Three boys Alpha, Beta and Gama planned to make a sports car. Alpha and Beta can finish the task in 18 days; Beta and Gama can do it in 24 days while Gama and Alpha can finish it in 36 days. In how many days will each one of them finish it?

Three boys Alpha, Beta and Gama planned to make a sports car. Alpha and Beta can finish the task in 18 days; Beta and Gama can do it in 24 days while Gama and Alpha can finish it in 36 days. In how many days will each one of them finish it? Correct Answer Alpha = 48 days; Beta = 144/5 days; Gama = 144 days

Given:

Alpha and Beta can finish the task in 18 days; Beta and Gama can do it in 24 days while Gama and Alpha can finish it in 36 days.

Calculation:

Time taken by (Alpha + Beta) to finish the work = 18 days

So, (Alpha + Beta)’s 1 day work = 1/18     ----(1)

Time taken by (Beta + Gama) to finish the task = 24 days

So, (Beta = Gama)’s 1 day work = 1/24     ----(2)

Time taken by (Gama + Alpha) to finish the work = 36 days

So, (Gama + Beta)’s 1 day work = 1/36      ----(3)

On adding eq. (1), (2) and (3), we have

2(Alpha + Beta + Gama)’s 1 day work = 1/18 + 1/24 + 1/36 = (4 + 3 + 2)/72 = 9/72 = 1/8

Also, (Alpha + Beta + Gama)’s 1 day work = 1/(2 × 8) = 1/16

So, Alpha, Beta, Gama together can make a sports car in 16 days.

Now, Alpha’s 1 day’s work = {(Alpha + Beta + Gama)’s 1 day’s work – (Beta + Gama)’s 1 day’s work} = 1/16 – 1/24 = (3 - 2)/48 = 1/48

Hence, Alpha alone can make the car in 48 days.

Beta 1 day’s work = {(Alpha + Beta + Gama)’s 1 day’s work – (Gama + Alpha)’s 1 day’s work}

= 1/16 – 1/36 = (9 - 4)/144 = 5/144

Hence, Beta alone can make the sports car in 144/5 days.

Gama’s 1 day’s work = {(Alpha + Beta + Gama)’s 1 day’s work – (Alpha + Beta)’s 1 day’s work}

= 1/16 – 1/18 = (9 - 8)/144 = 1/144

Hence, Gama alone can make the sports car in 144 days.

∴ Alpha, Beta and Gama alone can finish the task in 48 days, 144/5 days and 144 days respectively.