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Algebra / Linear inequalities in one or two variables Difficulty: Hard

Ken is working this summer as part of a crew on a farm. He earned $8 per hour for the first 10 hours he worked this week. Because of his performance, his crew leader raised his salary to $10 per hour for the rest of the week. Ken saves 90% of his earnings from each week. What is the least number of hours he must work the rest of the week to save at least $270 for the week?

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Explanation

Choice C is correct. Ken earned $8 per hour for the first 10 hours he worked, so he earned a total of $80 for the first 10 hours he worked. For the rest of the week, Ken was paid at the rate of $10 per hour. Let x be the number of hours he will work for the rest of the week. The total of Ken’s earnings, in dollars, for the week will be $10 x plus 80$ . He saves 90% of his earnings each week, so this week he will save $0 point 9 times, open parenthesis, 10 x plus 80, close parenthesis$ dollars. The inequality $0 point 9 times, open parenthesis, 10 x plus 80, close parenthesis, is greater than or equal to 270$ represents the condition that he will save at least $270 for the week. Factoring 10 out of the expression $10 x plus 80$ gives $10 times, open parenthesis, x plus 8, close parenthesis$ . The product of 10 and 0.9 is 9, so the inequality can be rewritten as $9 times, open parenthesis, x plus 8, close parenthesis, is greater than or equal to 270$ . Dividing both sides of this inequality by 9 yields $x plus 8, is greater than or equal to 30$ , so $x is greater than or equal to 22$ . Therefore, the least number of hours Ken must work the rest of the week to save at least $270 for the week is 22.

Choices A and B are incorrect because Ken can save $270 by working fewer hours than 38 or 33 for the rest of the week. Choice D is incorrect. If Ken worked 16 hours for the rest of the week, his total earnings for the week will be $80 dollars plus 160 dollars, equals 240 dollars$ , which is less than $270. Since he saves only 90% of his earnings each week, he would save even less than $240 for the week.