Question: A and B can complete a work in 12 days, B and C can complete in 20 days, C and A in 15 days.
(i) In how many days A, B and C together can complete the work?
(ii) In how many days A alone can complete the work?
(iii) In how many days B alone can complete the work?
(iv) In how many days C alone can complete the work?
Answer and Explanation:
Let’s solve the problem step by step using the LCM method.
Step 1: Calculate Total Work
Given:
- A + B can complete the work in 12 days.
- B + C can complete the work in 20 days.
- C + A can complete the work in 15 days.
The LCM of 12, 20, 15 is 60 units.
So, the total work is 60 units.
Step 2: Calculate the combined work rates
A + B’s efficiency = \displaystyle \frac{60}{12} ​= 5 units/day
B + C’s efficiency = \displaystyle \frac{60}{20} = 3 units/day
C + A’s efficiency = \displaystyle \frac{60}{15} ​= 4 units/day
Step 3: Find A+B+C’s 1-day work
We know:
(A + B)+(B + C)+(C + A) = 2(A + B + C)
Substitute the values:
5 + 3 + 4 = 2(A + B + C)
A + B + C = \displaystyle \frac{12}{2} = 6 units/day.
Step 4: Solve the questions
(i) In how many days A, B, and C together can complete the work?
Time taken = \displaystyle \frac{\text{Total work}}{\text{Work rate of A + B + C}} = \displaystyle \frac{60}{6} = 10 days.
(ii) In how many days A alone can complete the work?
From the equation A + B + C = 6, subtract B + C’s work rate:
A = (A + B + C) – (B + C) = 6 – 3 = 3 units/day.
Time taken by A = \displaystyle \frac{\text{Total work}}{\text{Work rate of A}} = \displaystyle \frac{60}{3} = 20 days.
(iii) In how many days B alone can complete the work?
From the equation A + B + C = 6, subtract C + A’s work rate:
B = (A + B + C) – (C + A) = 6 – 4 = 2 units/day.
Time taken by B = \displaystyle \frac{\text{Total work}}{\text{Work rate of B}} =\displaystyle \frac{60}{2} = 30 days.
(iv) In how many days C alone can complete the work?
From the equation A + B + C = 6, subtract A + B’s work rate:
C = (A + B + C) – (A + B) = 6 – 5 = 1 unit/day.
Time taken by C = \displaystyle \frac{\text{Total work}}{\text{Work rate of C}} =\displaystyle \frac{60}{1} = 60 days.
Final Answers:
(i) A, B, and C together can complete the work in 10 days.
(ii) A alone can complete the work in 20 days.
(iii) B alone can complete the work in 30 days.
(iv) C alone can complete the work in 60 days.
Similar Problems
- A and B can do a work in 3 days, B and C can do it in 4 days, A and C can do it 6 days. How long will it take to A to do it alone? [Ans: 8 days]
- A and B can do a work in 8 days, B and C can do it in 24 days, C and A can do it in \displaystyle 8\frac{4}{7} days. In how many days C alone can do the whole work? [Ans: 60 days]
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