Question 1

A quarter kilogram of potatoes costs 60 paise. At the same rate, how many paise will 200 grams of potatoes cost?

Accepted Answer

Introduction / Context:This is another direct proportion question using the unitary method, now with potatoes instead of carrots. It checks whether you can convert between kilograms and grams, and then correctly scale the cost according to the new quantity. These small numerical questions are typical in entry level aptitude tests.Given Data / Assumptions:Cost of one quarter kilogram of potatoes (250 grams) is 60 paise.The cost per gram of potatoes is constant.We want the cost of 200 grams of potatoes.Concept / Approach:The cost of a given weight is directly proportional to that weight when the price per gram is constant. We either compute the cost per gram and multiply by 200 grams, or use a ratio to relate the costs for 250 grams and 200 grams. Both are straightforward applications of the unitary method.Step-by-Step Solution:Step 1: One quarter kilogram equals 250 grams.Step 2: Cost for 250 grams is 60 paise.Step 3: Cost per gram = 60 / 250 paise.Step 4: Cost for 200 grams = (60 / 250) * 200 paise.Step 5: Simplify: 60 * 200 / 250 = 60 * (4 / 5) = 48 paise.Verification / Alternative check:Observe that 200 grams is four fifths of 250 grams. Therefore, cost for 200 grams must be four fifths of 60 paise. Compute 60 * 4 / 5 to obtain 48 paise directly, which matches the detailed calculation. This ratio based reasoning gives a quick mental check.Why Other Options Are Wrong:40 paise would correspond to a rate lower than given by the original data. 54 paise and 60 paise are too high for 200 grams compared to 250 grams, since cost must decrease in proportion. 72 paise is higher than the cost of the larger quantity, which is impossible without a higher rate per gram.Common Pitfalls:Using 200 as denominator and 250 as numerator in the ratio, effectively inverting the proportion.Incorrectly converting quarter kilogram, for example, taking it as 200 grams instead of 250 grams.Rounding prematurely instead of working with exact fractions until the end.Final Answer:The cost of 200 grams of potatoes is 48 paise.

Question 2

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

Accepted Answer

Introduction / Context:This question is about resource consumption and uses the concept of cow days similar to man days in work problems. It tests whether you can translate a group consumption rate into an individual rate, and then answer a question about a single cow and a single bag of husk.Given Data / Assumptions:40 cows consume 40 bags of husk in 40 days.The rate of consumption per cow per day is constant.We must find how many days 1 cow will take to eat 1 bag of husk.Concept / Approach:The key idea is that total consumption can be expressed in cow days of food per bag. We first determine the total consumption in cow days for 40 bags, then compute how many cow days correspond to one bag. Once we know the number of days for one cow to finish one bag, we have our answer.Step-by-Step Solution:Step 1: Total cow days of consumption in the given situation is 40 cows * 40 days = 1600 cow days.Step 2: These 1600 cow days correspond to 40 bags of husk.Step 3: Cow days per bag = 1600 / 40 = 40 cow days.Step 4: One cow consuming one bag needs 40 cow days.Step 5: Since one cow supplies one cow day per day, the number of days required = 40 days.Verification / Alternative check:Another way is to find the daily quantity of husk eaten. Forty cows eat 40 bags in 40 days, so they eat 1 bag per day as a group. Therefore, in one day each cow eats 1 / 40 of a bag. To finish one whole bag at that rate, a single cow takes 40 days, which matches the earlier calculation.Why Other Options Are Wrong:1 day and 10 days are far too small and ignore how slowly a single cow consumes compared to the group. 20 days still halves the correct time. 80 days is double the correct value and would imply that the initial data underestimates the group consumption rate.Common Pitfalls:Confusing bags per day with cow days per bag.Assuming that one cow will also take 40 days to eat 40 bags, which confuses per cow and per group quantities.Skipping the step of finding the total cow days and directly guessing an answer.Final Answer:One cow will take 40 days to eat one bag of husk.

Question 3

Four mat weavers can weave four mats in four days. At the same rate of working, how many mats will be woven by eight mat weavers in eight days?

Accepted Answer

Introduction / Context:This is a classic direct proportion and work rate problem. It relates the number of workers, the time they work and the amount of work done. Here, mat weavers produce mats, and we are asked to scale up the situation when both the number of workers and the number of days are doubled.Given Data / Assumptions:4 mat weavers weave 4 mats in 4 days.All weavers work at the same constant rate.The new situation involves 8 mat weavers working for 8 days.We must find how many mats they will produce.Concept / Approach:First, determine the rate of mat production per weaver per day. Then multiply that rate by the total number of weaver days in the new situation. This is a straightforward chain rule problem where work is proportional to workers multiplied by time.Step-by-Step Solution:Step 1: Total weaver days in the original case = 4 weavers * 4 days = 16 weaver days.Step 2: These 16 weaver days produce 4 mats.Step 3: So production rate = 4 mats / 16 weaver days = 0.25 mat per weaver day.Step 4: In the new case, we have 8 weavers working for 8 days, so number of weaver days = 8 * 8 = 64.Step 5: Total mats produced = 64 * 0.25 = 16 mats.Verification / Alternative check:Notice that we doubled both the number of weavers and the number of days compared with the original setup. Doubling weavers and doubling time multiplies the total work by 4. Originally 4 mats were woven, so in the new setup they should weave 4 * 4 = 16 mats. This quick proportional check matches the detailed calculation.Why Other Options Are Wrong:4 mats corresponds to the original arrangement, not the increased one. 8 and 12 mats underestimate the effect of doubling both workers and days. 20 mats would require a higher rate than what is given by the original information.Common Pitfalls:Forgetting to multiply both the number of weavers and the number of days when computing weaver days.Assuming that doubling workers doubles mats, but overlooking that days also doubled, which gives a factor of 4.Misinterpreting the phrase four weavers weave four mats in four days as one weaver per mat without verifying the rate.Final Answer:Eight mat weavers will weave 16 mats in eight days.

