MATH Question please.?

Question

Two cocnut markers, Mike and Nina, hire a worker each to assist them in putting an equal number of coconuts into their respective trucks. The two hired workers work at a constant rate, while Mike works twice as fast as Nina. Mike and his worker finish the job after Mike has put 27 coconuts into his truck. Nina and her worker finish their job after Nina has put 18 coconuts into their truck. How many coconuts were in the combined harvest?

Answer 1

Mike collected 50% more coconuts than Nina, but he works twice as fast as Nina, so his load took 3/4 (1.5/2) as long to finish as Nina's.

time(mike) = number(mike)/rate(mike)
time(nina) = number(nina)/rate(nina)

rate(mike) = 2 * rate(nina)

time(mike)/time(nina) = (number(mike)/rate(mike)) / (number(nina)/rate(nina))
time(mike)/time(nina) = (27/(2*rate(nina))) / (18/rate(nina))
time(mike)/time(nina) = 27/(2*18)
time(mike)/time(nina) = 3/4

Mike collected nine more coconuts than Nina, but each truck contains the same number. This means that Nina's assistant must have collected 9 fewer than Mike's assistant.

number(mike) + number(mike_assistant) = number(nina) + number(nina_assistant)

27 + number(mike_assistant) = 18 + number(nina_assistant)
number(nina_assistant) = 9 + number(mike_assistant)

Since Mike's assistant worked 1/4 less time than Nina's assistant, and the assistants work at the same rate... that 9 coconut difference represents 1/4 of the coconuts that Nina's assistant loaded:

time(mike_assistant) * rate(assistant) = number(mike_assistant)

time(nina_assistant) * rate(assistant) = number(nina_assistant)

number(mike_assistant)/number(nina_assistant) = (time(mike_assistant) * rate(assistant)) / (time(nina_assistant) * rate(assistant))
= (time(mike) / time(nina))
= 3/4 (from above)

9 = number(mike_assitant) - number(nina_assistant)
= (time(nina_assistant) * rate(assistant)) - (time(mike_assistant) * rate(assistant))
= (time(nina_assistant) - 3/4 time(nina_assistant)) * rate(assistant)
9 = 1/4 time(nina_assistant) * rate(assistant)

Thus:

9*4 = time(nina_assistant) * rate(assistant)
36 = time(nina_assistant) * rate(assistant)

... which is the number collected by Nina's assistant.

Nina's assistant would have loaded 36, plus Nina's 18, is 54 in Nina's truck. Doubling to account for the identical amount in Mike's truck yields 108 for the combined harvest.