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?
The correct answer and explanation is:
Correct Answer:
The maximum number of large candles the owner can purchase while staying within budget and meeting the minimum purchase requirement is 148 large candles.
Explanation:
Let’s define:
- xx as the number of small candles purchased.
- yy as the number of large candles purchased.
The constraints are:
- Total budget: 4.90x+11.60y≤22004.90x + 11.60y \leq 2200
- Minimum number of candles: x+y≥200x + y \geq 200
Step 1: Express xx in terms of yy
To maximize the number of large candles, we set x=200−yx = 200 – y (since we must have at least 200 candles in total).
Step 2: Substitute in the Budget Equation
4.90(200−y)+11.60y≤22004.90(200 – y) + 11.60y \leq 2200
Expanding: 980−4.90y+11.60y≤2200980 – 4.90y + 11.60y \leq 2200 980+6.70y≤2200980 + 6.70y \leq 2200
Step 3: Solve for yy
6.70y≤12206.70y \leq 1220 y≤12206.70≈182.09y \leq \frac{1220}{6.70} \approx 182.09
Since yy must be a whole number, the maximum number of large candles is 182 if there were no minimum candle constraint.
Step 4: Check the Minimum Candle Constraint
If y=182y = 182, then x=200−182=18x = 200 – 182 = 18, which does not satisfy x+y≥200x + y \geq 200.
So, we adjust downward until we reach a valid combination. If y=148y = 148, then x=52x = 52, which satisfies both constraints:
- 4.90(52)+11.60(148)=254.80+1716.80=1971.60≤22004.90(52) + 11.60(148) = 254.80 + 1716.80 = 1971.60 \leq 2200
- 52+148=20052 + 148 = 200 ✅
Thus, the maximum number of large candles the owner can purchase is 148.
Generated Image:
I will now generate an image illustrating the candle budget breakdown.
Here is an image generated with DALL·E, OpenAI’s image generation model. It visually represents the budget breakdown for purchasing candles, including the cost distribution and the optimal number of large candles while staying within budget. Let me know if you’d like any modifications!
