + (n - 1)d = -11d + (n - 1)d = (n - 12)d.Step 8: For a_n to be zero and d ≠ 0, we need (n - 12) = 0 → n = 12. Verification / Alternative check: By taking a specific value for d, say d = 1, we get a = -11. Then the 12th term is a_12 = -11 + 11 = 0, and you can numerically verify that the sum of the 3rd and 15th terms equals the sum of the 6th, 11th and 13th terms, confirming the algebraic derivation. Why Other Options Are Wrong: Terms like the 7th, 9th, 14th or 5th would give a_n = (n - 12)d ≠ 0 for n ≠ 12 (as long as d ≠ 0). They do not satisfy the requirement of being forced to zero by the given condition. Only the 12th term is guaranteed to be zero for any non-zero common difference d consistent with the condition. Common Pitfalls: Some learners forget the general term formula or miscalculate one of the indices, for example taking a_15 = a + 15d instead of a + 14d. Others incorrectly add the terms or simplify the equation, leading to a wrong relationship between a and d. Carefully writing each term and performing algebraic steps slowly avoids these mistakes. Final Answer: The 12th term of the arithmetic progression must be equal to zero.

Question 6

Some persons can complete a particular piece of work in 12 days. If twice that number of persons work at the same constant rate, in how many days will they complete half of that work?

Accepted Answer

Introduction / Context: This question highlights the relationship between number of workers, amount of work, and time. It is a direct application of the idea that time is inversely proportional to the number of workers and directly proportional to the amount of work for a fixed work rate. Given Data / Assumptions: - Some persons (let the number be n) can complete the full work in 12 days. - We consider twice as many persons, that is 2n workers. - These 2n workers need to complete only half of the original work. - All workers have the same constant efficiency. Concept / Approach: Total work can be thought of in terms of man-days. If W is the original work, then: - With n persons in 12 days: W = n * 12 man-days. Half the work is W / 2, and with 2n persons working, we can find the new time by equating man-days. Alternatively, use proportional reasoning: time ∝ work / persons. Step-by-Step Solution: Step 1: Express total work using the initial scenario: W = n * 12.Step 2: Half of this work is W / 2 = (n * 12) / 2 = n * 6 man-days.Step 3: Now we have 2n persons to do W / 2.Step 4: Let T be the required number of days with 2n persons.Step 5: Man-days for this scenario: 2n * T; this must equal n * 6.Step 6: So, 2n * T = n * 6 → divide both sides by n to get 2T = 6.Step 7: Solve for T: T = 6 / 2 = 3 days. Verification / Alternative check: Using proportional reasoning, time ∝ work / persons. The new time T2 compared to original time T1 is: T2 / 12 = (1/2 work) / (2n persons) divided by (1 work) / (n persons) = (1/2) / 2 = 1/4. So T2 = 12 * (1/4) = 3 days. This matches the previous result. Why Other Options Are Wrong: 4 or 6 days ignore the combined effect of doubling the workforce and halving the work. 12 days would be correct only if work and workers stayed the same. 8 days is not consistent with the proportional relationships and typically arises from partial or incorrect scaling. Common Pitfalls: Common mistakes include halving the time simply because the work is halved (giving 6 days), but forgetting the workforce has doubled, or conversely, halving time due to doubling workers but not adjusting for half work. The correct method considers both changes simultaneously: work is halved and workers are doubled, giving a quarter of the original time. Final Answer: Twice as many persons will complete half the work in 3 days.

Question 7

In a queue of boys, Shankar is 9th from the rear end, and Althaf is 8th from the front. Nitu is standing somewhere between these two boys. What is the minimum possible number of boys standing in the queue?

