Work Word Problem

[NJGx03x871]

The book I'm using shows me how to solve but I'm getting a little confused at the end:

Suppose one painter can paint the entire house in twelve hours, and the second painter takes eight hours. How long would it take the two painters together to paint the house?

First

Add together what they can do individually per hour: 1/12 + 1/8 = 5/24 (Makes sense)

Second

Let "t" stand for how long they take to do the job together. (Okay)

Third

Then they can do 1/t per hour, so 5/24 = 1/t. When for t = 24/5, t = 4.8 hours (WTF)

[soliloquy]

Does this make it more clear for you?

If the first painter can do the entire job in twelve hours and the second painter can do it in eight hours, then (this here is the trick!) the first guy can do 1/12 of the job per hour, and the second guy can do 1/8 per hour. How much then can they do per hour if they work together?

To find out how much they can do together per hour, I add together what they can do individually per hour: 1/12 + 1/8 = 5/24. They can do 5/24 of the job per hour. Now I'll let "t" stand for how long they take to do the job together. Then they can do 1/t per hour, so 5/24 = 1/t. Flip the equation, and you get that t = 24/5 = 4.8 hours. That is:
hours to complete job:

first painter: 12
second painter: 8
together: t

completed per hour:
first painter: 1/12
second painter: 1/8
together: 1/t

adding their labor:
1/12 + 1/8 = 1/t

5/24 = 1/t

24/5 = t

They can complete the job together in just under five hours.

[NJGx03x871]

Does this make it more clear for you?

If the first painter can do the entire job in twelve hours and the second painter can do it in eight hours, then (this here is the trick!) the first guy can do 1/12 of the job per hour, and the second guy can do 1/8 per hour. How much then can they do per hour if they work together?

To find out how much they can do together per hour, I add together what they can do individually per hour: 1/12 + 1/8 = 5/24. They can do 5/24 of the job per hour. Now I'll let "t" stand for how long they take to do the job together. Then they can do 1/t per hour, so 5/24 = 1/t. Flip the equation, and you get that t = 24/5 = 4.8 hours. That is:
hours to complete job:

first painter: 12
second painter: 8
together: t

completed per hour:
first painter: 1/12
second painter: 1/8
together: 1/t

adding their labor:
1/12 + 1/8 = 1/t

5/24 = 1/t

24/5 = t

They can complete the job together in just under five hours.

woah... do you have the same book? lol

that's almost verbatim the explanation given my my study booklet. Anyways, my question is why are we turning "t" into "1/t"...? logically it doesn't make sense to me.

[soliloquy]

(First painter's rate of 1 house per hour) + (Second painter's rate of 1 house per hour) = (Their combined painting of 1 house per hour)

Then,




Does that help? I'm not the best teacher. Basically it's because you are painting one kitchen PER whatever time it takes them to do it together. The wording of the question PER requires the fraction because it's 1 kitchen per t hours.

[NJGx03x871]

(First painter's rate of 1 house per hour) + (Second painter's rate of 1 house per hour) = (Their combined painting of 1 house per hour)

Then,




Does that help? I'm not the best teacher. Basically it's because you are painting one kitchen PER whatever time it takes them to do it together. The wording of the question PER requires the fraction because it's 1 kitchen per t hours.

All clear now, thanks!