The Chain Rule is a method used to solve problems involving two or more quantities that are related to each other through a series of proportional relationships.
It is especially useful when several factors such as workers, days, hours, machines, wages, quantity, speed or time change at the same time.
Direct Proportion → quantities change in the same direction.
Inverse Proportion → quantities change in opposite directions.
Suppose a known quantity is related to several other quantities. We start with the known situation and change each quantity one at a time until we reach the required situation.
Workers increase from 5 to 10.
Therefore, days decrease.
5 × 12 = 10 × x
x = 6 days
Two quantities are in direct proportion when an increase in one causes a proportional increase in the other.
x/y = constant
Therefore:
x₁/y₁ = x₂/y₂
More books → more cost.
Therefore, it is direct proportion.
4/200 = 7/x
4x = 1400
x = ₹350
Two quantities are in inverse proportion when an increase in one causes a proportional decrease in the other.
xy = constant
Therefore:
x₁y₁ = x₂y₂
More workers → fewer days.
Therefore, it is inverse proportion.
8 × 15 = 12 × x
120 = 12x
x = 10 days
The chain rule can often be understood through the unitary method. First find the value corresponding to one unit and then calculate the required value.
More machines → more production.
Production = 900 × 8/6
= 1200 units
For the same amount of work and equal efficiency:
Thus, men and days are in inverse proportion.
12 × 18 = 9 × x
216 = 9x
x = 24 days
When daily working hours also change, all three factors must be considered.
For the same work and same efficiency:
10 × 8 × 15 = 12 × 10 × x
1200 = 120x
x = 10 days
For equal working time and efficiency:
More workers → more production.
Required production = 240 × 15/8
= 450 units
If the number of workers and efficiency remain constant:
More days → more work.
Work = 300 × 16/10
= 480 units
If workers have different efficiencies, the more efficient worker takes less time for the same work.
Efficiency × Time = Constant
A's efficiency : B's efficiency = 2 : 1
Time ratio = 1 : 2
A's time = 20 × 1/2
= 10 days
M₁ × D₁ × H₁ = M₂ × D₂ × H₂
If machines work at the same rate, production is directly proportional to the number of machines and the working time.
Production ratio
= (10/6) × (12/8)
Required production
= 4800 × 10/6 × 12/8
= 12,000 items
When workers receive equal wages per day and work for the same number of hours:
Required wages
= 27,000 × 20/15 × 18/12
= ₹54,000
Speed = Distance / Time
Time = Distance / Speed
For a fixed distance, speed and time are inversely proportional.
50 × 6 = 75 × x
x = 4 hours
For the same quantity of food and equal consumption per person:
20 × 15 = 25 × x
300 = 25x
x = 12 days
If people consume different quantities of food per day, consumption must also be included.
10 × 20 × 2 = 16 × x × 1.5
400 = 24x
x = 16⅔ days
When several factors change simultaneously, multiply the corresponding ratios. For every factor, decide whether it is direct or inverse.
Days are inversely proportional to workers and hours.
Required days = 15 × 8/12 × 6/8
= 7.5 days
Required Value = Known Value × Product of Direct Ratios ÷ Product of Inverse Ratios.
| Situation | Relationship / Formula |
|---|---|
| Direct Proportion | x/y = constant |
| Inverse Proportion | xy = constant |
| Work | Workers × Days × Hours × Efficiency |
| Equal Efficiency | M₁D₁H₁ = M₂D₂H₂ |
| Production | Machines × Time × Efficiency |
| Wages | Workers × Days × Daily Wage |
| Distance | Speed × Time |
| Fixed Distance | Speed × Time = Constant |
| Food | Persons × Days × Consumption |
| Efficiency | Time ∝ 1/Efficiency |
8 × 24 = x × 16
x = 192/16
= 12 men
Required = 2000 × 8/5 × 6/4
= 4800 units
60 × 8 = 80 × x
x = 6 hours
12 × 7 × 20 = 15 × 8 × x
x = 1680/120
= 14 days
- 10 workers complete a job in 18 days. How many days will 15 workers take?
- 8 workers working 6 hours a day complete a job in 20 days. How many days will 12 workers working 8 hours a day take?
- 6 machines produce 3600 articles in 5 hours. How many articles will 10 machines produce in 8 hours?
- A car travels 300 km in 5 hours. How long will it take to travel the same distance at 75 km/h?
- Food is sufficient for 24 people for 15 days. How long will it last for 30 people?
- 15 men earn ₹18,000 in 12 days. How much will 20 men earn in 15 days at the same rate?
- 12 workers working 8 hours a day complete a job in 15 days. How many workers are required if they work 10 hours a day and must complete the job in 12 days?
- A machine produces 500 units in 4 hours. How many units will it produce in 9 hours at the same rate?
- A train covers a distance in 6 hours at 50 km/h. Find the time needed at 75 km/h.
- 10 people consume 30 kg of food in 5 days. How much food will 16 people consume in 8 days at the same rate?
2. 10 days
3. 9600 articles
4. 4 hours
5. 12 days
6. ₹30,000
7. 15 workers
8. 1125 units
9. 4 hours
10. 76.8 kg
1. Write the known values.
2. Write the required value as x.
3. Compare every changing quantity with the required quantity.
4. Decide Direct or Inverse.
5. Multiply the ratios.
6. Simplify and calculate x.
7. Check whether the answer logically increases or decreases.
