# Thread: Working together word problem

1. ## Working together word problem

TJ can brush his teeth 20 seconds faster than his girlfriend. If they work together they can brush his teeth in 30 seconds. How long does it take TJ by himself?

Ridiculous story, I know - but I cannot remember how to create an equation to say "
if they work together they can brush his teeth in 30 seconds." Obviously one equation would be:

T = G - 20

Failbait

2. Originally Posted by Failbait
[SIZE=3]TJ can brush his teeth 20 seconds faster than his girlfriend. If they work together they can brush his teeth in 30 seconds. How long does it take TJ by himself?
What a pathetic question. Let us change it.

A man can paint a table 20 seconds faster than his son. If they work together they take 30 seconds. How long does it take the man.

The general rule is the following: If it take a man $\displaystyle A$ seconds and his son $\displaystyle B$ seconds then together it takes $\displaystyle \frac{AB}{A+B}$ seconds. Here $\displaystyle A=x$ what we are trying to find so $\displaystyle B = x+20$. So together is takes 30 seconds and thus,
$\displaystyle \frac{x(x+20)}{x+(x+20)} = 30$.
Solve for $\displaystyle x$.

3. Given that I trust you with a simple word problem as such, I've calculated

A ≈ 51.622 seconds
B ≈ 71.622 seconds

I find this fairly hard to believe, as my teacher hardly ever assigns questions with such irregular answers. Do my answers look right to you?

Thanks again!

4. Originally Posted by Failbait
Given that I trust you with a simple word problem as such, I've calculated

A ≈ 51.622 seconds
B ≈ 71.622 seconds

I find this fairly hard to believe, as my teacher hardly ever assigns questions with such irregular answers. Do my answers look right to you?

Thanks again!
yup. those seem ok. here's another approach.

i will refer to brushing TJ's teeth as a job. If TJ can complete the job in
$\displaystyle x$ seconds, then it takes his girlfriend $\displaystyle x + 20$ seconds.

now we set up the rates, that is, the $\displaystyle \frac {\mbox {Job}}{ \mbox {Time}}$ ratios.

TJ does one job in $\displaystyle x$ seconds, so his rate is $\displaystyle \frac 1x$

His girlfriend does one job in $\displaystyle x + 20$ seconds, so her rate is $\displaystyle \frac 1{x + 20}$

Both do the job is 30 seconds, so together, their rate is $\displaystyle \frac 1{30}$

thus, the sum of their individual rates gives their rate working together, thus we have:

$\displaystyle \frac 1x + \frac 1{x + 20} = \frac 1{30}$

simplifying, we obtain the exact quadratic we would have using TPH's method

5. Thanks a ton, both of you!

6. Originally Posted by Failbait
Thanks a ton, both of you!
you're welcome