Question

a cartoonist has 150 pieces of original artwork to give to his publishers and some fans who won his online contest. he plans to send 30 drawings to his publishers. he is sending at least 3 pieces of artwork to each contest winner. how many contest winners could there be?

Explanation:

Step1: Determine the number of artworks for winners

The cartoonist has 150 pieces in total and sends 30 to publishers. So the number of artworks left for winners is $150 - 30 = 120$.

Step2: Set up inequality for number of winners

Let $x$ be the number of contest winners. Since each winner gets at least 3 pieces, we have the inequality $3x\leq120$.

Step3: Solve the inequality

Divide both sides of $3x\leq120$ by 3: $x\leq\frac{120}{3}=40$. Also, since the number of winners can't be negative, $x\geq1$. But considering the "at least 3" per winner, we can also think in terms of maximum. The maximum number of winners is when each gets exactly 3, so $x = \frac{120}{3}=40$. But there can be fewer, as long as each gets at least 3. For example, if each gets 4, then $x=\frac{120}{4} = 30$, etc. But the maximum possible number of winners is 40 (when each gets 3), and the minimum positive integer (since you can't have 0 winners if you are sending to "some fans") is 1 (but with 120 pieces, which is more than 3, so valid). But the problem is asking "how many contest winners could there be", so we can find the range or the maximum. Let's find the maximum first.

Answer:

The number of contest winners could be at most 40 (and any positive integer less than or equal to 40 as long as $3x\leq120$, i.e., $x\leq40$ and $x\geq1$). If we consider the maximum number of winners (when each gets 3 pieces), the number of contest winners could be 40.