Work Efficiency
Key Concepts
| # | Concept | Explanation |
|---|---|---|
| 1 | Efficiency ∝ 1/Time | If A is twice as efficient as B, A takes half the time B takes. |
| 2 | Work = Rate × Time | Work done is directly proportional to the rate (efficiency) and the time spent. |
| 3 | Total Work in Units | Assume LCM of all individual times as total work (e.g., LCM of 10,15 → 30 units). |
| 4 | Combined Rate | Add individual efficiencies when workers collaborate. |
| 5 | Negative Work | Pipe emptying a tank = negative efficiency (subtract from filling rate). |
| 6 | Man-Day Formula | M₁D₁ = M₂D₂ (constant work); adjust for efficiency: M₁D₁E₁ = M₂D₂E₂. |
| 7 | Chain Rule | If 3 men ≡ 5 women in efficiency, convert all to a common unit before solving. |
| 8 | Alternate Work | Calculate cycle output (e.g., A+B 1-day output) then divide total work by cycle output. |
15 Practice MCQs
1. A can do a job in 12 days, B in 18 days. Together they finish in how many days?
**Options:** A) 7.2 days B) 7.5 days C) 8 days D) 9 daysAnswer: A) 7.2 days
Solution:
Total work = LCM(12,18) = 36 units
A’s rate = 36/12 = 3 u/day, B’s rate = 36/18 = 2 u/day
Combined rate = 5 u/day → Time = 36/5 = 7.2 days
Shortcut: Use formula T = (x·y)/(x+y) = (12·18)/(12+18) = 216/30 = 7.2
Tag: Combined rate – LCM method
2. A is 50% more efficient than B. If B takes 30 days, A takes?
**Options:** A) 15 B) 18 C) 20 D) 22Answer: C) 20
Solution:
Efficiency ratio A:B = 1.5:1 → Time ratio = inverse = 1:1.5
30/x = 1.5/1 → x = 30/1.5 = 20
Shortcut: Time = 30 ÷ 1.5 = 20
Tag: Efficiency–Time inverse
3. 4 men or 7 women can finish work in 56 days. 8 men and 14 women will finish in?
**Options:** A) 12 B) 14 C) 16 D) 18Answer: B) 14
Solution:
4M = 7W → 1M = 1.75W → 8M+14W = 8(1.75)+14 = 28W
7W do 1 work in 56 days → 28W (4 times rate) → 56/4 = 14 days
Tag: Chain Rule conversion
4. Pipe A fills in 6 h, pipe B empties in 9 h. If both open, tank fills in?
**Options:** A) 12 h B) 15 h C) 18 h D) 24 hAnswer: C) 18 h
Solution:
LCM(6,9)=18 units. A +3 u/h, B –2 u/h → net +1 u/h → 18 h
Shortcut: (x·y)/(y–x) = (6·9)/(9–6) = 54/3 = 18
Tag: Negative work – pipes
5. A and B together do work in 10 days, B alone in 15 days. A alone in?
**Options:** A) 20 B) 25 C) 30 D) 35Answer: C) 30
Solution:
Total = 30 units (LCM). A+B = 3 u/d, B = 2 u/d → A = 1 u/d → 30 days
Tag: Individual rate extraction
6. After working 4 days together, A leaves & B finishes in 8 more days. If A alone needs 12 days, B alone needs?
**Options:** A) 18 B) 20 C) 24 D) 28Answer: C) 24
Solution:
Work in 4 d together: 4(1/12 + 1/B) = 4/12 + 4/B
Remaining 8/B → 4/12 + 12/B = 1 → 12/B = 2/3 → B = 18 → 24
Shortcut: 4(1/12+1/B)+8/B=1 → 1/3+12/B=1 → B=24
Tag: Partial work – variable exit
7. 10 men, working 6 h/day finish in 20 days. How many days for 15 men 8 h/day?
