Chain Rule Aptitude

The Chain Rule is a vital concept in arithmetic and algebra, often used in problems related to ratios and proportions. It involves understanding how a change in one quantity leads to a change in another, based on a fixed ratio or relationship. This concept is widely used in various fields, including finance, physics, and everyday problem-solving.

1. If 20 men can build a wall in 40 days, how long will 30 men take to build the same wall?

More men will take fewer days (inverse proportion).
20 men : 30 men = 40 days : x days
(20 × 40) = (30 × x)
x = (20 × 40) / 30 = 27 days.

2. A car can cover 600 km in 8 hours. How long will it take to cover 900 km?

More distance takes more time (direct proportion).
600 km : 900 km = 8 hours : x hours
(600 × 8) = (900 × x)
x = (600 × 8) / 900 = 12 hours.

3. If 5 machines can manufacture 200 widgets in 4 hours, how many widgets can 10 machines manufacture in 6 hours?

More machines and more hours, more widgets (both direct proportion).
(5 machines, 4 hours) : (10 machines, 6 hours) = 200 widgets : x widgets
(5 × 4 × 200) = (10 × 6 × x)
x = (5 × 4 × 200) / (10 × 6) = 600.

4. A garrison of 500 men had provisions for 27 days. After 3 days, 125 men leave. How long will the food last now?

Fewer men, more days (inverse proportion).
Initially, 500 men have provisions for 27 days.
After 3 days, 500 - 125 = 375 men remain.
500 men : 375 men = (27 - 3) days : x days
(500 × 24) = (375 × x)
x = (500 × 24) / 375 = 32 more days.
Total = 3 (already passed) + 32 = 35 days.

5. If 9 workers can complete a piece of work in 6 days, in how many days will 6 workers complete that work?

Fewer workers will take more days (inverse proportion).
9 workers : 6 workers = 6 days : x days
(9 × 6) = (6 × x)
x = (9 × 6) / 6 = 9 days.

6. If a car travels 360 km in 4 hours, how much distance will it cover in 5 hours at the same speed?

More time, more distance (direct proportion).
4 hours : 5 hours = 360 km : x km
(4 × 360) = (5 × x)
x = (4 × 360) / 5 = 450 km.

7. 16 pumps can empty a pool in 8 hours. How many pumps are needed to empty it in 6 hours?

Fewer hours, more pumps needed (inverse proportion).
8 hours : 6 hours = 16 pumps : x pumps
(8 × 16) = (6 × x)
x = (8 × 16) / 6 = 21 pumps.

8. A certain number of workers can do a job in 10 days. If there were 10 more workers, it could be finished in 8 days. How many workers were there originally?

More workers, fewer days (inverse proportion).
Let original number of workers = x.
x workers : (x + 10) workers = 8 days : 10 days
(8 × x) = (10 × (x + 10))
8x = 10x + 100
2x = 100
x = 50.

9. A pipe can fill a tank in 5 hours. After half the tank is filled, three more similar pipes are opened. What is the total time to fill the tank?

First pipe fills half the tank in 5/2 hours.
Four pipes (including the first) will fill the remaining half 4 times faster.
Time for the remaining half = (5/2) / 4 = 5/8 hours.
Total time = 5/2 + 5/8 = 2.5 hours.

10. If 6 printers can print 240 pages in 20 minutes, how many printers are needed to print 400 pages in 30 minutes?

More pages and more time, more printers needed (both direct proportion).
(240 pages, 20 minutes) : (400 pages, 30 minutes) = 6 printers : x printers
(240 × 20 × 6) = (400 × 30 × x)
x = (240 × 20 × 6) / (400 × 30) = 8 printers.

11. A group of men decided to do a job in 10 days, but 5 of them became absent. If the rest of the group did the job in 12 days, find the original number of men.

Let the original number of men be x.
x men : (x - 5) men = 10 days : 12 days
(x × 10) = ((x - 5) × 12)
10x = 12x - 60
2x = 60
x = 30.

