# Word problems (Deriving an equation from the problem)

1. Apr 25, 2016

### ryan001

1. The problem statement, all variables and given/known data
Working together, Bill and Tom painted a fence in 8 hours. Last year,
Tom painted the fence by himself. The year before, Bill painted it by himself,
but took 12 hours less than Tom took. How long did Bill and Tom take, when each was painting alone?
(represent the above with an equation without using rational equations)

2. Relevant equations
none

3. The attempt at a solution
I came up with the following equation.

(b = bill; t = tom)
b + t = 8
b = t - 12
However the above answer is incorrect

2. Apr 25, 2016

### SteamKing

Staff Emeritus
You have just come up with two equations. The problem asks you how long each person took to paint the fence when working by himself.

This means you have to come up with a numerical solution.

3. Apr 25, 2016

### haruspex

That equation appears to say that the time taken for Bill and Tom working together is the sum of the times each would take by himself. Doesn't sound like a very efficient team.

4. Apr 25, 2016

### mathdroid

It always helps to properly define your variables.
Let b = the time it takes Bill to paint the fence alone.
Let t = the time it takes Tom to paint the fence alone.
So you need two equations.

Since it takes Bill 12 hours less than Tom when both
work alone:

b = t - 12

I believe that if both of them working together can do the
job in 8 hours, then the sum of the times of their individual
efforts is twice what it takes them together:

b + t = 16

If you now solve this system, you will get two believable answers.

5. Apr 26, 2016

### Ray Vickson

When Bill and Tom work together, their rates add (not their times). Bill paints the fraction (1/b) of a fence in one hour; Tom paints the fraction (1/t) of a fence in one hour.

6. Apr 26, 2016

### ryan001

@mathdriod
t + b = 16
does not add up either thanks though
1.)t + b = 16
2). b = t - 12
16 - t = t - 12
2t = 28
t = 14
this would mean b = 12 which is incorrect.
@Ray Viskson
So is there a way to solve the equation without rational equations?

7. Apr 26, 2016

### haruspex

The right way is as Ray described, in terms of the rates. It is the two work rates that add to form a combined rate of working.
Equivalently, you can think in terms of fractions of a fence painted in an hour. If Bill takes b hours to paint the whole fence and Tom takes t hours to paint the whole fence, how much of the fence will Bill paint in an hour? How much of it will Tom paint in the same hour? How much will they have painted between them?

8. Apr 26, 2016

### Staff: Mentor

You should start off with more precise descriptions for your variables. How does b represent Bill and how does t represent Tom? By "more precise" I mean something like this:
Let b = Bill's hat size
Let t = the number of fingers on Tom's left hand
(or attributes more suitable for your problem...)

9. Apr 27, 2016

### Ray Vickson

I am not sure what a rational equation is, but I am guessing it is an equation which involves some fractions where variables are in the denominator. If so, I would not worry about it to start with; just get correct equations (whether rational or not), then later worry about how to make them non-rational if they start off being rational.

10. Apr 28, 2016

### ryan001

Thanks everyone