Accepted Answer

Introduction / Context: This is a position-in-queue or ranking-type reasoning question. It tests your ability to interpret information given from both ends of a line and to reason about the minimum possible total number of people when another person is said to be standing between two given persons. Given Data / Assumptions: - Shankar is 9th from the rear end of the queue. - Althaf is 8th from the front of the queue. - Nitu is standing between Shankar and Althaf. - We must find the minimum possible total number of boys in the queue consistent with this information. Concept / Approach: Let the total number of boys be N. If a person is k-th from the rear, his position from the front is N - k + 1. Knowing Shankar's and Althaf's positions from one end, we can express their positions from the front, calculate the distance between them, and ensure that there is at least one position strictly between them where Nitu can stand. We choose N to make this distance just large enough to allow at least one person in between, giving the minimum N. Step-by-Step Solution: Step 1: Let N be the total number of boys in the queue.Step 2: Shankar is 9th from the rear. His position from the front is: N - 9 + 1 = N - 8.Step 3: Althaf is 8th from the front, so his position from the front is 8.Step 4: For Nitu to be between them, the positions of Shankar and Althaf from the front must differ by at least 2 (so there is at least one position in between).Step 5: Assume Shankar is behind Althaf (farther from the front), so N - 8 > 8.Step 6: Then the number of positions strictly between them is (N - 8) - 8 - 1 = N - 17.Step 7: For at least one person between them, we need N - 17 ≥ 1 → N ≥ 18.Step 8: The minimum such N is 18, which allows exactly one position between Althaf and Shankar where Nitu can stand. Verification / Alternative check: Check with N = 18. Then Shankar's position from the front is 18 - 8 = 10. Althaf is 8th from the front. The positions are: Althaf at 8, Nitu at 9, Shankar at 10. Nitu is indeed between them and the total number of boys is 18, confirming that this is feasible and minimal. Why Other Options Are Wrong: For N = 14, 20, 22 or 24, you can place Nitu between them, but these are not minimal because N = 18 already satisfies the condition. For N < 18, there is either no valid position between them or the positions of Shankar and Althaf overlap or invert, which contradicts the given data. Common Pitfalls: Many students forget to convert the position from the rear to the position from the front correctly or miscount the number of positions between two given positions. Others ignore the “between” condition and just add positions from front and rear. Always write the positions explicitly and use inclusive/exclusive counting carefully. Final Answer: The minimum possible number of boys in the queue is 18.

Question 8

A person has Rs. 2000 and wants to distribute this amount among his five children so that each child receives Rs. 30 more than the immediately younger one. Under this arithmetic progression distribution, what will be the share of the youngest child?

Accepted Answer

Introduction / Context: This problem involves arithmetic progression and distribution of money. The father distributes Rs. 2000 among five children, with each child getting a fixed amount more than the younger one. The question tests your understanding of arithmetic sequences and how to sum them to match a total amount. Given Data / Assumptions: - Total money to be distributed = Rs. 2000. - There are five children, ordered by age: youngest, then four older children. - Each child gets Rs. 30 more than the immediately younger child. - We need the share of the youngest child. Concept / Approach: The distribution forms an arithmetic progression (A.P.) of 5 terms with common difference d = Rs. 30. Let the youngest child's share be x. Then the shares will be x, x + 30, x + 60, x + 90, x + 120. The sum of these five amounts must equal Rs. 2000. We set up that equation and solve for x. Step-by-Step Solution: Step 1: Let the youngest child's share be x rupees.Step 2: Then 2nd child gets x + 30, 3rd child x + 60, 4th child x + 90, and the eldest x + 120.Step 3: Total sum of all 5 shares: x + (x + 30) + (x + 60) + (x + 90) + (x + 120).Step 4: Combine like terms: 5x + (30 + 60 + 90 + 120) = 5x + 300.Step 5: Given that this total is Rs. 2000, write the equation: 5x + 300 = 2000.Step 6: Subtract 300 from both sides: 5x = 1700.Step 7: Solve for x: x = 1700 / 5 = 340. Verification / Alternative check: Check by listing the actual amounts: youngest = 340, then 370, 400, 430, 460. Sum = 340 + 370 + 400 + 430 + 460. First add 340 + 460 = 800, 370 + 430 = 800, plus 400 gives 800 + 800 + 400 = 2000. The distribution matches the total Rs. 2000, confirming the calculation. Why Other Options Are Wrong: Values like Rs. 175, Rs. 260, Rs. 290 or Rs. 325 would not produce a total of Rs. 2000 when you form a 5-term A.P. with common difference Rs. 30. They typically come from arithmetic mistakes, such as mis-summing the common difference or incorrectly dividing the remaining amount by 5. Common Pitfalls: Some students forget that there are five children and mis-apply the difference only four times. Others subtract 30 repeatedly from 2000 instead of setting up a proper A.P. sum. Writing the full list of terms and using basic algebra to sum them is the safest and clearest method. Final Answer: The youngest child's share is Rs. 340.

Question 9

Eight men can reap 80 hectares of a field in 24 days. Assuming all men work at the same constant rate, how many hectares can 36 men reap in 30 days under similar conditions?

