Help with a math problem please!?

To solve this question, let's first start with a slightly different problem. Let's say the problem is as follows: Farmer A can plow a field in 6 hours. Farmer B can plow a field in 10 hours. How long will it take them to plow a field together?

Now, the way I learned to do these problems is simply by thinking rationally and asking these simple questions:

1) How much of the field can Farmer A do in one hour? Well, if he/she can do the entire field in 6 hours, he/she can do one sixth of the field in one hour. Make sense?

2) How much of the field can Farmer B do in one hour? As before, if he/she can do the entire field in 10 hours, he/she can do one tenth of the field in one hour.

3) How many hours does it take for them to plow the entire field together? This, of course, is the initial question. However, this is the time to assign a variable. So, lets name the number of hours it takes them to plow together, x.

4) How much of the field do they want to plow? Obviously the entire field. Not half a field. Not a quarter, not 3 fields. Just one. So, the answer to this question is simply one.

Now, what do I do with all this? Simple. I'll now write an equation:

x ( 1/6 + 1/10) = 1

Make sense? The number of hours multiplied by what each farmer can do in a single hour equals one completed field. I'm not going to solve this equation, because this is not what the original question asks. However, the same principles will be used.

So back to the original. Let's answer the same questions as before:

1) Farmer A completes 1/6 of the field as hour.

2) Farmer B completes 1/x of the field as hour. (The definition of x has changed. It is now how long it takes Farmer B to plow the field.)

3) Together, it takes them 4.5 hours.

4) They still want to complete 1 field.

Set up the equation:

4.5 ( 1/6 + 1/x) = 1

Divide both sides by 4.5:

1/6 + 1/x = 2/9 (I multipled 1/4.5 by two to get rid of the decimal)

Subtract 1/6:

1/x = 1/18

So, x = 18. Farmer B can plow the field in 18 hours. Do you understand? :)

These are always done as inverse relationships. It would be written as

1/6 + 1/x = 1/4.5.......multiply all by (x)(4.5)(6)

4.5x + (4.5)(6) = 6x

27 = 6x - 4.5x

27 = 1.5x

x = 18...........answer

Proof

1/6 + 1/18 = 1/4.5.......multiply all by (6)(18)(4.5)

18(4.5) + (4.5)(6) = (6)(18)

81 + 27 = 108

108 = 108

Equation:

1/(1/6 + 1/x) = 4.5

Solution:

1/(1/6 + 1/x) = 4.5

1/(x/6x + 6/6x) = 4.5

1/([x + 6]/6x) = 4.5

6x/(x+6)= 4.5

6x = 4.5 (x +6)

6x = 4.5x + 27

6x - 4.5x = 27

1.5x = 27

x = 18 hours