2010 AMC 10A Problems/Problem 8
Tony works hours a day and is paid $ per hour for each full year of his age. During a six month period Tony worked days and earned $. How old was Tony at the end of the six month period?
Tony worked hours a day and is paid dollars per hour for each full year of his age. This basically says that he gets a dollar for each year of his age. So if he is years old, he gets dollars a day. We also know that he worked days and earned dollars. If he was years old at the beginning of his working period, he would have earned dollars. If he was years old at the beginning of his working period, he would have earned dollars. Because he earned dollars, we know that he was for some period of time, but not the whole time, because then the money earned would be greater than or equal to . This is why he was when he began, but turned sometime in the middle and earned dollars in total. So the answer is .The answer is . We could find out for how long he was and . . Then is and we know that he was for days, and for days. Thus, the answer is .
Let equal Tony's age at the end of the period. We know that his age changed during the time period (since does not evenly divide ). Thus, his age at the beginning of the time period is .
Let be the number of days Tony worked while his age was . We know that his earnings every day equal his age (since ). Thus, Since , . Then we know that and
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