A and B can complete a work in 12 days, B and C in 15 days, an... | Mathematics MCQ | Examoogle

MathematicsTime and Work

A and B can complete a work in 12 days, B and C in 15 days, and C and A in 20 days. In how many days can A alone complete the work?

$20 \text{ days}$

$30 \text{ days}$

$40 \text{ days}$

$60 \text{ days}$

Correct Answer

📐 MA • Math LCM MethodMathematicsTime and Work

Option B

$30 \text{ days}$

Quick Summary:

Use the LCM of 12, 15, and 20 (which is 60) as the total work units. Then, find total efficiency 2(A+B+C) = 5+4+3 = 12, so A+B+C = 6. A's efficiency = (A+B+C) - (B+C) = 6 - 4 = 2.

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📐MAMath SolutionLCM Method

📋 Given

Combined work rates: (A+B) = 12 days, (B+C) = 15 days, (C+A) = 20 days.

🔢 Formula Used

2(A+B+C)=∑Efficiencies=112+115+1202(A + B + C) = \sum \text{Efficiencies} = \frac{1}{12} + \frac{1}{15} + \frac{1}{20}2(A+B+C)=∑Efficiencies=121​+151​+201​

⚡ Exam Hall Shortcut / Speed Trick

Use the LCM of 12, 15, and 20 (which is 60) as the total work units. Then, find total efficiency 2(A+B+C) = 5+4+3 = 12, so A+B+C = 6. A's efficiency = (A+B+C) - (B+C) = 6 - 4 = 2.

⚠️ Common Student Trap / Pitfall

Students often forget to divide the sum of efficiencies by 2, leading to the sum for two pairs instead of one set of (A+B+C).

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Calculate LCM of work days

Assume total work as the LCM of 12, 15, and 20.

LCM(12,15,20)=60 unitsLCM(12, 15, 20) = 60 \text{ units}LCM(12,15,20)=60 units

Calculate daily work rates

Determine the efficiency per day for each pair.

(A+B)=6012=5,(B+C)=6015=4,(C+A)=6020=3(A+B) = \frac{60}{12} = 5, (B+C) = \frac{60}{15} = 4, (C+A) = \frac{60}{20} = 3(A+B)=1260​=5,(B+C)=1560​=4,(C+A)=2060​=3

Calculate total efficiency

Add the efficiencies and divide by 2 to get $A+B+C$ .

2(A+B+C)=5+4+3=12  ⟹  (A+B+C)=62(A+B+C) = 5+4+3 = 12 \implies (A+B+C) = 62(A+B+C)=5+4+3=12⟹(A+B+C)=6

Find efficiency of A

Subtract the efficiency of $(B+C)$ from the total efficiency of $(A+B+C)$ .

EfficiencyA=(A+B+C)−(B+C)=6−4=2Efficiency_A = (A+B+C) - (B+C) = 6 - 4 = 2EfficiencyA​=(A+B+C)−(B+C)=6−4=2

Calculate days for A

Divide total work by A's efficiency.

DaysA=602=30Days_A = \frac{60}{2} = 30DaysA​=260​=30

✅

B is correct because A's daily efficiency is 2 units, requiring 30 days to complete 60 units of work.

Core Concepts Used

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Work-Efficiency Relationship LCM Method for Time and Work Simultaneous Equation Systems

💡 EXAM TIP

This concept is identical to solving pipe and cistern problems where A, B, and C represent inflow rates, often encountered in fluid mechanics or pipe flow questions in competitive engineering exams.

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