Accepted Answer

Introduction / Context: This is a multi-step chain rule problem based on work and time. It examines your understanding of how work (here measured in hectares reaped) changes when both the number of workers and the number of days change. The situation uses direct proportion among work, men, and days. Given Data / Assumptions: - 8 men reap 80 hectares in 24 days. - All men work at the same constant rate. - We need to find how many hectares 36 men can reap in 30 days. - Work is directly proportional to (number of men) * (number of days). Concept / Approach: We can calculate the work rate per man-day in terms of hectares per man-day. Then, using that rate, we compute the total work (in hectares) done by 36 men over 30 days. This method ensures we handle both changes (men and days) correctly. Step-by-Step Solution: Step 1: Compute total man-days in the initial scenario: 8 men * 24 days = 192 man-days.Step 2: Total work done is 80 hectares, so the rate per man-day = 80 / 192 hectares.Step 3: Simplify 80 / 192: divide numerator and denominator by 16 → (80 / 16) / (192 / 16) = 5 / 12 hectares per man-day.Step 4: In the new scenario, we have 36 men working 30 days, so total man-days = 36 * 30 = 1080 man-days.Step 5: Total hectares reaped = 1080 * (5 / 12).Step 6: Compute 1080 / 12 = 90, so total hectares = 90 * 5 = 450 hectares. Verification / Alternative check: Use direct chain rule: hectares ∝ men * days. Factor change in men = 36 / 8 = 4.5, factor change in days = 30 / 24 = 1.25. Combined factor = 4.5 * 1.25 = 5.625. Therefore, new work = 80 * 5.625 = 450 hectares, confirming the previous calculation. Why Other Options Are Wrong: Values like 350, 400, 425 or 500 hectares come from incorrect scaling, such as adjusting only for extra men or only for extra days, or miscalculating the factor 1080 * (5 / 12). Only 450 hectares is consistent with the proportional change in both men and days. Common Pitfalls: Students often forget to divide 80 by the original man-days and instead try to adjust directly, leading to arithmetic errors. Another mistake is to treat men and days inconsistently, for example using a factor of 36 / 8 but forgetting to also include 30 / 24. Always compute the per man-day rate or carefully apply the chain rule with both ratios. Final Answer: 36 men can reap 450 hectares in 30 days.

Question 10

A fort has provisions of food sufficient for 150 men for 45 days. After 10 days, 25 men leave the fort. For how many additional days will the remaining food last for the men who remain?

Accepted Answer

Introduction / Context: This is a classic fort and food problem involving the concept of man-days of food. It tests your ability to account for consumption over time and adjust remaining supplies when the number of consumers changes during the period. Given Data / Assumptions: - Initial provisions are enough for 150 men for 45 days. - After 10 days, 25 men leave, so 125 men remain. - We must find how many more days the remaining food will last. - All men consume food at the same constant rate. Concept / Approach: The key idea is that total food can be measured in man-days: one man-day is the amount of food consumed by one man in one day. We first compute the total food available in man-days, subtract what is consumed in the first 10 days, and then divide the remaining man-days by the new number of men to find how long the food will last thereafter. Step-by-Step Solution: Step 1: Compute total food in man-days: 150 men * 45 days = 6750 man-days.Step 2: In the first 10 days, the fort still has 150 men.Step 3: Food consumed in first 10 days = 150 * 10 = 1500 man-days.Step 4: Remaining food in man-days = 6750 - 1500 = 5250 man-days.Step 5: After 10 days, 25 men leave, so remaining men = 150 - 25 = 125 men.Step 6: Let D be the number of days the remaining food will last for 125 men.Step 7: Then 125 * D = 5250 man-days.Step 8: Solve for D: D = 5250 / 125.Step 9: Compute: 125 * 40 = 5000, remainder 250, and 125 * 2 = 250, so D = 42 days. Verification / Alternative check: Total elapsed time from the start until the food ends would be 10 days (initial phase) + 42 days (after men leave) = 52 days. Check by total consumption: 150 * 10 + 125 * 42 = 1500 + 5250 = 6750 man-days, which matches the original total available, confirming the calculation. Why Other Options Are Wrong: Values like 29, 32 or 37 days underestimate the remaining time and would correspond to total consumption less than 6750 man-days. 54 days greatly overshoots the remaining man-days and would require more food than initially stored. Only 42 days balances the man-days exactly. Common Pitfalls: Common errors include forgetting to subtract the food consumed in the first 10 days, or using 150 men instead of 125 men in the second phase. Some students also miscalculate 5250 / 125. A step-by-step man-days approach keeps the logic clear and prevents such mistakes. Final Answer: The remaining food will last for 42 more days.

Question 11

If the cost of x metres of wire is d rupees, then, at the same uniform rate per metre, what will be the cost (in rupees) of y metres of the same wire?

