SOLUTION: Eight elves can pack 12 boxes in 30 minutes how long would it take for six elves to pack 24 boxes? 8 divided by 12 is 1.5 boxes per elf... Then I get stuck

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Question 1004262: Eight elves can pack 12 boxes in 30 minutes how long would it take for six elves to pack 24 boxes?
8 divided by 12 is 1.5 boxes per elf...
Then I get stuck
Found 3 solutions by josmiceli, Theo, rothauserc:
Answer by josmiceli(19441)   (Show Source):
You can put this solution on YOUR website!
You can approach this logically without any
algebra at all
---------------
1/4 as many elves would take 4 times as long
to do the same job, so
----------------------
2 elves can pack 12 boxes in 2 hrs
----------------------------
The same number of elves will takes twice
as long to pack twice as many boxes, so
----------------------------------
2 elves can pack 24 boxes in 4 hrs
-----------------------------
3 times as many elves can do the same job in
1/3 of the time, so
----------------------
6 elves can pack 24 boxes in 4/3 hour which is
1 hour and 20 minutes

Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
Eight elves can pack 12 boxes in 30 minutes how long would it take for six elves to pack 24 boxes?

formula is:

rate per elf * number of elves * time = quantity

rate per elf is equal to x, because we don't know what it is.

number of elves is 8
time = 30 minutes.
quantity = 12 boxes.

you get:

x * 8 * 30 = 12

solve for x to get:

x = 12 / (8 * 30) = 12 / 240 = 1/20

this means that each elf can pack 1/20 of a box in 1 minute.

this means that 8 * 1/20 * 30 = 12

that's 8 elves packing 1/20 of a box in one minute each and working for 30 minutes will be able to pack 12 boxes.

evaluate this equation to get 12 = 12.

this confirms the solution for the rate that each elf works at is good.

the problem then asks how long would it take for 6 elves to pack 24 boxes.

the assumption is that they work at the same rate each as they did in the original problem.

the same formula applies.

rate per elf * number of elves * time = quantity

replace this formula with known quantities of each to get:

1/20 * 6 * time = 24

solve for time to get:

time = 24 * 20 / 6 = 80 minutes.

did we get it right?

rate per elf * number of elves * minutes = quantity

1/20 * 6 * 80 = 1/20 * 480 = 24.

looks like we got it right.

you might have been able to solve this logically without knowing that formula.

you know that 8 elves working for 30 minutes can pack 12 boxes.

divide that by 30 and you get 8 elves working for 1 minute can pack 12/30 boxes.

divide that by 8 and you get 1 elve working for 1 minute can pack 12 / 30 / 8 boxes.

take 12 / 30 / 8 and you get 3 / 30 / 2 which becomes 1 / 20.

that tells you that 1 elve working for 1 minute can pack 1/20 of a box.

that tells you that 6 elves working for 1 minute can pack 6/20 of a box.

if you want to know how long it would take those elves to pack 24 boxes, you would have to divide 24 by (6/20) which is the same as 24 * (20/6) which is equal to 4 * 20 which is equal to 80 minutes.

you get the same answer.

the formula is just a formal way of representing it.

rate of each elf * number of elves * number of minutes = number of packed boxes.

Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website!
Convert this to elf-minutes
8 * 30 = 240 elf-minutes
now 24 boxes represents 240 * 2 = 480 elf-minutes, then
6 * x = 480
x = 80 minutes, therefore
it takes 6 elves 80 minutes to pack 24 boxes