Josephine started a business selling cosmetics. She spent $4500 to obtain her merchandise, and it costs her $200 per week for general expenses. She earned $550 per week in sales. What is the minimum number of weeks it will take for Josephine to make a profit? Write an inequality to model the problem. A. 550w > 4500 + 200w B. 200w > 4500 + 550w C. 550w < 4500 + 200w D. 200w ≥ 4500 + 550w
What she spends; \[$4500+$200/w\] What she earns; \[$550/w\] The inequality of choice A represents this problem; \[550w>4500+200w\] 550w representing $550 per week that she earns, 4500 representing the initial cost of $4500 for her cosmetics, and 200w representing $200 per week that she pays. We are going to use the greater than inequality symbol because we want to know profit, and to have profit you need more than the expenses. Now we solve to find how many weeks it will take her to make a profit; \[550w>4500+200w\] Subtract 200w from both sides to get w on only one side; \[550w-200w>4500+200w-200w\]\[350w>4500\]Divide both sides by 350; \[\frac{ 350w }{ 350 }>\frac{ 4500 }{ 350 }\]\[w>12.857\]So approx. 13 weeks to make a profit. Check; \[13\times350=4550\]Which is greater than 4500.
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