Accepted Answer

Introduction / Context: This question involves direct proportion and algebraic manipulation. It asks you to express the cost of y metres of wire in terms of x, y and d, given that x metres cost d rupees. Such algebraic unit-rate questions are common in aptitude and quantitative exams. Given Data / Assumptions: - x metres of wire cost d rupees. - The cost per metre is constant (uniform rate). - We need the cost of y metres of the same wire in rupees, expressed in terms of x, y, and d. Concept / Approach: The fundamental idea is cost = (length) * (cost per metre). First, we find the cost per metre from the given information. Since x metres cost d rupees, cost per metre is d / x. Then we multiply this rate by y metres to get the cost of y metres. Step-by-Step Solution: Step 1: Given: x metres cost d rupees.Step 2: Cost per metre = total cost / total length = d / x rupees per metre.Step 3: We want the cost of y metres at the same rate.Step 4: Cost of y metres = (cost per metre) * (number of metres) = (d / x) * y.Step 5: Rearrange the expression: (d / x) * y = (y d) / x.Step 6: Therefore, cost of y metres = Rs. (yd / x). Verification / Alternative check: As a quick check, substitute concrete numbers. Suppose x = 2 metres and d = 10 rupees. Then each metre costs 5 rupees. For y = 3 metres, cost should be 3 * 5 = 15 rupees. Using the formula Rs. (yd / x): (3 * 10) / 2 = 30 / 2 = 15 rupees, which matches perfectly. Why Other Options Are Wrong: Rs. (xy / d) inverts the relationship and would give nonsensical units. Rs. (xd) or Rs. (yd) ignore the need to divide by x and would not reduce to the correct numerical values when checked. Rs. (d / xy) is dimensionally wrong because it decreases as y increases, which contradicts the idea that more length costs more. Only Rs. (yd / x) correctly models cost ∝ length and fits the given rate. Common Pitfalls: Students often mix up which variables go in the numerator or denominator when setting up the rate. Another frequent mistake is to assume cost is proportional to the product xy without checking units. Always start from cost per unit (here, per metre) and then multiply by the desired quantity. This systematic approach prevents such algebraic errors. Final Answer: The cost of y metres of wire is Rs. (yd / x).

Question 12

If a quarter kilogram (250 g) of potatoes costs 60 paise, then, assuming the same rate per gram, how many paise will 200 grams of potatoes cost?

Accepted Answer

Introduction / Context: This is another unitary-method question involving direct proportion between weight and cost, this time using potatoes instead of carrots. It reinforces your understanding that for a fixed rate per gram, cost changes in direct proportion to weight. Given Data / Assumptions: - 250 grams (a quarter kilogram) of potatoes cost 60 paise. - We must find the cost of 200 grams of potatoes at the same rate. - The price per gram is constant. Concept / Approach: Because cost is directly proportional to quantity when rate per unit is fixed, we can either find the cost per gram and multiply by 200, or simply scale the cost in proportion to the weight ratio 200 / 250. These are equivalent methods and both rely on simple proportionality. Step-by-Step Solution: Step 1: Recognise that 250 g of potatoes cost 60 paise.Step 2: Cost per gram = 60 / 250 paise.Step 3: Cost for 200 g = (60 / 250) * 200 paise.Step 4: Compute (60 * 200) / 250 = 12,000 / 250.Step 5: 12,000 / 250 = 48, since 250 * 48 = 12,000.Step 6: Therefore, 200 g of potatoes cost 48 paise. Verification / Alternative check: The quicker approach: 200 g is 4/5 of 250 g. Hence, cost for 200 g is 4/5 of 60 paise = (4 * 60) / 5 = 240 / 5 = 48 paise. This matches the unitary method and confirms the answer. Why Other Options Are Wrong: 54 or 56 paise suggest incorrect fraction manipulation, such as using 200 / 220 or mis-simplifying 12,000 / 250. 72 paise would be correct if 250 g cost 90 paise, not 60. 40 paise is too low and does not maintain the given rate of 60 paise per 250 g. Only 48 paise fits the direct proportionality. Common Pitfalls: As with similar questions, students might miscalculate the ratio between 200 g and 250 g or perform the division 12,000 / 250 incorrectly. Also, mixing paise and rupees or dropping a zero can cause errors. Carefully writing each step and checking the final magnitude (less than 60 paise for less than 250 g) helps ensure correctness. Final Answer: The cost of 200 grams of potatoes is 48 paise.

Question 13

In a dairy farm, 40 cows eat 40 bags of husk in 40 days. Assuming the rate of eating is constant and the cows are identical in consumption, in how many days will one cow eat one bag of husk?