Question 4

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

Accepted Answer

Introduction / Context:This question involves equivalent consumption of meal by men and children. It tests your ability to convert between two types of consumers using a ratio and then work out how much of the resource remains. Such problems often appear in aptitude tests under chain rule or ratio and proportion topics.Given Data / Assumptions:The total stock of meal is enough for 120 men.The same stock is also enough for 200 children.So the feeding capacity for men and children is equivalent.150 children have already taken the meal.We must find how many men can be fed with the remaining meal.Concept / Approach:The total meal can be measured either in man meals or child meals. It is convenient to use child meals first. After some children have eaten, we determine the remaining child meal capacity and then convert that remaining capacity into an equivalent number of men using the known equivalence between men and children.Step-by-Step Solution:Step 1: Total meal is enough for 200 children, so think of it as 200 child meals.Step 2: 150 children have eaten, so child meals consumed = 150.Step 3: Remaining child meals = 200 - 150 = 50.Step 4: From the original equivalence, 120 men correspond to 200 children.Step 5: Therefore, 1 man corresponds to 200 / 120 = 5 / 3 children.Step 6: Inverting this, 1 child corresponds to 3 / 5 of a man.Step 7: Remaining child meals (50) correspond to 50 * (3 / 5) men = 30 men.Verification / Alternative check:You can also convert the total meal to man meals first. Since 120 men correspond to 200 children, each child is equivalent to 120 / 200 = 3 / 5 of a man for one meal. When 150 children eat, that equals 150 * 3 / 5 = 90 man meals consumed. The original stock was 120 man meals, so remaining man meals = 120 - 90 = 30, which confirms the earlier result.Why Other Options Are Wrong:20 or 25 men are too small, since more meal remains than that. 35 or 40 men are too many and would require more meal than is left after 150 children have eaten. Only 30 men correctly matches the equivalence relationship.Common Pitfalls:Assuming that 150 children equal 150 men without using the given conversion ratio.Mixing up which way to convert, for example multiplying by 5 / 3 when you should multiply by 3 / 5.Forgetting that the 150 children have already eaten, and instead trying to allocate the full capacity again.Final Answer:The remaining meal can feed 30 men.

Question 5

Ten men, working 6 hours a day, can complete a piece of work in 18 days. How many hours a day must 15 men work to complete the same work in 12 days?

Accepted Answer

Introduction / Context:This question is a multi factor work problem involving number of men, hours per day and number of days. It tests the chain rule concept where total work is proportional to men * hours per day * days. Several quantities change, and you must adjust the remaining variable to keep the total work constant.Given Data / Assumptions:10 men working 6 hours per day complete the work in 18 days.All men work at a constant rate.New scenario: 15 men must complete the same work in 12 days.We must find required working hours per day in the new situation.Concept / Approach:Total work can be expressed as the product of men, hours per day and days. Since the work does not change between scenarios, the total man hours in both cases must be equal. We compute total man hours from the first scenario, then divide by the product of the new number of men and days to obtain the required hours per day.Step-by-Step Solution:Step 1: Total work in man hours for the first case = 10 men * 6 hours per day * 18 days.Step 2: Compute: 10 * 6 * 18 = 1080 man hours.Step 3: In the new case, number of men = 15, number of days = 12.Step 4: Let required hours per day be H.Step 5: Total man hours in the new case = 15 * H * 12.Step 6: Since work is the same, 15 * H * 12 = 1080.Step 7: Simplify: 15 * 12 = 180, so 180H = 1080, giving H = 1080 / 180 = 6 hours per day.Verification / Alternative check:Check with proportions. The number of men increases from 10 to 15, a factor of 1.5. The days decrease from 18 to 12, a factor of 12 / 18 = 2 / 3. To keep total man hours unchanged, hours per day must adjust so that (men factor) * (day factor) * (hour factor) = 1. So 1.5 * (2 / 3) * hour factor = 1, giving hour factor = 1. Therefore, the hours per day remain the same at 6, which matches the detailed solution.Why Other Options Are Wrong:4 or 5 hours per day would reduce the total man hours below 1080. 7 or 8 hours per day would give more man hours than needed, meaning the work would finish earlier than 12 days, contradicting the requirement to exactly match the original work.Common Pitfalls:Multiplying or dividing the factors incorrectly when many quantities change at once.Trying to solve only with ratios and losing track of which factors are in the numerator or denominator.Forgetting that all three elements (men, hours per day, days) contribute equally to total man hours.Final Answer:The 15 men must work 6 hours a day to finish the work in 12 days.

