Shortcut Trick
Efficiency Ratio (Amrit : Bhagat) = 3 : 1
Time Ratio (Amrit : Bhagat) = 1 : 3 (Inversely proportional to efficiency)
Difference in time units = 3 − 1 = 2 units → 50 days
1 unit = 25 days ∴ Amrit = 25 days, Bhagat = 75 days
Together time = (Product of Days) ÷ (Sum of Days) = (25 × 75) ÷ 100 = 75 ÷ 4 = 18 3/4 days
∴ The correct answer is 18 3/4 days.
Alternate Method
Given:
Amrit is thrice as good as Bhagat. Let Bhagat's efficiency be 1 unit/day, then Amrit's efficiency is 3 units/day.
Time difference between Amrit and Bhagat = 50 days.
Formula Used:
Total Work = Efficiency × Time
Efficiency ∝ 1 / Time
| Person | Days | Total Work | Efficiency |
|---|
| Amrit | 25 | 75 | 3 |
| Bhagat | 75 | 1 | |
⇒ Let the time taken by Bhagat be 3x and Amrit be x.
⇒ 3x − x = 50 ⇒ 2x = 50 ⇒ x = 25.
⇒ Amrit finishes in 25 days and Bhagat in 75 days.
⇒ Total work (LCM of 25 and 75) = 75 units.
⇒ Combined efficiency of Amrit and Bhagat = 3 + 1 = 4 units/day.
⇒ Time taken together = 75 ÷ 4 = 18 3/4 days.
∴ The correct answer is 18 3/4 days.
Additional Information
Efficiency-Time Relationship
Efficiency is inversely proportional to the time taken to complete a task. If the efficiency ratio is a:b, the time ratio is b:a.
Combined Work Formula
For two people taking x and y days respectively, the combined time is (x × y) ÷ (x + y).
Work and Wages
Wages are always distributed in the ratio of the work done, which is equivalent to the ratio of their efficiencies if they work for the same duration.