Time and Work: LCM Method, Efficiency, Partial Work and Alternate Days
Time and Work: LCM Method, Efficiency, Partial Work and Alternate Days
Time and work questions become simple integer arithmetic with the LCM method: take the total work as the LCM of the given days, find each person's work per day, then add, subtract or divide. NTPC asks two or three people together, pair sums, someone leaving midway, efficiency ratios and alternate-day working.
1. The LCM Method
Total work = LCM of the given days Efficiency (work per day) = total work ÷ days
Time = work ÷ (sum of efficiencies working together)
| Person | Days alone | Work per day (total 36) |
|---|---|---|
| A | 12 | 3 |
| B | 18 | 2 |
| A + B | 36/5 = 7.2 days | 5 |
Fraction method (same answer): A does 1/12 per day, B 1/18, together 5/36 per day.
2. Standard Results
| Situation | Result |
|---|---|
| A in x days, B in y days | Together xy/(x + y) days |
| A, B, C in x, y, z days | Together xyz/(xy + yz + zx) days |
| A + B, B + C, C + A in p, q, r days | 2(A + B + C) = sum of the three pair efficiencies |
| A is k times as efficient as B | A takes 1/k of B's time; efficiencies k : 1 |
| A is r% more efficient than B | Efficiencies (100 + r) : 100; times 100 : (100 + r) |
| Wages | Shared in the ratio of work done (efficiency × days worked) |
| Men, women, children | Convert all to one unit (1 man = k women) using the given equal work |
| Chain rule | M₁D₁H₁/W₁ = M₂D₂H₂/W₂ |
✗ A takes 10 days and B 15 days, so together 12.5 days | ✓ Times never average; add efficiencies: 3 + 2 = 5 units of 30, so 6 days
✗ Alternate days, so the time is double the together time | ✓ Count complete cycles, then finish the leftover with whoever's turn it is; with 16 and 12 days, B starting, the answer is 13 2/3 days, not 2 × 48/7 = 13 5/7
हिंदी नोट: समय और कार्य में कुल कार्य को दिनों का ल.स. मानिए, फिर हर व्यक्ति का एक दिन का काम निकालिए। दिन कभी नहीं जोड़े जाते, एक दिन के काम (क्षमता) जोड़े जाते हैं।
Exam Pointer: Verified NTPC patterns: two or three people together (January and April 2021), pair sums (April 2016, June 2022 CBT-2), one person leaving after some days (June 2025 Graduate CBT-1, June 2026 UG CBT-1), efficiency ratios including the March 2026 Graduate CBT-1 "B is 75% more efficient" type, wages (January 2021), men-women equivalence and the chain rule (April 2016, January 2021), alternate days (June 2025 Graduate CBT-1) and workers leaving or joining midway (January 2021, June 2025 Graduate CBT-1).
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Two or three people working together
[PYQ: NTPC CBT-1 8-Apr-2021 Shift-1 | NTPC CBT-1 18-Jan-2021 Shift-1]
EXAM LEVEL
Q. A can finish a work in 12 days and B in 18 days. In how many days can they finish it together?
Total work = LCM(12, 18) = 36 units. A does 3 per day, B 2 per day, together 5. Time = 36/5 = 7.2 days (7 1/5 days).
Answer: 7 1/5 days
EXAMATLAS LEVEL
Q. A, B and C can finish a work in 10, 15 and 30 days respectively. They start together, but C leaves after 2 days. In how many days is the work finished?
Total = 30 units; A 3, B 2, C 1 per day. In 2 days all three do 12 units. Remaining 18 units at A + B = 5 per day take 3.6 days. Total = 5.6 days (5 3/5 days). If all three had stayed, it would take 30/6 = 5 days.
Answer: 5 3/5 days
Pattern 2: Pair sums: A + B, B + C, C + A given
[PYQ: NTPC CBT-1 22-Apr-2016 Shift-3 | NTPC CBT-2 16-Jun-2022 Shift-2]
EXAM LEVEL
Q. A and B can do a work in 10 days, B and C in 15 days, and C and A in 12 days. How long will A, B and C take together?
Total = LCM(10, 15, 12) = 60. Pair efficiencies: 6, 4, 5, total 15 = 2(A + B + C). So A + B + C = 7.5 per day and the time = 60/7.5 = 8 days.
Answer: 8 days
EXAMATLAS LEVEL
Q. A and B can do a work in 12 days, B and C in 15 days, and C and A in 20 days. How long would each take alone?
Total = 60. Pairs: 5, 4, 3, total 12, so A + B + C = 6 per day. A = 6 − (B + C) = 6 − 4 = 2 → 30 days; B = 6 − 3 = 3 → 20 days; C = 6 − 5 = 1 → 60 days.
Answer: A 30, B 20, C 60 days
Pattern 3: Someone works for some days and then leaves or joins
[PYQ: NTPC UG CBT-1 19-Jun-2026 Shift-3 | NTPC Graduate CBT-1 12-Jun-2025 Shift-3]
EXAM LEVEL
Q. A can do a work in 15 days and B in 10 days. They work together for 4 days, then A leaves. In how many days will B finish the remaining work?
Total = 30; A 2, B 3 per day. In 4 days they do 20 units. B finishes the remaining 10 units in 10/3 = 3 1/3 days.