Question 6

If 16 men working 7 hours per day can plough a field in 48 days, in how many days will 14 men working 12 hours per day plough the same field?

Accepted Answer

Introduction / Context: Chain rule problems in time and work test how changes in the number of workers and hours per day affect the time required to complete a fixed job. This question compares two different working situations and asks for the new time taken when both the number of men and the working hours per day are changed. Understanding the relation between men, hours, days, and total work is very important for aptitude exams and competitive tests. Given Data / Assumptions: 16 men working 7 hours per day can plough a field in 48 days. We assume all men work at the same constant rate. The total work, that is the quantity of field to be ploughed, remains the same. We need the number of days required when 14 men work 12 hours per day on the same field. Concept / Approach: The basic idea is that total work is proportional to men * hours per day * number of days. If the work is constant, then: men1 * hours1 * days1 = men2 * hours2 * days2 We will use this proportionality to find the unknown number of days in the second case, keeping the total work fixed between both situations. Step-by-Step Solution: Total work W in the first situation = 16 * 7 * 48 W = 16 * 7 * 48 In the second situation, work W = 14 * 12 * D, where D is the required days. Equate the two expressions for W: 16 * 7 * 48 = 14 * 12 * D Compute the left side: 16 * 7 = 112, so 112 * 48. On the right side, 14 * 12 = 168, so 168 * D. So 112 * 48 = 168 * D Divide both sides by 168: D = (112 * 48) / 168 Simplify: 112 / 168 = 2 / 3, so D = (2 / 3) * 48 D = 32 days Verification / Alternative check: We can also reason proportionally. The effective daily work is proportional to men * hours. In the first case effective factor is 16 * 7. In the second case it is 14 * 12. The ratio of second case to first case is (14 * 12) / (16 * 7) which simplifies to 168 / 112 = 3 / 2. That means the second team does work 1.5 times faster, so they take 2 / 3 of the time. Two thirds of 48 days is 32 days, which confirms the calculation. Why Other Options Are Wrong: 30 days is slightly less than the correct 32 days and comes from rough rounding instead of exact proportional calculation. 35 days is larger than 32 days and would correspond to a smaller productivity increase than actually given. 46 days is much too large and does not match the fact that the second arrangement has more total man hours per day than the first situation. 28 days is also incorrect because it assumes a stronger increase in work rate than what the numbers actually give. Common Pitfalls: A common mistake is to compare only the number of men and forget the change in hours per day. Another frequent error is to treat the relation as directly proportional in days without using the complete man hour product. Some students also mix up the direction of the ratio and mistakenly multiply instead of divide when moving from one scenario to another. Carefully grouping the terms as men * hours * days and equating total work helps avoid these errors. Final Answer: The number of days required when 14 men work 12 hours per day to plough the same field is 32 days.

Question 7

If the price of 6 toys is Rs. 264.37, what will be the approximate price of 5 toys at the same rate?

Accepted Answer

Introduction / Context: This is a direct application of the unitary method within the broader idea of the chain rule. The question asks you to find the approximate price of 5 toys when the price of 6 toys is given. Problems of this kind are very common in aptitude tests and help you understand proportional relationships and basic calculations involving money and quantities. Given Data / Assumptions: Cost of 6 toys = Rs. 264.37. We are asked to find the approximate cost of 5 toys. The price per toy is assumed to be constant, so the cost is directly proportional to the number of toys. We are allowed to approximate the final amount to a reasonable nearby value. Concept / Approach: When the quantity is directly proportional to cost, we can first find the cost of one unit and then scale it. This is called the unitary method. The steps are: calculate cost per toy by dividing the total cost by the number of toys, then multiply this cost per toy by the desired number of toys, which is 5 in this question. Step-by-Step Solution: Cost of 6 toys = Rs. 264.37 Cost of 1 toy = 264.37 / 6 Approximate 264.37 / 6 ≈ 44.06 rupees per toy Now cost of 5 toys = 44.06 * 5 44.06 * 5 ≈ 220.30 So the approximate cost of 5 toys is about Rs. 220.30 Rounding to the nearest reasonable option, we select Rs. 220 Verification / Alternative check: We can do a quick mental check. If each toy is around Rs. 44, then 6 toys would be near 6 * 44 = 264, which matches the given 264.37 fairly well. Then 5 toys would be about 5 * 44 = 220, which again points to Rs. 220 as the best approximate answer. This quick estimate confirms our more detailed computation. Why Other Options Are Wrong: Rs. 120 and Rs. 100 are far too low and would correspond to a cost of around Rs. 20 per toy, which is inconsistent with the given data. Rs. 200 is a bit closer but still significantly lower than the calculated approximate value. Rs. 240 is slightly higher than the calculated approximate cost of Rs. 220.30 and does not match the nearest rounded value. Therefore, Rs. 220 is the best approximation. Common Pitfalls: Students sometimes multiply instead of divide when finding the cost per unit, which completely reverses the logic. Another mistake is to ignore the word approximate and expect an exact match with the decimal value, even when the options are rounded. Some learners also attempt to directly scale 6 toys to 5 toys without passing through the per toy cost, which often leads to calculation errors. Using the unitary method step by step keeps the process clear and reliable. Final Answer: The approximate price of 5 toys at the same rate is Rs. 220.