Accepted Answer

Introduction / Context: This is a standard work-rate style question framed in terms of cows and husk consumption. It tests your understanding of unit consumption (one cow-day) and how to convert total consumption across many animals and days into the rate for a single animal and a single bag. Given Data / Assumptions: - 40 cows eat 40 bags of husk in 40 days. - All cows eat at the same constant rate. - We need the number of days in which one cow will eat one bag of husk. Concept / Approach: We treat consumption in cow-days and bag-units. Total consumption in the given scenario is 40 bags. We express this consumption as cow-days and then find how many cow-days are needed per bag. Finally, we convert cow-days per bag into days for one cow per bag. Step-by-Step Solution: Step 1: Compute total cow-days: 40 cows * 40 days = 1600 cow-days.Step 2: During these 1600 cow-days, the cows consume 40 bags of husk.Step 3: So cow-days per bag = 1600 cow-days / 40 bags = 40 cow-days per bag.Step 4: This means that one bag of husk requires 40 cow-days of eating.Step 5: If we have just one cow, each day contributes 1 cow-day.Step 6: Therefore, to reach 40 cow-days with one cow, we need 40 days. Verification / Alternative check: Consider the daily total consumption: 40 bags in 40 days means 1 bag per day by all 40 cows together. Thus, per cow per day, the consumption is 1/40 bag. To eat 1 bag at this rate, one cow needs time = 1 / (1/40) = 40 days, confirming the earlier calculation. Why Other Options Are Wrong: 1 day would mean all 40 cows together eat 40 bags per day, contradicting the data. 1/40 day is an extreme misinterpretation of the rate. 20 or 80 days do not match the cow-day calculation and arise from mistakes such as halving or doubling the result without reason. Only 40 days preserves the correct proportional consumption. Common Pitfalls: Students sometimes assume that if 40 cows eat 40 bags in 40 days, then one cow eats one bag in 1 day, incorrectly ignoring the time dimension in the original statement. Others miscalculate cow-days per bag. Always convert to a per cow per day rate or use cow-days systematically to avoid confusion. Final Answer: One cow will eat one bag of husk in 40 days.

Question 14

Four mat-weavers can weave four mats in four days. Working at the same constant rate, how many mats would be woven by eight mat-weavers in eight days?

Accepted Answer

Introduction / Context: This question is another work-rate problem, but framed in terms of mat-weavers and mats. It checks your understanding of direct proportion between the number of workers, the time they work, and the number of items produced, assuming a constant rate of production per worker. Given Data / Assumptions: - 4 mat-weavers weave 4 mats in 4 days. - All weavers have the same constant weaving rate. - We need the number of mats woven by 8 mat-weavers in 8 days. Concept / Approach: We can express total work in weaver-days and mats. First, we compute how many mats are produced per weaver-day. Then we multiply this rate by the total weaver-days in the new scenario (8 weavers for 8 days) to find the total mats produced. Step-by-Step Solution: Step 1: Compute total weaver-days in the initial situation: 4 weavers * 4 days = 16 weaver-days.Step 2: Total mats produced = 4 mats.Step 3: Mats per weaver-day = 4 mats / 16 weaver-days = 1/4 mat per weaver-day.Step 4: In the new situation, we have 8 weavers working for 8 days.Step 5: Total weaver-days now = 8 * 8 = 64 weaver-days.Step 6: Total mats produced = 64 weaver-days * (1/4 mat per weaver-day) = 64 * 1/4 = 16 mats. Verification / Alternative check: Use simple chain rule: compare the second situation to the first. The number of weavers doubles (4 → 8), and the number of days also doubles (4 → 8). So the total work should increase by a factor of 2 * 2 = 4. Originally, 4 mats were made. 4 times that is 16 mats. This matches the weaver-day method. Why Other Options Are Wrong: 8 mats would be correct if only one of the two factors (either weavers or days) doubled, but both increased. 12 or 24 mats do not match the precise factor of 4 increase and typically arise from partial reasoning or arithmetic errors. 4 mats would mean no increase in production despite more weavers and more time, which is clearly inconsistent. Common Pitfalls: Some learners forget to consider both changes (workers and days) or mix up whether the relationship is direct or inverse. Others miscalculate the factor of increase. Always think in terms of total worker-days and then scale production accordingly. Final Answer: Eight mat-weavers will weave 16 mats in eight days.

Question 15

Ten men, working 6 hours a day, can complete a certain piece of work in 18 days. At the same efficiency, how many hours per day must fifteen men work to complete the same work in 12 days?

