A small business owner budgets $ 2,200 to purchase candles. The owner must purchase a minimum of 200 candles to

Question

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?

Accepted Answer

The correct answer is 182 . Let s represent the number of small candles the owner can purchase, and let l represent the number of large candles the owner can purchase. It&rsquo;s given that the owner pays $ 4.90 per candle to purchase small candles and $ 11.60 per candle to purchase large candles. Therefore, the owner pays 4.90 s dollars for s small candles and 11.60 l dollars for l large candles, which means the owner pays a total of 4.90 s + 11.60 l dollars to purchase candles. It&rsquo;s given that the owner budgets $ 2,200 to purchase candles. Therefore, 4.90 s + 11.60 l d 2,200 . It&rsquo;s also given that the owner must purchase a minimum of 200 candles. Therefore, s + l e 200 . The inequalities 4.90 s + 11.60 l d 2,200 and s + l e 200 can be combined into one compound inequality by rewriting the second inequality so that its left-hand side is equivalent to the left-hand side of the first inequality. Subtracting l from both sides of the inequality s + l e 200 yields s e 200 - l . Multiplying both sides of this inequality by 4.90 yields 4.90 s e 4.90 200 - l , or 4.90 s e 980 - 4.90 l . Adding 11.60 l to both sides of this inequality yields 4.90 s + 11.60 l e 980 - 4 . 90 l + 11 . 60 l , or 4.90 s + 11.60 l e 980 + 6.70 l . This inequality can be combined with the inequality 4.90 s + 11.60 l d 2,200 , which yields the compound inequality 980 + 6.70 l d 4.90 s + 11.60 l d 2,200 . It follows that 980 + 6.70 l d 2,200 . Subtracting 980 from both sides of this inequality yields 6.70 l d 2,200 . Dividing both sides of this inequality by 6.70 yields approximately l d 182.09 . Since the number of large candles the owner purchases must be a whole number, the maximum number of large candles the owner can purchase is the largest whole number less than 182.09 , which is 182 .

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