Question 8

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

Accepted Answer

Introduction / Context: This question is a direct application of the chain rule and the unitary method in the context of speed and time. The loom weaves a certain length of cloth every second, and we need to determine how long it will take to weave a specific total length. Such questions are very common in aptitude tests and help to build comfort with division, rates, and approximations. Given Data / Assumptions: The loom weaves 1.14 metres of cloth every second. The total length of cloth to be woven is 52 metres. The weaving rate is constant for the entire process. The final answer is to be chosen approximately from the given options. Concept / Approach: The relationship between distance, speed, and time is: time = distance / speed Here, distance is the total length of cloth, speed is the weaving rate in metres per second, and time will naturally come out in seconds. Since the speed is given per second, no unit conversion is needed. We simply divide 52 by 1.14 and then compare the result to the options, rounding as needed. Step-by-Step Solution: Weaving rate = 1.14 metres per second Required length = 52 metres time in seconds = 52 / 1.14 Compute the division: 52 / 1.14 ≈ 45.61 seconds So the loom will take approximately 45.61 seconds to weave 52 metres of cloth Among the options, 45.61 seconds matches exactly and is therefore the correct choice Verification / Alternative check: As a rough check, if the speed were 1 metre per second, 52 metres would take 52 seconds. Since the loom actually weaves slightly more than 1 metre per second, the time required should be somewhat less than 52 seconds. Our calculated value of about 45.61 seconds is indeed less than 52 seconds but not extremely small, so it fits this intuitive check quite well. Why Other Options Are Wrong: 29.32 seconds and 39.25 seconds are too low and would imply a much higher weaving speed than 1.14 metres per second. 42.51 seconds is closer but still a bit below the calculated approximate time. 60.00 seconds is greater than the 52 second rough upper bound and would imply a speed lower than 1 metre per second. Only 45.61 seconds aligns with the exact calculation using the given rate. Common Pitfalls: A typical mistake is to multiply 52 by 1.14 instead of dividing, which reverses the roles of speed and time. Another error is ignoring the units and misinterpreting centimetres for metres or vice versa. Some students also round too early during calculation, which can push them toward an incorrect option. Keeping the formula time = distance / speed clear in mind prevents most of these errors. Final Answer: The loom will take approximately 45.61 seconds to weave 52 metres of cloth.

Question 9

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 at this constant rate?

Accepted Answer

Introduction / Context: This problem is another example of using the relationship between distance, speed, and time, which is a core part of chain rule style questions. We are given how many metres of cloth an industrial loom weaves each second, and we must compute the total time required to weave a specified total length. The focus here is careful division and sensible approximation to match a multiple choice option. Given Data / Assumptions: Weaving speed of the loom = 0.128 metres per second. Total cloth required = 25 metres. The weaving rate is assumed to be constant for the entire duration. We need the approximate time in seconds, selecting from the provided options. Concept / Approach: Using the standard formula, time = distance / speed we substitute the total distance in metres and the speed in metres per second to get time in seconds. Because the numbers are fractional, we expect a non integer result, which we then approximate to the closest option given. This is a straightforward numerical calculation with a clear proportional meaning. Step-by-Step Solution: Speed of loom = 0.128 metres per second Required distance = 25 metres time in seconds = 25 / 0.128 Compute the division: 25 / 0.128 ≈ 195.31 seconds This is slightly above 195 and below 196, so the closest integer is 195 seconds Therefore, the approximate time required is 195 seconds Verification / Alternative check: As a quick check, multiply 0.128 by 200 seconds and see the distance. 0.128 * 200 = 25.6 metres So in 200 seconds the loom weaves about 25.6 metres, which is a bit more than 25 metres. Therefore, in slightly less than 200 seconds it should weave exactly 25 metres. The value 195 seconds fits this reasoning nicely, since 195 is a little less than 200 and 25.6 metres is just a little more than 25 metres. Why Other Options Are Wrong: 194 seconds gives a smaller distance than 25 metres because it is slightly too short compared to our precise calculation. 196 seconds and 197 seconds are both larger than the calculated 195.31 seconds and therefore represent slightly longer times than needed. 205 seconds is significantly larger and would overshoot the required distance by a wider margin. Among all options, 195 seconds is the closest to the exact computed value. Common Pitfalls: A frequent error is inverting the formula and computing 25 * 0.128 instead of 25 / 0.128, which would incorrectly reduce the time instead of increasing it. Some learners also misread 0.128 as 0.0128 or 1.28, leading to large errors. Rounding too early in the computation, for example using 0.13 instead of 0.128, can also shift the final answer away from the correct choice. Careful handling of decimals and clear use of the time formula are essential. Final Answer: The loom will take approximately 195 seconds to weave 25 metres of cloth.

