Suppose that Hortense and Mort can clean their entire house in 7 ​hours, while their​ toddler, Mimi, just by being​ around, can completely mess it up in only 5 hours. If Hortense and Mort clean house while Mimi is at her​ grandmother's, and then start cleaning up after Mimi the minute she gets​ home, how long does it take from the time Mimi gets home until the whole place is a​ shambles?

To determine how long it takes from the time Mimi gets home until the whole place is a shambles, we need to consider the rates at which Hortense, Mort, and Mimi can clean or mess up the house.

First, let's calculate the work rate of Hortense and Mort. We know that they can clean the entire house in 7 hours, so their work rate is 1/7 of the house per hour.

Next, let's calculate Mimi's work rate. We are given that Mimi can mess up the entire house in 5 hours, so her work rate is 1/5 of the house per hour.

If Hortense and Mort start cleaning up after Mimi as soon as she gets home, the combined work rate of Hortense, Mort, and Mimi is the sum of their individual work rates.

Combined work rate = Hortense and Mort's work rate + Mimi's work rate
= 1/7 + 1/5
= (5 + 7) / (5 * 7)
= 12 / 35

Now, we want to find the time it takes for the combined work rate to mess up the entire house, which is equivalent to completing 1 whole unit of work. To do this, we can use the formula: Time = Work / Rate.

Time = 1 / (12/35)
= 35/12
= 2.92 hours (approximately)

Therefore, it takes approximately 2.92 hours from the time Mimi gets home until the whole place is a shambles.

assuming the house is clean when Mimi gets home, consider that in one hour,

1/5 - 1/7 = 2/35 of the house gets messed up each hour.