math challenge?

Question

jen cleans the kitchen in 20 minutes, her brother ken does it in 12 minutes, but ben, her 2 year old brother, can mess up the kitchen in 10 minutes, how many minutes does it take three of them to clean the kitchen

Answer 1

in 1 min, Jen can clean 1/20 of the kitchen
in 1 min, her brother can clean 1/12 of the kitchen
in 1 min, her 2 yo brother can mess 1/10 of the kitchen

the rates are:
jen = + 1/20
her brother = +1/12
2 yo brother: -1/10

let t be the time it takes the 3 of them to clean the kitchen
(1/20)t + (1/12)t - (1/10)t = 1
(1/30)t = 1
t = 30 min

Answer 2

Each minute, Jen does 1/20 of the cleaning, Ken does 1/12 and Ben does -1/10 of the "cleaning".

Add them up and see when they will complete 1 job:
(1/20)n + (1/12)n - (1/10)n = 1

The common denominator is 60:
(3/60)n + (5/60)n - (6/60)n = 1
(2/60)n = 1
(1/30)n = 1
n = 30

So it will take them 30 minutes.

(*and in response to the first answer... if Ben is kept out of the kitchen you have (8/60)n = 1, so they will finish in 60/8 minutes which is 7½ minutes, not 8.)

Answer 3

In one minute 1/20+1/12-1/10 kitchen is cleaned = 2/60 = 1/30 kitchen. Therefore in 30 min whole kitchen will be cleaned.

Answer 4

After all the 2 year old will remain victor the kitchen will never be cleaned up! so it will take forever and ever!

Answer 5

If they lock Ben in the closet, then they can do it in 8 minutes