Question 10

Running at the same constant rate, 6 identical machines can produce a total of 270 bottles per minute. At this rate, how many bottles could 10 such machines produce in 4 minutes?

Accepted Answer

Introduction / Context: This question tests the concept of proportionality and the chain rule in the context of production rates. Machines working at a constant rate produce a certain number of bottles per minute, and we are asked to find how many bottles will be produced when the number of machines and the time both change. This type of problem appears frequently in aptitude examinations to check reasoning about direct proportion. Given Data / Assumptions: 6 identical machines together produce 270 bottles per minute. All machines have the same constant rate of production. We need the number of bottles produced by 10 machines running for 4 minutes at the same per machine rate. There is no downtime or change in efficiency during the production period. Concept / Approach: If 6 machines produce 270 bottles per minute, we can first find the production rate of one machine. Since the relationship is directly proportional, we then multiply by the new number of machines (10) to find the total bottles per minute for the second scenario. Finally, we multiply by the total time in minutes, which is 4, to get the total number of bottles produced in that duration. Step-by-Step Solution: Production by 6 machines in 1 minute = 270 bottles Production by 1 machine in 1 minute = 270 / 6 = 45 bottles Production by 10 machines in 1 minute = 10 * 45 = 450 bottles Time given = 4 minutes Total production in 4 minutes = 450 * 4 450 * 4 = 1800 bottles Therefore, 10 machines will produce 1800 bottles in 4 minutes Verification / Alternative check: We can also use a direct proportion approach without explicitly finding the single machine rate. The number of bottles is proportional to both the number of machines and the time. So, New bottles = 270 * (10 / 6) * 4 10 / 6 simplifies to 5 / 3 New bottles = 270 * (5 / 3) * 4 270 * (5 / 3) = 270 * 5 / 3 = 90 * 5 = 450 Then 450 * 4 = 1800 bottles This confirms the earlier result. Why Other Options Are Wrong: 648 bottles is too low and would correspond to ignoring the full 4 minutes or misusing the ratios. 2700 bottles and 10800 bottles are far too high compared to the correct proportional scaling of the original scenario. 1350 bottles is somewhat close but still lower than the correct result and arises from incomplete multiplication of the machine or time factor. Only 1800 bottles match the proportional reasoning based on the given data. Common Pitfalls: A common pitfall is to scale only the number of machines and forget to scale the time, or vice versa. Some students mistakenly multiply 270 directly by 4 without adjusting for the increase in machine count. Another error is confusing direct proportion with inverse proportion. Remember that more machines and more time both increase total production in a directly proportional way, which is why we multiply by both factors. Final Answer: At the given constant rate, 10 machines running for 4 minutes will produce 1800 bottles.

Question 11

If the total price of 23 toys is Rs 276, then what will be the total price of 12 such toys bought at the same rate?

Accepted Answer

Introduction / Context: This question is a basic application of the unitary method and direct proportion in arithmetic. The idea is that if the cost of a certain number of toys is known, then the cost of any other number of the same toys at the same rate can be found by first finding the cost of one toy and then scaling up or down. Such questions are very common in aptitude tests because they check a candidate's understanding of proportional reasoning and careful calculation. Given Data / Assumptions: The total price of 23 toys is Rs 276. All toys are identical and are sold at the same constant rate. We need to find the price of 12 toys at this same rate. There are no discounts or extra charges involved. Concept / Approach: The concept used is direct proportion and the unitary method. When the price is directly proportional to the number of items, we can write: price per toy = total price / number of toys Once the price of one toy is known, the price of any other number of toys is: price of required toys = price per toy * number of required toys All calculations are done with simple division and multiplication. Step-by-Step Solution: Step 1: Total price of 23 toys = Rs 276. Step 2: Find the price of one toy using unitary method. price per toy = 276 / 23 276 divided by 23 is 12, so price per toy = Rs 12. Step 3: Now we need the price of 12 toys. price of 12 toys = price per toy * 12 price of 12 toys = 12 * 12 = 144 So the required cost of 12 toys is Rs 144. Verification / Alternative check: We can verify quickly. If one toy costs Rs 12, then 23 toys should cost: 23 * 12 = 276 This matches the given total price, so our price per toy is correct. Scaling from 23 to 12 toys is also consistent because the relation between quantity and cost is linear for a constant rate. Why Other Options Are Wrong: Option b (148): This would imply a non integer or incorrect price per toy and does not match the original total of Rs 276 for 23 toys. Option c (156): This would correspond to a higher unit price and would not give 276 when multiplied back by 23. Option d (186): This is much larger than the correct value and clearly does not fit the proportional relationship. Common Pitfalls: Students sometimes directly try to form a shortcut without first finding the price per toy, which can cause errors. Another common mistake is doing 23 * 12 instead of 276 / 23, or mis dividing 276 by 23. Careless multiplication, such as computing 12 * 12 incorrectly, may also lead to wrong answers in an otherwise simple question. Always ensure that the unit price is found correctly before scaling up or down. Final Answer: The total price of 12 toys at the same rate is Rs 144.

