Sam is hired for a 20-day period. On days that he works, he earns $\$$60. For each day that he does not work, $\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\$$660. How many days did he not work?
Call $x$ the number of days Sam works and $y$ the number of days he does not. We can set up the following system of equations to represent the given information: \begin{align*}
x+y &= 20 \\
60x - 30y &= 660 \\
\end{align*} The first equation represents the total number of days Sam works, and the second equation represents his total profit. Solving for $x$ in the first equation yields $x = 20 - y$. Substituting into the second equation gives $60(20-y) - 30y = 660$. Canceling a factor of $10$ and multiplying out gives $120 - 6y - 3y = 66$. This simplifies to $-9y = -54$, or $y = 6$. Thus, Sam did not work for $\boxed{6}$ days.
6
Level 3
Algebra
Problem: Sam is hired for a 20-day period. On days that he works, he earns $\$$60. For each day that he does not work, $\$$30 is subtracted from his earnings. At the end of the 20-day period, he received $\$$660. How many days did he not work?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>Let the number of days Sam worked be denoted as W. Since the total period is 20 days, the number of days he did not work is 20 - W. 

Sam earns $60 for each day he works, so his total earnings from working for W days is 60W. For each day he does not work, he loses $30, leading to a total deduction of 30(20 - W) for the days he did not work.

The equation representing his total earnings after 20 days can be set up as follows:

Total earnings = Earnings from working - Deductions for not working
660 = 60W - 30(20 - W)

Now, simplify the equation:

660 = 60W - 600 + 30W
660 = 90W - 600

Next, add 600 to both sides to isolate the term with W:

660 + 600 = 90W
1260 = 90W

Now, divide both sides by 90 to solve for W:

W = 1260 / 90
W = 14

Now that we have the number of days Sam worked, we can find the number of days he did not work:

Days not worked = 20 - W
Days not worked = 20 - 14
Days not worked = 6

Thus, the number of days Sam did not work is 6.<|END_OF_LATENT|> The final answer is $6$.