12. A fort has enough food for 45 days for 175 soldiers. If 25 soldiers leave the fort, how long will the food last now?

Fewer soldiers, more days (inverse proportion).
175 soldiers : 150 soldiers = 45 days : x days
(175 × 45) = (150 × x)
x = (175 × 45) / 150 = 52.5 days ≈ 53 days.

13. A factory has resources to operate 40 machines for 8 hours a day for 30 days. If the factory operates 20 machines for 12 hours a day, for how many days will the resources last?

Fewer machines and more hours, more days (inverse and direct proportion).
(40 machines, 8 hours) : (20 machines, 12 hours) = 30 days : x days
(40 × 8 × 30) = (20 × 12 × x)
x = (40 × 8 × 30) / (20 × 12) = 60 days.

14. If 12 men or 18 women can complete a work in 14 days, how many days will 8 men and 16 women take to complete the same work?

12 men = 18 women (work equivalence).
1 man = 1.5 women.
8 men = 8 × 1.5 = 12 women.
Total equivalent workers = 12 women + 16 women = 28 women.
18 women take 14 days, so 28 women will take (18/28) × 14 = 9 days.

15. 40 liters of a mixture contains milk and water in the ratio 7:1. How much water must be added to make the ratio 3:1?

Initial ratio of milk to water = 7:1. Total initial quantity = 40 liters.
Milk = (7/8) * 40 = 35 liters, Water = 40 - 35 = 5 liters.
To make the ratio of milk to water 3:1, new water quantity = 35/3 = 11.67 liters.
Water to be added = 11.67 - 5 = 6.67 liters.

16. A train 120 meters long passes a man, running at 5 km/hr in the same direction in which the train is going, in 10 seconds. The speed of the train is:

Relative speed = Distance / Time = 120/10 m/s = 12 m/s.
Relative speed in km/hr = (12 * 18/5) km/hr = 43.2 km/hr.
Speed of train = Speed of man + Relative speed = 5 + 43.2 = 48.2 km/hr ≈ 50 km/hr.

17. If 15 carpenters working 6 hours a day can make 10 tables in 12 days, how many carpenters working 8 hours a day will be required to make 20 tables in 15 days?

More tables and more days, more carpenters needed (both direct proportion).
(10 tables, 12 days, 6 hours) : (20 tables, 15 days, 8 hours) = 15 carpenters : x carpenters
(10 × 12 × 6 × 15) = (20 × 15 × 8 × x)
x = (10 × 12 × 6 × 15) / (20 × 15 × 8) = 9.

18. A can do a piece of work in 4 days. B can do it in 12 days. With the help of C, they finish the work in 2 days. C alone can do the work in:

A's 1 day work = 1/4, B's 1 day work = 1/12.
(A + B + C)'s 1 day work = 1/2.
C's 1 day work = 1/2 - (1/4 + 1/12) = 1/6.
C alone can do the work in 6 days.

19. A garrison of 2000 men has provisions for 54 days. At the end of 18 days, a reinforcement arrives, and it is found that now the provisions will last only for 20 days more. What is the number of men in the reinforcement?

For 2000 men, provisions for 54 days. After 18 days, provisions left for 36 days.
Let reinforcement be x men. Now, total men = 2000 + x.
2000 men: (2000 + x) men = 36 days : 20 days
2000/36 = (2000 + x)/20
x = 2500.

20. If 9 engines consume 24 metric tonnes of coal when each is running 16 hours a day, how much coal will be required for 8 engines, each running 13 hours a day?

Less engines and less hours, less coal (both inverse proportion).
(9 engines, 16 hours) : (8 engines, 13 hours) = 24 tonnes : x tonnes
(9 × 16 × 24) = (8 × 13 × x)
x = (9 × 16 × 24) / (8 × 13) = 20.77 ≈ 20 tonnes.