Question 12

The cost of $16$ packets of salt, each weighing $900$ grams is ₹ $28$. What will be the cost of $27$ packets, if each packet weighs $1$ kg?

Accepted Answer

Concept & Chain Rule: The total cost of goods is directly proportional to the total weight of the goods (number of items multiplied by the weight per item). \frac{Cost_1}{N_1 × W_1} = \frac{Cost_2}{N_2 × W_2} Step-by-Step Solution: • Given: - Case 1: N_1 = 16, W_1 = 900 g, Cost_1 = 28 - Case 2: N_2 = 27, W_2 = 1000 g (converted from 1 kg), Cost_2 = x • Calculation: (28/16 × 900) = (x/27 × 1000) Simplify the denominators: (28/14400) = (x/27000) Solve for x: x = (28 × 27000/14400) = (28 × 270/144) x = 28 × (15/8) = 7 × (15/2) = (105/2) = 52.50 Exam Strategy & Shortcut: Compare the multipliers. Quantity goes from 16 to 27 (factor of (27/16)). Weight goes from 900 to 1000 (factor of (10/9)). New cost = 28 × (27/16) × (10/9) = 28 × (3/16) × 10 = (840/16) = 52.50. Common Pitfall: Forgetting to convert 1 kg to 1000 grams, resulting in a severe mismatch of units and an incorrect decimal answer. Final Answer: Therefore, the correct answer is ₹ 52.50.

Question 13

49 pumps can empty a reservoir in $6 \frac{1}{2}$ days, working 8 hours a day. If 196 pumps are used for 5 hours each day, then the same work will be completed in

Accepted Answer

Concept & The Chain Rule: This problem involves three variables affecting total work: Number of workers/machines (M), Days (D), and Hours per day (H). Because the total amount of work (emptying one reservoir) is constant, the product of these variables must remain constant across different scenarios. M_1 D_1 H_1 = M_2 D_2 H_2 Step-by-Step Solution: • Identify the variables for the first scenario: M_1 = 49 pumps, D_1 = 6 (1/2) = (13/2) days, H_1 = 8 hours. • Identify the variables for the second scenario: M_2 = 196 pumps, H_2 = 5 hours, D_2 = x days. • Substitute the values into the formula: 49 × (13/2) × 8 = 196 × x × 5. • Simplify the left side: The 2 in the denominator cancels with 8 to give 4. The equation becomes 49 × 13 × 4 = 196 × 5 × x. • Notice that 49 × 4 = 196. Substitute this into the equation: 196 × 13 = 196 × 5 × x. • Cancel 196 from both sides: 13 = 5x. • Solve for x: x = (13/5) = 2 (3/5) days. Exam Strategy & Shortcut: Look for multiples before multiplying large numbers. Seeing 49 and 196 should immediately trigger the realization that 196 = 49 × 4. Keeping the equation as factors (49 × 13 × 4) rather than computing the product (2548) allows for instant cancellation, saving valuable time. Common Pitfall: A common mistake is incorrectly converting the mixed fraction 6 (1/2) into an improper fraction, or forgetting to multiply by the hours per day when setting up the initial total work equivalence. Final Answer: Therefore, the correct answer is 2 (3/5) days.

Question 14

30 labourers, working 7 hours a day can finish a piece of work in 18 days. If the labourers work 6 hours a day, then the number of labourers to finish the same piece of work in 30 days, will be:

Accepted Answer

Concept & Work Equivalence Theorem: When the total amount of work is identical in two different situations, the total man-hours required remains constant. We can equate the product of Men, Days, and Hours for both scenarios using the foundational Chain Rule formula. M_1 D_1 H_1 = M_2 D_2 H_2 Step-by-Step Solution: • Scenario 1 data: M_1 = 30 labourers, D_1 = 18 days, H_1 = 7 hours/day. • Scenario 2 data: M_2 = ? labourers, D_2 = 30 days, H_2 = 6 hours/day. • Set up the equivalence equation based on the formula: 30 × 18 × 7 = M_2 × 30 × 6. • Divide both sides by 30 to immediately simplify the equation: 18 × 7 = M_2 × 6. • Divide both sides by 6: 3 × 7 = M_2. • Calculate the final value: M_2 = 21. Exam Strategy & Shortcut: Write down the equation as a fraction to solve for the unknown directly: M_2 = (30 × 18 × 7/30 × 6). From this vantage point, you can visually cross out the 30 in the numerator and denominator, leaving (18 × 7/6). Since (18/6) = 3, the answer is simply 3 × 7 = 21. Common Pitfall: Multiplying out the entire left side (30 × 18 × 7 = 3780) before dividing wastes time and increases the probability of basic arithmetic calculation errors. Always cancel factors first. Final Answer: Therefore, the correct answer is 21.

Question 15

A wall of 100 metres can be built by 7 men or 10 women in 10 days. How many days will 14 men and 20 women take to build a wall of 600 metres?

