Time & Work — Advanced Exam Problems

🎯 Learning Objective

Master phase-wise counting — workers leaving mid-way, alternate-day duty and mixed men-women teams.

💡 Concept

• Phase problems: someone joins or leaves mid-way — break the timeline into phases and count units in each
• Alternate-day work: A and B form a two-day cycle; finish full cycles, then the leftover goes to whoever's turn it is
• 'Worked x days then left' → the remaining fraction and remaining days fix the finisher's rate
• Mixed workforce (men + women): equate the total work of two teams to link the two unit rates
• RRB twist: the LAST unit is done at ONE person's rate — fraction-of-a-day answers are normal

🧮 Key Formulas

Alternate days: cycle work = (a + b) units per 2 days

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Phase method: work done = rate × days, phase by phase

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Team equality: D₁ × (team₁ rate) = D₂ × (team₂ rate) = total work

✏️ Easy Example

Q. A can do a work in 12 days and B in 18 days. They start together but A leaves after 4 days. In how many more days will B finish the remaining work?

1. Total work = LCM(12, 18) = 36 units — LCM keeps every rate a whole number
2. A = 3 units/day, B = 2 units/day; together 5 units/day
3. In 4 days together they finish 5 × 4 = 20 units, leaving 36 − 20 = 16
4. B alone at 2 units/day needs 16/2 = 8 days

Answer: 8 days

🇮🇳 Real-Life Example

On construction sites labour changes daily — contractors mentally re-run this phase maths every time a mason leaves early.

📝 Exam-Level Example

Q. A can do a piece of work in 9 days and B in 12 days. They work on alternate days with A starting. In how many days will the work be completed?

1. Work = LCM(9, 12) = 36 units; rates A = 4, B = 3 — alternate-day questions demand unit rates
2. One full A+B cycle (2 days) = 4 + 3 = 7 units; never average alternate work, count in cycles
3. 5 cycles = 10 days = 35 units — stop at the largest total BELOW 36, never overshoot
4. 1 unit is left; day 11 belongs to A (he started, so odd days are his), and A does 4 units in a full day
5. A needs 1/4 of a day for 1 unit → total time = 10 + 1/4 = 10¼ days

Answer: 10¼ days (10.25 days)

📝 Exam-Level Example

Q. A and B together can finish a work in 30 days. They worked together for 20 days, then B left. A finished the remaining work in 20 more days. In how many days can A alone do the whole work?

1. Together they do 1/30 of the work per day, so in 20 days they finish 20/30 = 2/3 — track the finished fraction first
2. Remaining work = 1 − 2/3 = 1/3
3. A alone did this 1/3 in 20 days — that single fact fixes A's rate
4. A's rate = (1/3) ÷ 20 = 1/60 of the work per day
5. So A alone needs 60 days for the full job

Answer: 60 days

📝 Exam-Level Example

Q. 2 men and 3 women can do a work in 10 days, while 3 men and 2 women can do the same work in 8 days. In how many days will 2 women alone do it?

1. Let 1 man do m units/day and 1 woman w units/day — the same job gives one equation from each team
2. Equal work: 10(2m + 3w) = 8(3m + 2w) → 20m + 30w = 24m + 16w
3. 14w = 4m → m = 3.5w — one man produces as much as 3.5 women here
4. Total work = 10(2m + 3w) = 10(7w + 3w) = 100w units, substituting m = 3.5w
5. 2 women work at 2w per day → 100w ÷ 2w = 50 days

Answer: 50 days

🪄 Memory Trick

In alternate-day problems, clear the largest full-cycle total below the work, then hand the leftover to whoever's turn it is — most wrong answers come from blindly finishing the cycle.

⚠️ Common Mistakes

• ❌ Completing the last cycle fully when the work ends mid-day (counting 42 units when only 36 exist)
• ❌ Using the combined rate for the phase AFTER a worker has left
• ❌ Treating men and women as the same unit before the equation links m and w

🏆 Exam Tips

• ✅ LCM units first, always — every advanced pattern reduces to unit counting
• ✅ Fraction tracking (2/3 done, 1/3 left) works even when individual rates are unknown
• ✅ Write each phase on its own line: who worked, at what rate, for how long

📌 Summary

• Advanced = phases; solve each phase with unit rates
• Alternate days → full cycles first, then the leftover turn
• Someone leaves → split the timeline at that moment
• Two team equations → link m and w, then count units