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A and B can do a piece of work in 30 days and 45 days respectively. They work together for 10 days and after that A goes on leave for 10 days and B works on every alternate day for the next 10 days and then goes on leave for the next 10 days after which A starts the work on every alternate day for the next 10 days. After that A and B together complete the remaining work. Find in how many days the work was completed.

A and B can do a piece of work in 30 days and 45 days respectively. They work together for 10 days and after that A goes on leave for 10 days and B works on every alternate day for the next 10 days and then goes on leave for the next 10 days after which A starts the work on every alternate day for the next 10 days. After that A and B together complete the remaining work. Find in how many days the work was completed. Correct Answer 33 days

Given:

A can do a piece of work = 30 days

B can do a piece of work = 45 days

They work together = 10 days

A goes on leave = 10 days

B works alternatively = 10 days

B goes on leave = 10 days

A works alternatively = 10 days

Concept used:

Total work = LCM

Work done in unit value of time is known as efficiency

Formula used:

Efficiency = total work / total days

 

Calculations:

Let the total work be x

Efficiency of A = total work / total days = x / 30

Efficiency of B = total work / total days = x / 45

∵ They starts the work together

∴ The work done them in one day = (x / 30) + (x / 45) = 5x / 90 = x / 18

∴ A, B work for ten days together so work completed by them = 10 × (x / 18) = 5x / 9

∵ After 10 days A goes on leave and B works on every alternate days for the next 10 days

∴ We can say that B works for only 5 days

∴ Work done by B in 5 days = (x / 45) × 5 = x / 9

∵ After 10 days B goes on leave and A works on every alternate days for the next 10 days

∴ We can say that A works for only 5 days

∴ Work done by A in 5 days = (x / 30) × 5 = x / 6

∴ The total work completed by A and B = 5x / 9 + x / 9 + x / 6 = 15x / 18 = 5x / 6

∵ Remaining work is done by A and B together

∴ Remaining work = x – (5x / 6) = x / 6

∴ Time taken by A and B for completing the remaining work = remaining work / efficiency

⇒ (x / 6) ÷ (x / 18) = 3 days

Hence total time taken by A and B for completing the total work = 10 + 10 + 10 + 3 = 33 days

Alternate method

Total work = LCM = 90

Efficiency of A = total work / total days = 90 / 30 = 3

Efficiency of B = total work / total days = 90 / 45 = 2

∵ They starts the work together

∴ The work done them in one day = 3 + 2 = 5

∴ A, B work for ten days together so work completed by them = 10 × 5 = 50

∵ After 10 days A goes on leave and B works on every alternate days for the next 10 days

∴ We can say that B works for only 5 days

∴ Work done by B in 5 days = 2 × 5 = 10

∵ After 10 days B goes on leave and A works on every alternate days for the next 10 days

∴ We can say that A works for only 5 days

∴ Work done by A in 5 days = 3 × 5 = 15

∴ The total work completed by A and B = 50 + 10 + 15 = 75

∵ Remaining work is done by A and B together

∴ Remaining work = 90 – 75 = 15

∴ Time taken by A and B for completing the remaining work = remaining work / efficiency

⇒ 15 / 5 = 3

Hence total time taken by A and B for completing the total work = 10 + 10 + 10 + 3 = 33 days

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