Accepted Answer

Concept & Extended Chain Rule: When "or" establishes an efficiency relationship and the total work output (meters of wall) changes, we combine the efficiency conversion with the extended chain rule. (M_1 D_1/W_1) = (M_2 D_2/W_2) Step-by-Step Solution: • Find the efficiency relationship: 7 men = 10 women. • Determine the workforce for scenario 2: 14 men and 20 women. • Convert the target workforce into a single unit (men): Since 10 women = 7 men, then 20 women = 14 men. • Total equivalent men for scenario 2: 14 men + 14 men = 28 men. • Identify variables for the formula: M_1 = 7 men, D_1 = 10 days, W_1 = 100 m M_2 = 28 men, D_2 = x days, W_2 = 600 m • Substitute into the formula: (7 × 10/100) = (28 × x/600) • Simplify the left side: (70/100) = 0.7. • Simplify the right side denominator: multiply both sides by 600. 0.7 × 600 = 28 × x 420 = 28x • Solve for x: x = (420/28) = 15 days. Exam Strategy & Shortcut: Observe the direct multipliers. The workforce changes from 7 men to an equivalent of 28 men (which is a 4x increase in workforce). 4x the workforce means they work 4x as fast, or do 4x the work in the same time. In 10 days, this 4x workforce would build 100 m × 4 = 400 m. To build 600m (which is 1.5x of 400m), they need 10 days × 1.5 = 15 days. Common Pitfall: A frequent error is forgetting to put the length of the wall (the "work") in the denominator and treating it as a direct multiplier in the numerator instead. Final Answer: Therefore, the correct answer is 15.

Question 16

If 10 men or 20 boys can make 260 mats in 20 days, then how many mats will be made by 8 men and 4 boys in 20 days?

Accepted Answer

Concept & Proportional Output: When the time (days) remains constant, the work done (mats made) is directly proportional to the total effective workforce. We convert the mixed group into a single unit to see how it compares to the original group. Work \propto Workforce Step-by-Step Solution: • First, establish the equivalency from the "or" statement: 10 men = 20 boys. • Simplify this ratio: 1 man = 2 boys. • Evaluate the target group: 8 men and 4 boys. • Convert the boys to men to match our primary unit: Since 2 boys = 1 man, then 4 boys = 2 men. • Find the total equivalent workforce of the new group: 8 men + 2 men = 10 men. • The new workforce is exactly 10 men. • From the initial conditions, we know that 10 men make 260 mats in 20 days. • Since the new group is mathematically equivalent to 10 men, and they are also working for 20 days, they will produce the exact same amount of work. • Mats made = 260. Exam Strategy & Shortcut: Always calculate the total equivalent workforce *first* before plugging numbers into complex chain rule formulas. Here, a quick mental check (4 boys = 2 men \rightarrow 8+2 = 10 men) reveals the workforce is identical to the given scenario. No calculation is needed; the output is identical. Common Pitfall: Blindly using the formula (M_1 D_1/W_1) = (M_2 D_2/W_2) without checking if M_1 and M_2 are actually the same. While the formula will yield the correct answer, it consumes precious time that could be saved through simple observation. Final Answer: Therefore, the correct answer is 260.

Question 17

If 600 men dig a 5.5 m wide, 4 m deep and 405 m long canal in half an hour, then how long a canal will 2500 men, working for 6 hours, dig if it is 10 m wide and 8 m deep?

Accepted Answer

Concept & Volumetric Work in Chain Rule: In this variation of the chain rule, the "Work" (W) is the physical volume of the canal dug. Volume is calculated as Length × Width × Depth. We substitute this product into the denominator of the standard formula. (M_1 H_1/V_1) = (M_2 H_2/V_2) Step-by-Step Solution: • Establish Scenario 1 variables: M_1 = 600 men, H_1 = 0.5 hours, V_1 = 5.5 × 4 × 405 = 8910 m^3. • Establish Scenario 2 variables: M_2 = 2500 men, H_2 = 6 hours, V_2 = 10 × 8 × L = 80L m^3 (where L is the unknown length). • Apply the chain rule equation: (600 × 0.5/5.5 × 4 × 405) = (2500 × 6/10 × 8 × L) • Simplify the numerators: (300/8910) = (15000/80L) • Simplify the fractions by cancelling zeros: (30/891) = (1500/8L) • Cross-multiply to isolate L: 30 × 8L = 1500 × 891 240L = 1336500 • Solve for L: L = (1336500/240) = (133650/24) • Divide to get the final decimal/fraction: 133650 \div 24 = 5568.75, which equals 5568 (3/4) m. Exam Strategy & Shortcut: To prevent massive multiplication, keep numbers factored and cancel them across the equals sign. From (600 × 0.5/5.5 × 4 × 405) = (2500 × 6/10 × 8 × L), immediately cancel the 600 × 0.5 (300) with the 2500 × 6 (15000) to leave a 50 on the right side. The equation shrinks rapidly to (1/5.5 × 4 × 405) = (50/80L). This heavily reduces the arithmetic burden. Common Pitfall: A major pitfall is converting "half an hour" into 30 minutes for H_1, but keeping H_2 as 6 hours. Units of time must be strictly consistent (both in hours or both in minutes). Final Answer: Therefore, the correct answer is 5568 (3/4) m.

