To determine the minimum number of cups of lemonade Malcolm sold to achieve at least $456 in profits, we can set up an equation.

Step 1: Define the variables

• Let $c$ be the number of cups of lemonade Malcolm sold.
• Each cup is sold for $5.50.
• Malcolm deducts $50 from his total earnings to account for supplies.

Step 2: Set up the profit equation

The total earnings from selling $c$ cups is $5.50c$.

After deducting $50 for supplies, the profit is given by: $\text{Profit} = 5.50c - 50$

Step 3: Set up the inequality for profit

We know that Malcolm's profit for the week is at least $456, so we can write the inequality: $5.50c - 50 \geq 456$

Step 4: Solve the inequality for $c$

First, add 50 to both sides of the inequality: $5.50c \geq 506$

Next, divide both sides by 5.50 to solve for $c$: $c \geq \frac{506}{5.50}$ $c \geq 92$

Step 5: Conclusion

Malcolm needs to sell at least 92 cups of lemonade to make at least $456 in profit.

Would you like further details or have any questions?

Here are five related questions that you might find helpful:

1. How would the minimum number of cups change if Malcolm deducted $100 instead of $50 for supplies?
2. If the price per cup increased to $6, how many cups would Malcolm need to sell to make $456 in profit?
3. How much profit would Malcolm make if he sold 100 cups?
4. What would happen to the profit if Malcolm reduced the price to $5 per cup?
5. How can Malcolm increase his profit margin without increasing the price per cup?

Tip: When solving inequalities, make sure to isolate the variable by performing operations systematically on both sides of the inequality.