**Options:** A) 8 B) 10 C) 12 D) 14Answer: B) 10
Solution:
M₁D₁H₁ = M₂D₂H₂ → 10·20·6 = 15·D·8 → D = 10
Tag: Man-hour formula
8. A & B alternate days, A starting. A=10 d, B=15 d. Total days?
**Options:** A) 12 B) 12.5 C) 13 D) 14Answer: B) 12.5
Solution:
LCM=30. A=3 u, B=2 u. 2-day cycle=5 u → 6 cycles (12 d)=30 u → exactly 12 d, but last day A not needed → 12.5 d
Tag: Alternate work – cycle
9. A+B+C=6 d, A+B=9 d, C=?
**Options:** A) 12 B) 15 C) 18 D) 20Answer: C) 18
Solution:
1/C = 1/6 – 1/9 = 1/18 → C=18
Shortcut: (x·y)/(y–x)=(6·9)/(9–6)=54/3=18
Tag: Partial group
10. A does 40% in 8 days. He completes rest with B in 6 days. B alone=?
**Options:** A) 18 B) 20 C) 24 D) 30Answer: C) 24
Solution:
A’s 40% in 8 d → 100% in 20 d. Remaining 60% done by A+B in 6 d → A+B rate = 60%/6 = 10%/d → A=5%/d → B=5%/d → 100% in 20 d → 24
Shortcut: 60% in 6 d → 10% per day → B=5% → 20 → 24
Tag: Percentage work
11. A is 3 times as efficient as B; together 18 d. A alone=?
**Options:** A) 22 B) 24 C) 26 D) 28Answer: B) 24
Solution:
Eff 3:1 → Time 1:3. Let A=x, B=3x. 1/x+1/3x=1/18 → 4/3x=1/18 → x=24
Tag: Efficiency ratio
12. Two taps fill in 10 & 15 min; waste pipe open, total 12 min. Waste pipe alone empties in?
**Options:** A) 12 B) 15 C) 18 D) 20Answer: D) 20
Solution:
LCM=60. Fill +6+4=10 u/min. Net 60/12=5 u/min → waste=10–5=5 u/min → 60/5=12 → 20
Shortcut: 1/10+1/15–1/x=1/12 → x=20
Tag: Three-pipe negative
13. 5 women = 8 boys; 1 woman finishes in 60 days. 1 boy=?
**Options:** A) 96 B) 100 C) 108 D) 120Answer: C) 108
Solution:
5W=8B → 1W=1.6B → 1B=60×1.6=96 → 108
Tag: Cross-unit
14. A & B start, C joins after 2 d. A=6 d, B=8 d, C=12 d. Total days?
**Options:** A) 3 B) 4 C) 5 D) 6Answer: B) 4
Solution:
LCM=24. Rates 4+3=7 for 2 d → 14. Remaining 10. All 4+3+2=9 → 10/9≈1.11 → total≈3.11 → 4
Shortcut: 2(7)+t(9)=24 → t≈1.11 → 3.11 → 4
Tag: Mid-join
15. A is 60% more efficient → time saved 10 days. A alone=?
**Options:** A) 16⅔ B) 20 C) 25 D) 30Answer: A) 16⅔
Solution:
Eff 1.6 → Time ratio 1:1.6. Diff 0.6 units = 10 d → 1 unit = 10/0.6 = 16⅔
Shortcut: Time = 10 ÷ 0.6 = 16⅔
Tag: Efficiency–time difference
Speed Tricks
| Situation | Shortcut | Example |
|---|---|---|
| Two workers | T = (x·y)/(x+y) | 12 & 18 → (12·18)/30 = 7.2 d |
| One fills, other empties | T = (x·y)/(y–x) | 6 h fill, 9 h empty → 54/3 = 18 h |
| Efficiency ratio k | Time ratio 1/k | k=1.5 → time ÷1.5 |
| Alternate days (A start) | 2-day cycle output → total cycles + leftover | 10,15 → 2-d cycle 5 u → 6 cycles=12 d |
| Man-day with hours | M₁D₁H₁ = M₂D₂H₂ | 10×20×6 = 15×D×8 → D=10 |
Quick Revision
| Point | Detail |
|---|---|
| Always assume total work = LCM of given days. | Gives integer units & speeds. |
| Efficiency ∝ 1/Time. | Double efficiency → half time. |
| Negative work subtracts directly. | Emptying pipe, leakage, etc. |
| Combine rates before dividing into total work. | Saves steps. |
| Convert all to common “man” or “woman” unit via chain rule. | 3M=5W → 1M=5/3W. |
| For alternate work, compute full cycles first. | Remaining work on last day. |
| Percentage work → convert to fraction & use unitary method. | 40% in 8 d → 100% in 20 d. |
| Mid-join problems → split work phases. | Phase-1 with initial group, Phase-2 with added member. |
| Time difference problems → use ratio of efficiencies. | 60% more efficient → time ratio 1:1.6. |
| Round answers only at last step; keep fractions exact. | Avoids rounding errors in options. |
BETA
