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?

To determine the maximum number of large candles the owner can purchase while staying within the budget of \( \$ 2,200 \) and maintaining the minimum purchase of 200 candles, we can set up the following equations. Let: - \( x \) = number of small candles purchased - \( y \) = number of large candles purchased The cost of the candles can be expressed as: \[ 4.90x + 11.60y \leq 2200 \] The total number of candles must be at least 200: \[ x + y \geq 200 \] We want to maximize \( y \) (the number of large candles). ### Step 1: Express \( x \) in terms of \( y \) From the inequality \( x + y \geq 200 \), we can express \( x \) as: \[ x \geq 200 - y \] ### Step 2: Substitute \( x \) into the cost equation Substituting \( x \) into the cost equation gives: \[ 4.90(200 - y) + 11.60y \leq 2200 \] ### Step 3: Simplify the equation Expanding the left side: \[ 980 - 4.90y + 11.60y \leq 2200 \] Combining like terms: \[ 980 + 6.70y \leq 2200 \] ### Step 4: Solve for \( y \) Subtract \( 980 \) from both sides: \[ 6.70y \leq 1220 \] Now, divide by \( 6.70 \): \[ y \leq \frac{1220}{6.70} \approx 181.34 \] Since \( y \) must be a whole number, the maximum number of large candles \( y \) can be is \( 181 \). ### Step 5: Calculate \( x \) Now, substituting \( y = 181 \) back to find \( x \): \[ x + 181 \geq 200 \implies x \geq 200 - 181 = 19 \] ### Step 6: Verify the budget Now we check if this combination stays within the budget: \[ x = 19, \quad y = 181 \] Calculating the total cost: \[ 4.90(19) + 11.60(181) = 93.10 + 2097.60 = 2190.70 \] Since \( 2190.70 \leq 2200 \), this combination is valid. ### Conclusion Thus, the maximum number of large candles the owner can purchase while staying within the budget and maintaining the minimum purchase requirement is: \[ \boxed{181} \]