### Instruction: One caterer charges a basic fee of $\$100$ plus $\$15$ per person. A second caterer charges a basic fee of $\$200$ plus $\$12$ per person. What is the least number of people for which the second caterer is cheaper?### Response: I need to compare the total costs of the two caterers for different numbers of people.
Let n be the number of people.
The first caterer's total cost is $100 + 15n$.
The second caterer's total cost is $200 + 12n$.
I want to find the smallest n such that the second caterer's cost is less than the first caterer's cost.
That means I need to solve the inequality $200 + 12n < 100 + 15n$.
I can simplify this by subtracting 100 from both sides and then subtracting 12n from both sides.
That gives me $100 < 3n$.
Then I can divide both sides by 3 to get $n > \frac{100}{3}$.
Since n has to be a whole number, I need to round up $\frac{100}{3}$ to the next integer.
That means n has to be at least 34.
So the least number of people for which the second caterer is cheaper is 34.