Chain Rule
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Solved Examples
Study MaterialSolved Examples – Chain Rule
Solved examples help students understand the practical application of Chain Rule concepts in competitive examinations. These examples are designed from basic to advanced level and cover important chain rule problems frequently asked in SSC, Banking, Railway, CDS, NDA, UPSC, CAT, and placement aptitude tests.
Topics Covered in Solved Examples
- Men-Days-Hours Problems
- Machine and Production Problems
- Speed-Time-Distance Applications
- Work and Wage Problems
- Direct and Inverse Proportion
- Pipes and Tanks Applications
- Business Production Problems
- Multi-Variable Chain Rule Problems
- Advanced Productivity Calculations
- Time and Work Applications
Example 1: Men and Days Problem
Question: 12 men can complete a work in 15 days. How many men are required to complete the same work in 10 days?
Solution:
Men and days are inversely proportional.
12 × 15 = x × 10
180 = 10x
x = 18
Therefore:
Required men = 18
Example 2: Men, Days, and Hours Problem
Question: 8 men working 6 hours daily complete a work in 15 days. In how many days will 12 men working 8 hours daily complete the same work?
Solution:
Using:
Men × Hours × Days = Constant
8 × 6 × 15 = 12 × 8 × x
720 = 96x
x = 7.5
Therefore:
Required days = 7.5
Example 3: Machine Production Problem
Question: 5 machines produce 1000 units in 8 days. How many units will 10 machines produce in 12 days?
Solution:
Production is directly proportional to:
- Machines
- Days
Required production:
= 1000 × (10/5) × (12/8)
= 1000 × 2 × 3/2
= 3000
Therefore:
Required production = 3000 units
Example 4: Speed and Time Problem
Question: A car travelling at 60 km/h takes 8 hours to complete a journey. How much time will it take if speed is increased to 80 km/h?
Solution:
Speed and time are inversely proportional.
60 × 8 = 80 × x
480 = 80x
x = 6
Therefore:
Required time = 6 hours
Example 5: Pipes and Tanks Problem
Question: 4 pipes can fill a tank in 12 hours. How much time will 6 pipes take to fill the same tank?
Solution:
Pipes and time are inversely proportional.
4 × 12 = 6 × x
48 = 6x
x = 8
Therefore:
Required time = 8 hours
Example 6: Wages Problem
Question: If 12 workers earn 18000 in 15 days, how much will 18 workers earn in 20 days?
Solution:
Wages are directly proportional to:
- Workers
- Days
Required wages:
= 18000 × (18/12) × (20/15)
= 18000 × 3/2 × 4/3
= 36000
Therefore:
Required wages = 36000
Example 7: Production and Workers Problem
Question: 15 workers produce 450 articles in 6 days. How many articles will 20 workers produce in 9 days?
Solution:
Production is directly proportional to:
- Workers
- Days
Required production:
= 450 × (20/15) × (9/6)
= 450 × 4/3 × 3/2
= 900
Therefore:
Required articles = 900
Example 8: Multi-Variable Work Problem
Question: 10 men working 8 hours daily complete a work in 12 days. In how many days will 15 men working 6 hours daily complete the same work?
Solution:
Using:
Men × Hours × Days = Constant
10 × 8 × 12 = 15 × 6 × x
960 = 90x
x = 10.67
Therefore:
Required days ≈ 10.67
Example 9: Travel Distance Problem
Question: A train travels 300 km in 5 hours. How much distance will it cover in 8 hours at the same speed?
Solution:
Distance is directly proportional to time.
Required distance:
= 300 × (8/5)
= 480 km
Therefore:
Required distance = 480 km
Example 10: Men and Wages Problem
Question: 25 men receive 50000 as wages for 20 days. How much will 40 men receive for 15 days?
Solution:
Wages are directly proportional to:
- Men
- Days
Required wages:
= 50000 × (40/25) × (15/20)
= 50000 × 8/5 × 3/4
= 60000
Therefore:
Required wages = 60000
Example 11: Workers and Workload Problem
Question: 18 workers complete 540 units of work in 9 days. How much work will 24 workers complete in 15 days?
Solution:
Work is directly proportional to:
- Workers
- Days
Required work:
= 540 × (24/18) × (15/9)
= 540 × 4/3 × 5/3
= 1200
Therefore:
Required work = 1200 units
Example 12: Machine-Time Problem
Question: 8 machines complete a work in 18 days. How many days will 12 machines take?
Solution:
Machines and days are inversely proportional.
8 × 18 = 12 × x
144 = 12x
x = 12
Therefore:
Required days = 12
Example 13: Advanced Production Problem
Question: 16 workers working 7 hours daily produce 560 units in 10 days. How many units will 20 workers working 8 hours daily produce in 12 days?
Solution:
Production is directly proportional to:
- Workers
- Hours
- Days
Required production:
= 560 × (20/16) × (8/7) × (12/10)
= 560 × 5/4 × 8/7 × 6/5
= 960
Therefore:
Required production = 960 units
Example 14: Water Supply Problem
Question: 5 taps fill a tank in 24 minutes. How long will 8 taps take to fill the same tank?
Solution:
Taps and time are inversely proportional.
5 × 24 = 8 × x
120 = 8x
x = 15
Therefore:
Required time = 15 minutes
Example 15: Workers and Hours Problem
Question: 20 workers working 9 hours daily complete a work in 18 days. In how many days will 30 workers working 6 hours daily complete the same work?
Solution:
Using:
Men × Hours × Days = Constant
20 × 9 × 18 = 30 × 6 × x
3240 = 180x
x = 18
Therefore:
Required days = 18
Example 16: Direct Proportion Problem
Question: If 12 books cost 720, what will be the cost of 20 books?
Solution:
Cost is directly proportional to number of books.
Required cost:
= 720 × (20/12)
= 1200
Therefore:
Required cost = 1200
Example 17: Inverse Proportion Problem
Question: A journey takes 12 hours at 50 km/h. How much time will it take at 75 km/h?
Solution:
Speed and time are inversely proportional.
50 × 12 = 75 × x
600 = 75x
x = 8
Therefore:
Required time = 8 hours
Example 18: Advanced Men-Days-Hours Problem
Question: 24 men working 8 hours daily complete a work in 20 days. How many men working 10 hours daily are required to complete the work in 16 days?
Solution:
Using:
Men × Hours × Days = Constant
24 × 8 × 20 = x × 10 × 16
3840 = 160x
x = 24
Therefore:
Required men = 24
Important Exam Tips
- Always identify direct and inverse relationships correctly.
- Use table method for complicated questions.
- Memorize work and production formulas.
- Simplify ratios before multiplication.
- Practice men-days-hours problems regularly.
- Verify proportional relations carefully.
- Avoid calculation mistakes during multiplication.
Practicing solved examples regularly improves conceptual clarity, logical thinking, and calculation speed in solving Chain Rule questions in competitive examinations.