Q1. A can complete a work in 10 days. How much of the work will he complete in 4 days?
Concept: The work done is directly proportional to the time taken if rate is constant.
Formula: Work done = Time taken ÷ Total time required
Substitution: 4 ÷ 10 = 0.4
Calculation: Simplify → 2 ÷ 5 = 2/5
Interpretation: A completes two-fifths of the total work in 4 days.
Final Answer: 2/5 of the work
Q2. A can do a piece of work in 12 days, and B can do it in 8 days. How long will they take to finish it together?
Concept: When two people work together, their efficiencies add up.
Formula: 1/Together = 1/A + 1/B
Substitution: 1/T = 1/12 + 1/8
Finding LCM: LCM = 24 → (2 + 3)/24 = 5/24
Calculation: T = 24/5 = 4.8 days = 4⅘ days
Final Answer: 4⅘ days
Q3. A and B together can finish a work in 15 days. A alone can do it in 25 days. In how many days can B alone do it?
Concept: Subtract A’s rate from the combined rate to find B’s rate.
Formula: 1/B = 1/Together − 1/A
Substitution: 1/B = 1/15 − 1/25
LCM: 75 → (5−3)/75 = 2/75
Inverse: B = 75/2 = 37.5 days
Final Answer: 37½ days
Q4. A can do a job in 20 days, B in 30 days. They work together for 5 days, then A leaves. How long will B take to finish the remaining work?
Step 1: A’s rate = 1/20, B’s rate = 1/30
Step 2: Combined rate = 1/20 + 1/30 = 1/12
Step 3: Work done in 5 days = 5 × 1/12 = 5/12
Step 4: Remaining work = 1 − 5/12 = 7/12
Step 5: B’s time = (7/12) ÷ (1/30) = 17.5 days
Final Answer: 17½ days
Q5. A is twice as efficient as B. Together they can complete a work in 12 days. In how many days can A alone finish it?
Concept: Efficiency ∝ 1/Time.
Step 1: If A:B = 2:1 ⇒ Time ratio = 1:2
Step 2: 1/A + 1/(2A) = 1/12 → 3/2A = 1/12
Step 3: A = 18 days
Final Answer: 18 days
Q6. A can do a piece of work in 15 days, B in 20 days. With the help of C, they finish it in 6 days. How long would C take to do it alone?
Step 1: Combined rate = 1/6
Step 2: A+B = 1/15 + 1/20 = 7/60
Step 3: C = 1/6 − 7/60 = 3/60 = 1/20
Final Answer: 20 days
Q7. A and B together can do a work in 18 days. B alone can do it in 27 days. In how many days can A alone do it?
Formula: 1/A = 1/Together − 1/B
Substitution: 1/18 − 1/27 = (3−2)/54 = 1/54
Final Answer: 54 days
Q8. A, B, and C can complete a job in 12, 15, and 20 days respectively. How long will they take together?
Formula: 1/T = 1/12 + 1/15 + 1/20
LCM: 60 → (5+4+3)/60 = 12/60 = 1/5
Final Answer: 5 days
Q9. A can do a work in 9 days and B in 18 days. A works for 3 days alone, then B joins. How many more days will they take?
A’s work in 3 days: 3 × 1/9 = 1/3
Remaining: 2/3
Together’s rate: 1/9 + 1/18 = 1/6
Time: (2/3) ÷ (1/6) = 4 days
Final Answer: 4 days
Q10. A can do a work in 8 days, B in 12 days. With the help of C, they complete it in 4 days. Find C’s efficiency.
Formula: 1/C = 1/Together − (1/A + 1/B)
Calculation: 1/4 − (1/8 + 1/12) = 1/4 − 5/24 = 1/24
Final Answer: C alone → 24 days
Q11. A is 50% more efficient than B. If A can finish a work in 18 days, find how many days B will take.
Efficiency ratio: A:B = 3:2 → Time ratio = 2:3
Substitution: A = 18 → B = (3/2)×18 = 27
Final Answer: 27 days
Q12. A can complete a work in 5 days less than B. Together they do it in 6 days. Find their individual times.
Formula: 1/A + 1/(A+5) = 1/6
Simplify: (2A+5)/A(A+5) = 1/6 → 6(2A+5)=A²+5A
Solve: A²−7A−30=0 → A=10, B=15
Final Answer: A=10 days, B=15 days
Q13. A does ¼ of work in 5 days. How many days will he take to complete the whole work?
Formula: Total = (Given × Total work) / Work done
Substitution: (5 × 1) / (¼) = 20
Final Answer: 20 days
Q14. A can finish a job in 16 days, B in 24 days. They work together for 4 days, then A leaves. How many more days will B need?
A+B/day = 1/16 + 1/24 = 5/48 → 4 days = 5/12 done
Remaining = 7/12 → B’s rate = 1/24
Time = (7/12) ÷ (1/24) = 14 days
Final Answer: 14 days
Q15. If A and B together can finish a work in 10 days, and A alone in 15 days, how many days for B alone?
1/B = 1/10 − 1/15 = (3−2)/30 = 1/30
Final Answer: 30 days
Q16. A can complete a work in 25 days. He works for 5 days and leaves. B completes remaining in 10 days. Find B’s time alone.
A’s work = 5 × 1/25 = 1/5
Remaining = 4/5
B’s rate = (4/5) ÷ 10 = 2/25
Final Answer: 12½ days
Q17. 4 men can complete a work in 12 days. How many men are required to finish it in 8 days?
M₁D₁ = M₂D₂ → 4×12 = M₂×8 → M₂=6
Final Answer: 6 men
Q18. If 10 men can finish a work in 15 days, how many days will 6 men take?
M₁D₁ = M₂D₂ → 10×15 = 6×D₂ → D₂=25
Final Answer: 25 days
Q19. A pipe fills a tank in 8 hours and another in 12 hours. How long will both take together?
1/T = 1/8 + 1/12 = 5/24 → T = 24/5 = 4⅘ hr
Final Answer: 4⅘ hours
Q20. One pipe fills a tank in 6 hours and another empties it in 9 hours. In how much time will the tank be full?
1/T = 1/6 − 1/9 = 1/18
Final Answer: 18 hours
Q21. Two pipes fill a tank in 15 and 20 hours. A third pipe empties it in 30 hours. How long to fill the tank when all are opened together?
1/T = 1/15 + 1/20 − 1/30 = (4+3−2)/60 = 5/60 → T=12 hr
Final Answer: 12 hours
Q22. A and B together can do a piece of work in 4 days. A alone can do it in 12 days. How long will B take to finish it alone?
1/B = 1/4 − 1/12 = (3−1)/12 = 1/6
Final Answer: 6 days
Q23. A can finish a job in 9 days, B in 18 days. If they work on alternate days starting with A, find total days to complete the job.
2 days work = 1/9 + 1/18 = 1/6
6 days → 3 pairs = 3/6 = ½ work
12 days → full work
Final Answer: 12 days
Q24. A can do a work in 15 days and B in 10 days. They work together for 2 days, then A leaves. Find total time taken.
Work in 2 days = 2(1/15+1/10)=2(1/6)=1/3
Remaining = 2/3 → B alone = (2/3)÷(1/10)=20/3=6⅔ days
Total = 2+6⅔=8⅔ days
Final Answer: 8⅔ days
Q25. A and B can finish a job in 9 days. A alone can do it in 15 days. After working together for 3 days, A leaves. How long will B take?
Work done in 3 days = 3(1/9)=1/3
Remaining=2/3
B’s rate=1/9−1/15=2/45
Time=(2/3)÷(2/45)=15 days
Final Answer: 15 days
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