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16 Sep 2014, 00:59
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A retailer sells 100 identical items each month, earning a $20 profit on each item. This profit constitutes 10% of the item's cost to the retailer. The retailer is considering offering a 5% discount on the items. What is the minimum number of items the retailer must sell each month to at least maintain the same level of monthly profit as without the discount?
A. 220
B. 221
C. 222
D. 223
E. 224
A. 220
B. 221
C. 222
D. 223
E. 224
16 Sep 2014, 00:59
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Official Solution:
A retailer sells 100 identical items each month, earning a $20 profit on each item. This profit constitutes 10% of the item's cost to the retailer. The retailer is considering offering a 5% discount on the items. What is the minimum number of items the retailer must sell each month to at least maintain the same level of monthly profit as without the discount?
A. 220
B. 221
C. 222
D. 223
E. 224
To solve this problem, we start by calculating the current monthly profit, which is 20 dollars*100 = 2000 dollars.
Next, we need to find the cost of each item to the retailer. Since the profit of 20 dollars is 10% of the item's cost, the item's cost is 200 dollars. Therefore, the current selling price of one item is 200 dollars + 20 dollars = 220 dollars.
Now, we can determine the new selling price with the 5% discount. Which would be 220 - 5% of 220 = 209 dollars. This means that the new profit on each item is 209 dollars - 200 dollars = 9 dollars. Therefore, the new net monthly profit is \(9x\) dollars, where \(x\) is the number of items the retailer will sell after the discount is announced.
The problem asks us to find the minimum value of \(x\) that will result in a new net monthly profit of at least 2000 dollars. To solve this, we set up the inequality \(9x ≥ 2000\) and solve for \(x\). Dividing both sides by 9, we get \(x ≥ \frac{2000}{9}\), which is approximately 222.22. Since \(x\) has to be an integer, we round up to the smallest integer greater than or equal to 222.22, which is 223. Therefore, the retailer must sell at least 223 items each month to maintain a monthly profit of at least 2000 dollars.
Answer: D
A retailer sells 100 identical items each month, earning a $20 profit on each item. This profit constitutes 10% of the item's cost to the retailer. The retailer is considering offering a 5% discount on the items. What is the minimum number of items the retailer must sell each month to at least maintain the same level of monthly profit as without the discount?
A. 220
B. 221
C. 222
D. 223
E. 224
To solve this problem, we start by calculating the current monthly profit, which is 20 dollars*100 = 2000 dollars.
Next, we need to find the cost of each item to the retailer. Since the profit of 20 dollars is 10% of the item's cost, the item's cost is 200 dollars. Therefore, the current selling price of one item is 200 dollars + 20 dollars = 220 dollars.
Now, we can determine the new selling price with the 5% discount. Which would be 220 - 5% of 220 = 209 dollars. This means that the new profit on each item is 209 dollars - 200 dollars = 9 dollars. Therefore, the new net monthly profit is \(9x\) dollars, where \(x\) is the number of items the retailer will sell after the discount is announced.
The problem asks us to find the minimum value of \(x\) that will result in a new net monthly profit of at least 2000 dollars. To solve this, we set up the inequality \(9x ≥ 2000\) and solve for \(x\). Dividing both sides by 9, we get \(x ≥ \frac{2000}{9}\), which is approximately 222.22. Since \(x\) has to be an integer, we round up to the smallest integer greater than or equal to 222.22, which is 223. Therefore, the retailer must sell at least 223 items each month to maintain a monthly profit of at least 2000 dollars.
Answer: D
General Discussion
23 Feb 2023, 02:57
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I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.