Accepted Answer

Introduction / Context: This is a time and work question involving three factors: number of men, hours per day, and days. It tests your ability to set up and use the relationship that total work is proportional to men * hours per day * number of days when everyone works at a constant rate. Given Data / Assumptions: - 10 men working 6 hours per day complete the work in 18 days. - The work rate per man per hour is constant. - We must find the number of hours per day, h, that 15 men must work to complete the same work in 12 days. Concept / Approach: We represent total work as the product of men, hours per day and number of days. The total work is the same in both situations. So, we equate the expressions for work in the two cases and solve for the unknown h. This is a direct extension of the man-day concept to man-hours. Step-by-Step Solution: Step 1: Express total work W using the first scenario.Step 2: W = number of men * hours per day * days = 10 * 6 * 18.Step 3: Compute: 10 * 6 * 18 = 10 * 108 = 1080 man-hours.Step 4: For the second scenario, we have 15 men working h hours per day for 12 days.Step 5: Work done in the second scenario = 15 * h * 12 man-hours.Step 6: Set total work equal: 15 * h * 12 = 1080.Step 7: Simplify: 15 * 12 = 180, so 180h = 1080.Step 8: Solve for h: h = 1080 / 180 = 6 hours per day. Verification / Alternative check: Check the total man-hours in the second scenario with h = 6 hours. 15 men * 6 hours/day * 12 days = 15 * 72 = 1080 man-hours, matching the original total. Thus the work equality holds, confirming that 6 hours per day is correct. Why Other Options Are Wrong: 4 or 5 hours per day would give fewer total man-hours than 1080, so the work would remain incomplete. 7 or 8 hours per day would exceed 1080 man-hours, implying the work would finish before 12 days at that effort level. Only 6 hours per day maintains exactly the same total man-hours. Common Pitfalls: Some students forget to include all three factors (men, hours, days) and mistakenly equate only two. Others miscalculate 10 * 6 * 18 or 15 * 12. Keeping track of units (man-hours) and performing multiplications step by step will greatly reduce these errors. Final Answer: Fifteen men must work 6 hours a day to complete the work in 12 days.

Question 16

In a work and time (chain rule) problem, if 16 men working 7 hours a day can plough a field in 48 days, then in how many days will 14 men working 12 hours a day be able to plough the same field, assuming that all men work at the same constant rate?

Accepted Answer

Introduction: This question is a classic example of the chain rule in aptitude, combining men, hours per day, and number of days. The underlying idea is that total work remains constant, and it is the product of the number of workers, the hours they work per day, and the total days they work. By equating the total work in two different scenarios, we can find the unknown time required. Given Data / Assumptions: Number of men in first case = 16. Working hours per day in first case = 7 hours. Number of days in first case = 48 days. Number of men in second case = 14. Working hours per day in second case = 12 hours. Work (ploughing the same field) is assumed to be fixed and identical in both cases. Concept / Approach: Total work is proportional to men * hours per day * days. Since the field is the same, both expressions for work must be equal. We use the relation: 16 * 7 * 48 = 14 * 12 * D where D is the required number of days in the second case. Solving this simple proportion gives the answer. Step-by-Step Solution: Step 1: Compute total work in the first case (in man hours). Work = 16 * 7 * 48. 16 * 7 = 112. 112 * 48 = 5376 man hours. Step 2: Express total work for the second case. In the second case, daily man hours = 14 * 12 = 168 man hours per day. Let required days = D. So total work = 168 * D. Step 3: Equate total work for both cases. 5376 = 168 * D. D = 5376 / 168. Compute the division: 5376 / 168 = 32. Therefore, 14 men working 12 hours a day will plough the field in 32 days. Verification / Alternative check: We can simplify the proportion directly: (16 * 7 * 48) / (14 * 12) = D. Cancel 2 from 16 and 14 to get 8 and 7. Now D = (8 * 7 * 48) / (7 * 12) = (8 * 48) / 12 = 8 * 4 = 32 days. The same answer confirms our calculation. Why Other Options Are Wrong: 46, 35, 30, 28 days do not satisfy the equality 16 * 7 * 48 = 14 * 12 * D, so they either overestimate or underestimate total work. For example, if D were 30, total work would be 14 * 12 * 30 = 5040 man hours, which is less than 5376, meaning the field would not be fully ploughed. Common Pitfalls: Many learners forget to include working hours per day and only compare men and days, which leads to a wrong result. Others set up the proportion incorrectly, inverting one of the ratios. Always remember that total work is men * hours * days and equate the complete expressions for both scenarios. Final Answer: The field will be ploughed in 32 days when 14 men work 12 hours a day.

Question 17

If the price of 6 identical toys is Rs. 264.37, what will be the approximate total price of 5 such toys (rounded to a convenient whole rupee value)?