Question 18

64 persons can dig a trench 50 m long, 2 m wide and 2 m deep in 5 days, working 12 hours daily. In how many days, working 8 hours daily, will 80 persons dig another trench 75 m long, 4 m wide and 3 m deep?

Accepted Answer

Concept & 4-Variable Chain Rule: This is a comprehensive chain rule problem utilizing all variables: Men, Days, Hours, and Work (calculated as a 3D volume: Length × Width × Depth). (M_1 D_1 H_1/W_1) = (M_2 D_2 H_2/W_2) Step-by-Step Solution: • Scenario 1 parameters: M_1 = 64, D_1 = 5, H_1 = 12. The work W_1 is the volume: 50 × 2 × 2 = 200 m^3. • Scenario 2 parameters: M_2 = 80, D_2 = x, H_2 = 8. The work W_2 is the volume: 75 × 4 × 3 = 900 m^3. • Plug these values into the master formula: (64 × 5 × 12/200) = (80 × x × 8/900) • Simplify the denominators by cancelling two zeros across the equation (multiply both sides by 100): (64 × 5 × 12/2) = (80 × x × 8/9) • Simplify the left side by computing the math: (64/2) = 32. Then 32 × 5 × 12 = 160 × 12 = 1920. • The equation is now: 1920 = (640 × x/9). • Notice that 1920 is exactly 3 times 640 (640 × 3 = 1920). Divide both sides by 640: 3 = (x/9) • Multiply by 9 to find x: x = 27. Exam Strategy & Shortcut: Always arrange the formula and ruthlessly cancel numbers on the top and bottom before multiplying. Write it as: x = (64 × 5 × 12 × 900/80 × 8 × 200). Cancel 900 and 200 to 9 and 2. Cancel 80 × 8 (640) with 64 × 10 from the top. The math will beautifully collapse to 3 × 9 = 27 in seconds without ever dealing with a four-digit number. Common Pitfall: Calculating the total product of the numerator and denominator before dividing (3840/200 = 640x/900 \rightarrow 19.2 = 0.711x). Introducing decimals into chain rule problems makes the math significantly harder to track and solve without a calculator. Final Answer: Therefore, the correct answer is 27.

Question 19

21 binders can bind 1400 books in 15 days. How many binders will be required to bind 800 books in 20 days?

Accepted Answer

Concept & Chain Rule Formula: This problem requires the fundamental Chain Rule (or unitary method) relating Men (binders), Days, and Work (books). (M_1 × D_1/W_1) = (M_2 × D_2/W_2) Step-by-Step Solution: 1. Identify the variables: - Initial scenario: M_1 = 21 binders, W_1 = 1400 books, D_1 = 15 days. - Target scenario: W_2 = 800 books, D_2 = 20 days. Let the required binders be M_2. 2. Apply the formula: - (21 × 15/1400) = (M_2 × 20/800) 3. Simplify the equation: - Simplify the denominators by dividing both by 200: - (21 × 15/7) = (M_2 × 20/4) - 3 × 15 = M_2 × 5 - 45 = 5 × M_2 4. Solve for M_2: - M_2 = (45/5) = 9 Exam Strategy & Shortcut: You can set up a direct proportion chain: Binders needed = Initial Binders × (Ratio of Books) × (Inverse Ratio of Days). M_2 = 21 × ((800/1400)) × ((15/20)) = 21 × (4/7) × (3/4) = 3 × 3 = 9. Common Pitfall: A frequent error is misplacing the "Work" parameter in the numerator instead of the denominator. Always remember that work done is inversely proportional to the time and effort required. Final Answer: Therefore, the correct answer is 9.

Question 20

A certain number of artisans can complete a shoe fabrication consignment in 16 days. 8 additional artisans had to be deployed for the same consignment and together they completed it in 4 days less than the earlier estimate. The number of artisans in…

Accepted Answer

Concept & Work Equivalence Principle: When the total amount of work remains constant, the number of workers is inversely proportional to the number of days taken. M_1 × D_1 = M_2 × D_2 Step-by-Step Solution: 1. Define the initial state: Let the initial number of artisans be x. They take 16 days. - Total Work = x × 16 2. Define the new state: 8 additional artisans are deployed, making the total number (x + 8). They complete the work in 4 days less, meaning it takes 16 - 4 = 12 days. - Total Work = (x + 8) × 12 3. Equate the work: - 16x = 12(x + 8) 4. Solve for x: - 16x = 12x + 96 - 4x = 96 - x = 24 Exam Strategy & Shortcut: Use the ratio method. The ratio of days taken is 16 : 12, which simplifies to 4 : 3. Since days and workers are inversely proportional, the ratio of workers must be 3 : 4. The difference in the ratio is 1 unit, which corresponds to the 8 additional artisans. Initial workers = 3 units = 3 × 8 = 24. Common Pitfall: Misinterpreting "4 days less" as completing the work in exactly 4 days. Always subtract the saved days from the original estimate (16 - 4 = 12). Final Answer: Therefore, the correct answer is 24.

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