Word Problem: Build a cabinet

[silverdragon316]

Beth can build a cabinet 3 times as fast as Ashley. If they work together, it takes them 4 hr to build the cabinet. How long does it take each of them, working alone?

I tried to do it from another example but it doesn't seem right. :?

I let a= the number of hours it takes beth to build the cabinets
then 3a= the number of hours it takes Ashley to build the cabinets

Beth I came up with 2 and 2/3 hours and Ashley 8.
I know I didn't do it right.

 

[jwpaine]

EDIT: deleted so it doesn't cause anyone else confusion.

 

[silverdragon316]

Beth can build a cabinet 3 times as fast as Ashley. If they work together, it takes them 4 hr to build the cabinet. How long does it take each of them, working alone?

I tried to do it from another example but it doesn't seem right. :?

I let a= the number of hours it takes beth to build the cabinets
then 3a= the number of hours it takes Ashley to build the cabinets

Beth I came up with 2 and 2/3 hours and Ashley 8.
I know I didn't do it right.

Then it would take beth 3 hr and Ashley 9 hrs.?

 

[soroban]

Hello, silverdragon316!

Beth can build a cabinet 3 times as fast as Ashley.
If they work together, it takes them 4 hr to build the cabinet.
How long does it take each of them, working alone?

I prefer to break it down to an hourly work-rate.

Let $\displaystyle a$ = hours for Beth to build a cabinet.
. . In one hour, Beth can build $\displaystyle \frac{1}{a}$ of a cabinet.

Let $\displaystyle 3a$ = hours for Ashley to build a cabinet.
. . In one hour, Ashley can build $\displaystyle \frac{1}{3a}$ of a cabinet.

Together, in one hour, they can build: $\displaystyle \,\frac{1}{a}\,+\,\frac{1}{3a}\:=\:\frac{4}{3a}$ of a cabinet. .[1]


But we are told that, working together, it takes them 4 hours to build a cabinet.
. . That is, in one hour, they can build $\displaystyle \frac{1}{4}$ of a cabinet. .[2]


There is our equation!

Equate [1] and [2]: \(\displaystyle \L\:\frac{4}{3a}\:=\:\frac{1}{4}\)

. . and solve for \(\displaystyle a:\;\L a \:=\:\frac{16}{3}\)


Therefore, it takes Beth $\displaystyle 5\frac{1}{3}$ hours to build a cabinet
. . . . .and it take Ashley $\displaystyle 16$ hours to build a cabinet.

 

[jwpaine]

Ohh... I'm sorry! I miss read your problem.

I wrote the problem as 4 cabinets......and how many each girl built... INSTEAD of 4 hours for 1 cabinet

I apologize for the confusion I caused you.