Question 1004262
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.