It takes Joey 10 hours to clean his room. It takes his mom 5 to clean his room. If Joey starts cleaning at 1 pm and mother starts helping at 2pm,

Question

at what time time will joeys room be clean. Solve

Answer 1

There are a few ways to solve this, but let me explain a way that makes a lot of sense.  With problems like these, I begin by writing down known information.  The rates for cleaning are (where J = Joey and M = mom):

 

J = 10 hours/room

M = 5 hours/room

 

Now, to get an equation, I will pick some numbers to get a clearer picture of what is going on:

 

Room is 1000 sq feet

 

I just picked an arbitrary number for the room size.  Now, if the room is 1000 sq feet, that means that Joey cleans at 100 sq feet per hour, and his mother at 200 square feet per hour:

 

J = 100 sq feet/hour

 

M = 200 sq feet/hour

 

 

Now, with this information, where x = hours, we can start to set up an equation.  To clean 1000 square feet, we have Joey cleaning at 100 square feet per hour plus his mother cleaning at 200 square feet per hour.  BUT, she starts one hour later than Joey.  So, the equation looks like this:

 

1000 = 100x + 200(x - 1)

 

Now, x is simply the total number of hours required for the entire room to be cleaned.  Solving for x, we get:

 

1000 = 100x + 200x - 200

1200 = 300x

x = 4

 

It will take 4 hours to clean the room.  Therefore, the room will be clean by 5 pm.

Answer 2

Joey cleans 1/10 of his room in one hour

Mom cleans 1/5 of his room in one hour

let x=# of hours Joey cleans

let x-1=# of hours mom cleans

They clean together.

(x)(1/10)+(x-1)(1/5)=1 (job finished; room cleaned)

Multiply both sides of the equation by 10.

x+2(x-1)=10

x+2x-2=10

3x-2=10

3x=12

x=4 hours that Joey cleans

x-1=4-1=3 hours that mom cleans

The room will be cleaned at 5 PM. (1+4 or 2+3)

check: 4(1/10)+3(1/5)=(4/10)+(3/5)=(2/5)+(3/5)=(5/5)=1 (simplify 4/10 to 2/5)