Question:
A and B together can complete a task in 20 days. B and C together can complete the same task in 30 days. A and C together can complete the same task in 40 days. What is the respective ratio of the number of days taken by A when completing the same task alone to the number of days taken by C when completing the same task alone?
A and B together can complete a task in 20 days. B and C together can complete the same task in 30 days. A and C together can complete the same task in 40 days. What is the respective ratio of the number of days taken by A when completing the same task alone to the number of days taken by C when completing the same task alone?
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Add all three pairwise equations, halve the sum to get A+B+C, then subtract each pair to isolate A and C.
Updated On: Jul 16, 2026
- 2 : 5
- 2 : 7
- 3 : 7
- 1 : 5
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The Correct Option is D
Solution and Explanation
Step 1: Take the total work as the LCM of the given days.
Let the total work be \(LCM(20,30,40) = 120\) units.
Combined rate of A and B \( = \dfrac{120}{20} = 6\) units/day.
Combined rate of B and C \( = \dfrac{120}{30} = 4\) units/day.
Combined rate of A and C \( = \dfrac{120}{40} = 3\) units/day.
Step 2: Add all three equations to find A + B + C.
\(2(A+B+C) = 6 + 4 + 3 = 13 \Rightarrow A+B+C = 6.5\) units/day.
Step 3: Subtract each pair's rate from the total to isolate A and C individually.
\(A = 6.5 - (B+C) = 6.5 - 4 = 2.5\) units/day.
\(C = 6.5 - (A+B) = 6.5 - 6 = 0.5\) units/day.
(Check: \(B = 6.5 - 3 = 3.5\), and \(A+B = 2.5+3.5=6\), which matches Step 1.)
Final Answer:
Days taken by A alone \( = \dfrac{120}{2.5} = 48\), days taken by C alone \( = \dfrac{120}{0.5} = 240\).
Ratio \( = 48:240 = 1:5\), so option D is correct. \[ \boxed{1:5} \]
Let the total work be \(LCM(20,30,40) = 120\) units.
Combined rate of A and B \( = \dfrac{120}{20} = 6\) units/day.
Combined rate of B and C \( = \dfrac{120}{30} = 4\) units/day.
Combined rate of A and C \( = \dfrac{120}{40} = 3\) units/day.
Step 2: Add all three equations to find A + B + C.
\(2(A+B+C) = 6 + 4 + 3 = 13 \Rightarrow A+B+C = 6.5\) units/day.
Step 3: Subtract each pair's rate from the total to isolate A and C individually.
\(A = 6.5 - (B+C) = 6.5 - 4 = 2.5\) units/day.
\(C = 6.5 - (A+B) = 6.5 - 6 = 0.5\) units/day.
(Check: \(B = 6.5 - 3 = 3.5\), and \(A+B = 2.5+3.5=6\), which matches Step 1.)
Final Answer:
Days taken by A alone \( = \dfrac{120}{2.5} = 48\), days taken by C alone \( = \dfrac{120}{0.5} = 240\).
Ratio \( = 48:240 = 1:5\), so option D is correct. \[ \boxed{1:5} \]
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