A company that sells widgets has to pay $\$500$ in maintenance fees each day and then it pays each worker $\$15$ per hour. Each worker makes 5 widgets per hour, which are sold at $\$3.10$ each. What is the least number of workers the company has to hire in order to make a profit in an 8-hour workday?
We look for the least number of workers $n$ such that the cost is less than the revenue in a day of work. The cost of each worker is $\$15$ per hour, while the revenue that comes from each worker hired is $\$3.10\times5$ widgets per hour. \begin{align*}
500+8(15n)&<(8)(3.1)(5)n=124n\quad\Rightarrow\\
500+120n&<124n\quad\Rightarrow\\
500&<4n\quad\Rightarrow\\
125&<n.
\end{align*}The smallest integer greater than 125 is 126, so the company has to hire at least $\boxed{126}$ workers to make a profit.
126
Level 5
Prealgebra
Problem: A company that sells widgets has to pay $\$500$ in maintenance fees each day and then it pays each worker $\$15$ per hour. Each worker makes 5 widgets per hour, which are sold at $\$3.10$ each. What is the least number of workers the company has to hire in order to make a profit in an 8-hour workday?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To determine the least number of workers needed to make a profit, we first calculate the total daily costs incurred by the company. The fixed maintenance fee is \$500. Each worker is paid \$15 per hour, and with an 8-hour workday, the daily wage for one worker is:

Daily wage per worker = \$15/hour * 8 hours = \$120.

If we let \( n \) represent the number of workers, the total wage cost for all workers is:

Total wage cost = \( n \times 120 \).

Thus, the total daily cost for the company is:

Total daily cost = Maintenance fee + Total wage cost = \$500 + n \times 120.

Next, we calculate the revenue generated from selling widgets. Each worker produces 5 widgets per hour, so in an 8-hour workday, one worker produces:

Widgets produced per worker = 5 widgets/hour * 8 hours = 40 widgets.

For \( n \) workers, the total number of widgets produced is:

Total widgets produced = \( n \times 40 \).

The selling price for each widget is \$3.10, so the total revenue from selling all widgets is:

Total revenue = Total widgets produced * Selling price = \( n \times 40 \times 3.10 \).

To achieve profitability, the total revenue must exceed the total daily cost:

Total revenue > Total daily cost.

Substituting the expressions we derived:

\( n \times 40 \times 3.10 > 500 + n \times 120 \).

Now, simplifying the inequality:

\( 124n > 500 + 120n \).

Subtracting \( 120n \) from both sides gives:

\( 4n > 500 \).

Dividing both sides by 4 results in:

\( n > 125 \).

Since \( n \) must be a whole number, the least integer satisfying this inequality is \( n = 126 \). Thus, the company must hire at least 126 workers to ensure profitability in an 8-hour workday.<|END_OF_LATENT|> The final answer is $126$.