Answer: 3 1/3 days
EXAMATLAS LEVEL
Q. A, B and C can do a work in 20, 30 and 60 days. All three start together; A leaves after 4 days and B leaves 2 days later. How many days does the whole work take?
Total = 60; A 3, B 2, C 1 per day. First 4 days: 6 × 4 = 24 units. Next 2 days (B and C): 3 × 2 = 6 units, total 30. C alone does the remaining 30 units in 30 days. Total time = 4 + 2 + 30 = 36 days.
Answer: 36 days
Pattern 4: Efficiency ratios
[PYQ: NTPC CBT-2 17-Jun-2022 Shift-1 | NTPC CBT-1 30-Dec-2020 Shift-2 | NTPC Graduate CBT-1 19-Mar-2026 Shift-1]
EXAM LEVEL
Q. A is twice as efficient as B, and together they finish a work in 12 days. In how many days can B finish it alone?
Efficiencies A : B = 2 : 1, together 3 units per day for 12 days = 36 units. B alone = 36/1 = 36 days (A alone 18 days).
Answer: 36 days
EXAMATLAS LEVEL
Q. A is 50% more efficient than B, so B takes 10 days more than A to finish a work. In how many days can they finish it together?
Efficiencies 3 : 2, so times are 2 : 3. The difference of 1 part = 10 days, so A takes 20 days and B 30 days. Together: 20 × 30/50 = 12 days.
Answer: 12 days
Pattern 5: Wages shared by work done
[PYQ: NTPC CBT-1 8-Jan-2021 Shift-2]
EXAM LEVEL
Q. A can do a job in 6 days and B in 8 days. Together they finish it and are paid ₹2,800. Find each share.
Efficiencies (total 24): A 4, B 3. Wages in the ratio 4 : 3: A ₹1,600, B ₹1,200.
Answer: A ₹1,600, B ₹1,200
EXAMATLAS LEVEL
Q. A alone can do a job in 12 days and B alone in 15 days. With the help of C they finish it in 5 days, and the payment is ₹5,400. Find C's share.
In 5 days A does 5/12 and B does 5/15 = 4/12 of the work, so C does 1 − 9/12 = 3/12 = 1/4. C's share = 5,400 × 1/4 = ₹1,350.
Answer: ₹1,350
Pattern 6: Men, women and the chain rule
[PYQ: NTPC CBT-1 2-Apr-2016 Shift-2 | NTPC CBT-1 9-Jan-2021 Shift-2]
EXAM LEVEL
Q. 30 men working 6 hours a day can do a work in 16 days. In how many days will 24 men working 8 hours a day do it?
30 × 6 × 16 = 24 × 8 × D, so D = 2,880/192 = 15 days.
Answer: 15 days
EXAMATLAS LEVEL
Q. 4 men or 6 women can do a work in 18 days. How long will 4 men and 3 women take?
4 men = 6 women, so 1 man = 1.5 women. Work = 6 × 18 = 108 woman-days. 4 men and 3 women = 6 + 3 = 9 women, so the time = 108/9 = 12 days.
Answer: 12 days
Pattern 7: Working on alternate days
[PYQ: NTPC Graduate CBT-1 13-Jun-2025 Shift-1]
EXAM LEVEL
Q. A can do a work in 10 days and B in 15 days. They work on alternate days, A starting. In how many days is the work finished?
Total = 30; A 3, B 2. A 2-day cycle does 5 units, so 6 cycles = 12 days complete exactly 30 units.
Answer: 12 days
EXAMATLAS LEVEL
Q. A can do a work in 16 days and B in 12 days. They work on alternate days, B starting. In how many days is the work finished?
Total = 48; A 3, B 4. One 2-day cycle = 7 units. 6 cycles (12 days) = 42 units, leaving 6. Day 13 is B's: 4 units, leaving 2. Day 14 is A's: 2 units take 2/3 of a day. Total = 13 2/3 days.
Answer: 13 2/3 days
Pattern 8: Workers leave or join midway
[PYQ: NTPC Graduate CBT-1 9-Jun-2025 Shift-2 | NTPC CBT-1 9-Jan-2021 Shift-1]
EXAM LEVEL
Q. 12 men can finish a work in 20 days. After 8 days, 4 men leave. In how many more days will the rest finish the work?
Work = 240 man-days. In 8 days 96 man-days are done, leaving 144. 8 men take 144/8 = 18 more days.
Answer: 18 days
EXAMATLAS LEVEL
Q. 10 men or 15 women can do a job in 24 days. 6 men and 6 women work for 8 days; then the women leave and 4 more men join. In how many more days is the job finished?
10 men = 15 women, so 1 woman = 2/3 man. Work = 240 man-days. 6 men + 6 women = 6 + 4 = 10 man-equivalents, doing 80 man-days in 8 days. Remaining 160 man-days with 10 men take 16 days.
Answer: 16 days
60-Second Revision
- Total work = LCM of days; add efficiencies, never days.
- Pair sums: add all three pairs and halve for A + B + C.
- r% more efficient: efficiencies (100 + r) : 100, times reversed.
- Wages follow work done; convert men, women, children to one unit.
- Alternate days: complete cycles first, then finish with whoever's turn it is.