If 8 men can do a piece of work in 12 days, how many days will 6 men take to do the same work?

16 days. Men and days are inversely proportional: 8 12 = 6 D, so D = 16 days.

15 workers build a wall in 48 hours. How many workers are needed to build the same wall in 30 hours?

24. 15 48 = W 30, so W = 72030 = 24 workers.

36 men can complete a job in 18 days. After 6 days, 12 more men join them. In how many more days is the job finished?

9 days. Total work = 36 18 = 648 man-days. In 6 days, 36 6 = 216 is done, leaving 432. With 36 + 12 = 48 men this takes 432 48 = 9 more days.

6 pumps working 8 hours a day empty a tank in 2 days. How many hours a day must 4 pumps work to empty it in 3 days?

8. Pump-hours are constant: 6 8 2 = 4 H 3, so 96 = 12H and H = 8 hours a day.

If 10 spiders spin 10 webs in 10 days, how many days does 1 spider take to spin 1 web?

10. 10 spiders make 10 webs in 10 days, so each spider makes one web in 10 days. One spider therefore takes 10 days for one web.

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Quantitative Aptitude · Averages, Ratios & Percentages

Chain Rule — Questions & Answers

Chain-rule problems link several quantities — workers, days, hours, output — where some vary directly and some inversely. Set up one equation by multiplying each factor on the correct side.

5 solved questions
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Key formulas included

Key formulas

$\dfrac{M_1 D_1 H_1}{W_1} = \dfrac{M_2 D_2 H_2}{W_2}$ (men $\times$ days $\times$ hours per unit work is constant)
Directly proportional: more of one means more of the other (work, output)
Inversely proportional: more of one means less of the other (men vs days)

5 solved Chain Rule questions

Try each question first — tap an option to check your answer, or open Learn step by step for a guided walkthrough.

01
If 8 men can do a piece of work in 12 days, how many days will 6 men take to do the same work?
5 steps · reveal one at a time
1. Step 1: Understand the Relationship
  This problem involves work and time. When more men work on a task, it takes less time. This means the number of men and the number of days are inversely proportional.
2. Step 2: Set Up the Equation
  For inversely proportional quantities, the product of men and days remains constant. So, if $M_1$ men take $D_1$ days and $M_2$ men take $D_2$ days, then $M_1 \times D_1 = M_2 \times D_2$ .
3. Step 3: Calculate the Unknown
  We are given $M_1 = 8$ men and $D_1 = 12$ days. We want to find $D_2$ for $M_2 = 6$ men. Substitute these values into the equation: $8 \times 12 = 6 \times D_2$ .
4. Step 4: Solve for Days
  First, calculate the product: $8 \times 12 = 96$ . So, the equation becomes $96 = 6 \times D_2$ . To find $D_2$ , divide 96 by 6: $D_2 = 96 \div 6 = 16$ days.
5. Step 5: The answer
  So the answer is (B) 16 days, because 6 men will take 16 days to complete the same amount of work.
02
15 workers build a wall in 48 hours. How many workers are needed to build the same wall in 30 hours?
5 steps · reveal one at a time
1. Step 1: Understand the Relationship
  This problem involves an inverse relationship between the number of workers and the time taken to complete a task. If you have more workers, it takes less time, and vice versa, for the same amount of work.
2. Step 2: Formulate the Equation
  For a fixed amount of work, the product of the number of workers and the time taken is constant. We can write this as $W_1 \times T_1 = W_2 \times T_2$ , where $W$ is workers and $T$ is time.
3. Step 3: Substitute Known Values
  We are given $W_1 = 15$ workers and $T_1 = 48$ hours. We need to find $W_2$ when $T_2 = 30$ hours. Substituting these into our equation gives $15 \times 48 = W_2 \times 30$ .
4. Step 4: Calculate the Workers
  First, calculate the product on the left side: $15 \times 48 = 720$ . So, the equation becomes $720 = W_2 \times 30$ . To find $W_2$ , divide 720 by 30: $W_2 = \frac{720}{30} = 24$ .
5. Step 5: The answer
  So the answer is (A) 24. We need 24 workers to build the same wall in 30 hours, maintaining the constant work rate.
03
36 men can complete a job in 18 days. After 6 days, 12 more men join them. In how many more days is the job finished?
4 steps · reveal one at a time
1. Step 1: Total Work Needed
  First, find the total amount of work required. If 36 men can complete the job in 18 days, the total work is $36 \times 18 = 648$ man-days. This is the total effort needed to finish the job.
2. Step 2: Work Done So Far
  For the first 6 days, 36 men worked. The amount of work completed during this period is $36 \times 6 = 216$ man-days. We need to find the remaining work.
3. Step 3: Remaining Work & Men
  Subtract the work done from the total work to find the remaining work: $648 - 216 = 432$ man-days. Now, 12 more men join, so the new team size is $36 + 12 = 48$ men.
4. Step 4: The answer
  So the answer is (B) 9 days. To find how many more days are needed, divide the remaining work by the new number of men: $432 \div 48 = 9$ days. The job will be finished in 9 more days.
04
6 pumps working 8 hours a day empty a tank in 2 days. How many hours a day must 4 pumps work to empty it in 3 days?
5 steps · reveal one at a time
1. Step 1: Understand the Goal
  The task is to empty the same tank in two different scenarios. This means the total work done in both scenarios must be equal. We need to find the hours per day for the second scenario.
2. Step 2: Calculate Total Work
  The total work is proportional to the number of pumps, the hours they work per day, and the number of days. For the first scenario, the total 'pump-hours' is $6 \text{ pumps} \times 8 \text{ hours/day} \times 2 \text{ days}$ .
3. Step 3: Set Up the Equation
  Let $H$ be the unknown hours per day for the second scenario. The total 'pump-hours' for the second scenario is $4 \text{ pumps} \times H \text{ hours/day} \times 3 \text{ days}$ . Since the total work is constant, we can set up the equation: $6 \times 8 \times 2 = 4 \times H \times 3$ .
4. Step 4: Solve for H
  Now, simplify and solve for $H$ : $96 = 12H$ . To find $H$ , divide both sides by 12: $H = 96 / 12 = 8$ . So, 4 pumps must work 8 hours a day.
5. Step 5: The answer
  So the answer is (C) 8. This means 4 pumps working 8 hours a day will empty the tank in 3 days, doing the same total amount of work as in the first scenario.
05
If 10 spiders spin 10 webs in 10 days, how many days does 1 spider take to spin 1 web?
4 steps · reveal one at a time
1. Step 1: Understand the Setup
  We are given that 10 spiders spin 10 webs in a total of 10 days. This type of problem often tests your understanding of work rate and proportionality.
2. Step 2: Consider Each Spider
  Think about what this means for an individual spider. If there are 10 spiders and 10 webs are made, it implies that, on average, each spider is responsible for spinning one web.
3. Step 3: Determine Individual Time
  Since each of the 10 spiders contributes one web to the total of 10 webs, and this entire process takes 10 days, it means that one spider takes 10 days to spin one web.
4. Step 4: The answer
  So the answer is (B) 10. The number of spiders and webs are directly proportional, meaning the time for one spider to spin one web remains constant.

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