Algebra Problem - Painting a Car It takes $50$ minutes for $10$ people to paint $1$ car. How many minutes will it take $15$ people to paint the car?
I think I have this, I just wanted to make sure. 
I have $50\text{ min}/10\text{ ppl}$ for $1$ car. So I multiply by $10$ to get it for $1$ person $\longrightarrow$ $500\text{ min}/\text{person}$ for $1$ car. Then multiply by $1/15$ to get the number of minutes for $15$ people. The answer was $500/15 \approx 33.3$ minutes. 
Is this correct?
Thanks.
 A: Yes! You are correct. 
$50*10=500$minutes. Then, $500/15=33.33...$minutes.
A: Your reasoning is perfectly correct. Since a different number of people are doing the same job, you could just do a cross-multiplication like this :
$$
\begin{array}{c|c|c}
&\text{Case 1} &\text{Case 2}\\
\hline
\frac1{\text{Nb. persons}} & \frac1{10} & \frac1{15}\\
\text{Time needed} & 50 & \text{?} \\
\end{array}
$$
Note that when you do this, you should have the second line increasing when the first line does, that's why we couldn't keep $\text{Nb. persons}$ and $\text{Time needed}$ but had to take the inverse of one of them.
The cross-multiplication tells you that in an array like this :
$$
\text{bottom-left} \times \text{upper-right} = \text{bottom-right} \times \text{upper-left}
$$
In our problem, that gives :
$$
50 \times \frac1{15} = \text{?} \times \frac1{10}
$$
Hence the number we are looking for is :
$$
\frac{50 \times \frac1{15}}{\frac1{10}} = \frac{50 \times 10}{15} = \frac{10 \times 10}{3} = 33.333...
$$

What you need to remember is that for problems like this, you can always do a cross-multiplication, just pay attention to the numbers you put in, there must not be one increasing while the other decreases.

Other really important fact, in your particular problem, you need to make sure that the people are doing the same job. Otherwise you could fall into a nice well-known trap : If it takes $1$ day for $8$ hens to lay $8$ eggs, how long does it take for $80$ hens to lay $80$ eggs ?
Here the jobs (laying eggs) are different since the number of eggs to lay is not the same. Therefore you cannot directly apply the cross-multiplication.

This will all become very natural with a bit of practice, I guarantee it ;)
Have fun !
