Questions: A small business owner budgets 2,200 to purchase candles. The owner must purchase a minimum of 200 candles to maintain the discounted pricing. If the owner pays 4.90 per candle to purchase small candles and 11.60 per candle to purchase large candles, what is the maximum number of large candles the owner can purchase to stay within the budget and maintain the discounted pricing?

Solution Steps

To solve this problem, we need to determine the maximum number of large candles that can be purchased while ensuring that at least 200 candles are bought in total and the total cost does not exceed $2,200. We can set up a system of inequalities to represent these constraints and solve for the maximum number of large candles.

1. Let \( x \) be the number of small candles and \( y \) be the number of large candles.
2. The total number of candles must be at least 200: \( x + y \geq 200 \).
3. The total cost must not exceed $2,200: \( 4.90x + 11.60y \leq 2200 \).
4. We want to maximize \( y \).

Let \( x \) be the number of small candles and \( y \) be the number of large candles. The constraints based on the problem are:

1. Total number of candles: \( x + y \geq 200 \)
2. Total cost: \( 4.90x + 11.60y \leq 2200 \)

We want to maximize \( y \). To do this, we can reformulate the problem as minimizing \( -y \) (since maximizing \( y \) is equivalent to minimizing \( -y \)).

The inequalities can be expressed as:

• Cost constraint: \( 4.90x + 11.60y \leq 2200 \)
• Minimum candles constraint: \( x + y \geq 200 \)

After solving the inequalities, we find that the maximum number of large candles \( y \) that can be purchased is approximately \( 182.0896 \). Since the number of candles must be a whole number, we round down to the nearest whole number.

Final Answer