Accepted Answer

Introduction: This question tests basic direct proportion and unit price calculation, which is a fundamental concept in arithmetic and commercial aptitude. We know the total cost for 6 toys, and we need to estimate the cost for 5 toys of the same type at the same rate, giving an approximate value in rupees. Given Data / Assumptions: Total price for 6 toys = Rs. 264.37. All toys are identical and have the same price. We need the approximate price of 5 toys. Rounding to a user friendly whole rupee answer is acceptable. Concept / Approach: When items are identical, total price is proportional to the number of items. First find the price of 1 toy by dividing the total amount by 6. Then multiply this unit price by 5 to get the cost of 5 toys. Finally, round the result to the nearest convenient rupee amount and match it with the options. Step-by-Step Solution: Step 1: Compute the price of 1 toy. Price per toy = 264.37 / 6. 264.37 / 6 ≈ 44.06 rupees per toy. Step 2: Compute the price of 5 toys. Price for 5 toys = 44.06 * 5. 44.06 * 5 ≈ 220.3 rupees. Step 3: Approximate to the nearest useful rupee. 220.3 rupees is very close to Rs. 220. Therefore, the approximate price of 5 toys is Rs. 220. Verification / Alternative check: We can back check by finding the cost of 10 toys at this approximate rate. If 1 toy is roughly Rs. 44, 10 toys would cost about Rs. 440. Since 6 toys cost about Rs. 264 (6 * 44), this is consistent with the given Rs. 264.37. This confirms that our approximation is reasonable and that Rs. 220 is a sensible estimate for 5 toys. Why Other Options Are Wrong: Rs. 120 and Rs. 100 are far too low. They imply a unit price of around Rs. 24 or Rs. 20, which contradicts the given data. Rs. 200 is closer but still underestimates the cost by about Rs. 20, which is a significant error compared to the actual calculation. Rs. 180 is also too low and inconsistent with the unit price of approximately Rs. 44. Common Pitfalls: A frequent mistake is to perform proportional reasoning incorrectly, such as scaling down the price linearly without first finding the unit price, or rounding too early in the calculation. Another error is misreading the data and thinking 264.37 is the price for a different number of toys. Always identify the per unit cost first and then scale up or down as required. Final Answer: The approximate total cost of 5 toys is Rs. 220.

Question 18

An industrial loom weaves cloth at a constant rate of 1.14 metres of cloth every second. Approximately how many seconds will it take this loom to weave a total of 52 metres of cloth?

Accepted Answer

Introduction: This problem is a straightforward application of the formula time = distance / speed, where distance is the length of cloth to be woven and speed is the weaving rate of the loom. Such questions are very common in quantitative aptitude under the topic of speed, time, and distance, or rate and work problems. Given Data / Assumptions: Weaving speed of the loom = 1.14 metres per second. Required length of cloth = 52 metres. The loom is assumed to operate at a constant speed without interruption. We need an approximate time in seconds. Concept / Approach: The relation between distance, speed, and time is: time = distance / speed. Here, distance is the total cloth length, speed is metres per second, and time will naturally come out in seconds. We just substitute the given values and compute the division carefully, then round the result to two decimal places to match the options. Step-by-Step Solution: Step 1: Write the formula. time = distance / speed. Step 2: Substitute the given values. time = 52 / 1.14 seconds. Step 3: Perform the division. 52 / 1.14 ≈ 45.6140 seconds. Step 4: Round to two decimal places. Rounded value ≈ 45.61 seconds. Therefore, the loom takes approximately 45.61 seconds to weave 52 metres of cloth. Verification / Alternative check: We can check quickly by multiplying the result back by the speed: 1.14 * 45.61 ≈ 52.0 metres (approximately). This confirms that our computed time is consistent with the original distance when multiplied by the weaving rate. Why Other Options Are Wrong: 29.32 sec and 39.25 sec correspond to weaving speeds that are too high, implying more cloth would be woven than 52 metres in those times. 42.51 sec is closer but still underestimates the time, which would produce less than 52 metres. 52.14 sec would overshoot the distance, producing more than 52 metres at the given speed. Common Pitfalls: Learners sometimes invert the formula and accidentally compute speed / distance, or they round intermediate values too aggressively. It is better to keep sufficient decimal precision until the final step, then round only once to match the required accuracy. Another common error is misreading the units and mixing up seconds and minutes, which does not happen here if you carefully track the units as metres per second and seconds. Final Answer: The loom will take approximately 45.61 seconds to weave 52 metres of cloth.

Question 19

In a camp there is enough meal to feed either 120 men or 200 children for one meal. If 150 children have already taken their meal, how many men can still be catered with the remaining meal supply?

