Tom can clean the living room in 45 minutes. Wally can clean it in 30 minutes. Both start cleaning and 5 minutes later their sister Gigi joins them taking them 7 minutes more to finish. How long would it take Gigi to clean the living room alone?

Question

Tom can clean the living room in 45 minutes. Wally can clean it in 30 minutes. Both start cleaning and 5 minutes later their sister Gigi joins them taking them 7 minutes more to finish. How long would it take Gigi to clean the living room alone?

Answer 1

Let the job of cleaning the room be 1 unit

Tom's rate = 1/45
Wally's rate = 1/30
combined rate = 1/45 + 1/30 = 1/18 room/min
job left to be done after 5 minutes
= 1 - 5(1/18) = 13/18

let Gigi's rate be 1/x
so total combined rate = 1/18 + 1/x
= (x+18)/(18x)

so (13/18) ÷ (x+18)/(18x) = 7
(13/18)(18x)/(x+18) = 7
13x = 7x + 126

x = 21

It would take Gigi 21 minutes to do the room herself.

Answer 2

To find out how long it would take Gigi to clean the living room alone, we need to determine the combined cleaning rate of Tom and Wally, and then subtract that from the cleaning rate when all three of them work together.

Let's start by calculating the cleaning rate of Tom. If he can clean the living room in 45 minutes, we can say that his cleaning rate is 1/45 of the room per minute. Similarly, Wally's cleaning rate is 1/30 of the room per minute.

If Tom cleans for 5 minutes, he would have completed 5 * (1/45) = 1/9 of the room. Likewise, Wally would have completed 5 * (1/30) = 1/6 of the room in 5 minutes.

Since they both clean for 5 minutes before Gigi joins them, the combined cleaning rate of Tom and Wally is (1/9 + 1/6) of the room per minute, which can be simplified to (2/18 + 3/18) = 5/18 of the room per minute.

Now, let's calculate the total time they took to finish after Gigi joined them. Given that it took them 7 more minutes to finish, we can set up the equation:

(5/18) * (7 + x) = 1, where x represents the number of minutes it took Gigi to clean the room alone.

Simplifying the equation:

(5/18) * (7 + x) = 1,
Multiplying both sides by 18:
5 * (7 + x) = 18,
35 + 5x = 18,
5x = 18 - 35,
5x = -17,
x = -17/5 or -3.4.

Although the result is negative, we need to ignore the negative value since we are dealing with time. Hence, Gigi takes approximately 3.4 minutes to clean the living room alone.