Accepted Answer

Introduction: This question illustrates a typical chain rule or proportion concept involving equivalent consumption of resources by men and children. The key idea is that the total quantity of meal is fixed, and the meal that can feed a certain number of children can be converted into an equivalent number of men based on the given ratios. Given Data / Assumptions: Total meal is sufficient for 120 men. The same meal is also sufficient for 200 children. 150 children have already taken the meal. We assume each man and each child consumes a fixed amount of meal per person. We need to find how many men can be fed with the remaining meal. Concept / Approach: Since the same meal can feed 120 men or 200 children, the total quantity of meal can be expressed in man meals or child meals. We can first work in terms of children. After 150 children have eaten, we find how many children equivalent meals remain. Then we convert that remaining quantity into an equivalent number of men using the ratio between men and children consumption. Step-by-Step Solution: Step 1: Treat the entire meal as 200 child meals. Total capacity = 200 children (for one meal). Step 2: Subtract the consumption of 150 children. Children already fed = 150. Remaining child meals = 200 − 150 = 50 child meals. Step 3: Relate child meals to man meals using the given information. 120 men are equivalent to 200 children. So 1 child meal = 120 / 200 man meals = 3 / 5 man meals. Step 4: Convert remaining child meals to man meals. Remaining man meals = 50 * (3 / 5) = 30 man meals. Therefore, 30 men can be fully fed with the remaining meal. Verification / Alternative check: We can also express the total meal in man meals. Since 120 men can be fed, total capacity is 120 man meals. One man meal = 200 / 120 child meals = 5 / 3 child meals. If 150 children have eaten, that is 150 child meals, which equals 150 * (3 / 5) = 90 man meals. Remaining man meals = 120 − 90 = 30, confirming the same result. Why Other Options Are Wrong: 50 men would require 50 man meals, which is more than the 30 man meals left. 40 and 20 men do not match the equivalence calculation when we convert remaining child meals into man meals. 10 men underutilise the available food, since the remaining quantity is enough for 30 men. Common Pitfalls: Many learners confuse the ratio and try to use direct proportion incorrectly, such as treating men and children as interchangeable without a conversion factor. Another common mistake is subtracting 150 directly from 120 or 200 without recognising that we must convert between child meals and man meals properly. Always convert the total resource into one consistent unit first and then perform the calculations. Final Answer: With the remaining meal, the camp can still cater to 30 men.

Question 20

An industrial loom weaves cloth at a uniform rate of 0.128 metres of cloth every second. Approximately how many seconds will it take for this loom to weave 25 metres of cloth in total?

Accepted Answer

Introduction: This question involves a direct application of the time = distance / speed formula, where the distance is the length of cloth to be woven and the speed is the weaving rate of the industrial loom. It checks your familiarity with rate, time, and distance type problems in a slightly different context of cloth weaving. Given Data / Assumptions: Weaving speed of the loom = 0.128 metres per second. Required cloth length = 25 metres. The loom is assumed to operate steadily at this constant rate. We are asked for the approximate time in seconds. Concept / Approach: The fundamental relation between distance, speed, and time is: time = distance / speed. Here distance is measured in metres, speed in metres per second, so time will naturally be in seconds. We substitute the given numerical values and then round the result to the nearest whole second to match the options. Step-by-Step Solution: Step 1: Write down the formula. time = distance / speed. Step 2: Substitute distance and speed. time = 25 / 0.128 seconds. Step 3: Perform the division. 25 / 0.128 = 195.3125 seconds. Step 4: Round to the nearest whole number. 195.3125 seconds is approximately 195 seconds. Therefore, the loom takes about 195 seconds to weave 25 metres of cloth. Verification / Alternative check: We can reverse check by multiplying the approximate time back by the speed: 0.128 * 195 = 24.96 metres (approximately). This is extremely close to 25 metres, and the small difference is due to rounding. If we use the exact time 195.3125 seconds, the cloth length becomes exactly 25 metres. This confirms that 195 seconds is a good approximation. Why Other Options Are Wrong: 194 seconds corresponds to 0.128 * 194 ≈ 24.83 metres, which is slightly short of the required 25 metres. 196 seconds gives 0.128 * 196 ≈ 25.09 metres, which is slightly more than needed. 197 seconds is even further away at roughly 25.22 metres. 190 seconds would produce only about 24.32 metres, clearly insufficient. The closest and most reasonable approximation among the options is 195 seconds. Common Pitfalls: A common mistake is to invert the formula and compute speed / distance instead of distance / speed. Another issue is mishandling decimal division by 0.128, which can lead to calculation errors. Breaking the division into smaller steps or using long division carefully reduces such errors. Also, rounding too early in the computation can cause inaccuracies; always round only at the end when you have the exact numerical result. Final Answer: The loom will take approximately 195 seconds to weave 25 metres of cloth.

Chain Rule Questions