SOLUTION: A math teacher’s salary increases 5% each year. Approximately, what is the percent increase in salary after 10 years? Answer a. 63 b. 48 c. 71 d. 41 e. 55

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Question 511348: A math teacher’s salary increases 5% each year. Approximately, what is the percent increase in salary after 10 years?
Answer
a. 63
b. 48
c. 71
d. 41
e. 55
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Let's call the math teacher's pay S at the start of the first year.
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At the start of the first year, gets paid S. So at the end of the first year the teacher's new rate of pay becomes S (starting salary) times 1.05. This can be written as:
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S%5B1%5D+=+S%281.05%29 with S%5B1%5D meaning new salary at the end of year 1 which is the starting salary for year 2.
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At the end of the second year, the teacher's salary will be again increased by 5%. To do this multiply the pay at the end of the first year S%5B1%5D by 1.05. The result will be:
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S%5B2%5D+=+%28S%281.05%29%29%281.05%29
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At the end of the third year, the teacher's salary will be again increased by 5%. To do this multiply S%5B2%5D by 1.05. The result will be:
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S%5B3%5D+=+%28S%281.05%29%281.05%29%29%281.05%29
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At the end of the fourth year, the teacher's salary will be again increased by 5%. To do this multiply S%5B3%5D by 1.05. The result will be:
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S%5B4%5D+=+%28S%281.05%29%281.05%29%281.05%29%29%281.05%29
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By now the pattern may be evident to you. The factor 1.05 appears as a multiplier the same number of times as the number of the year that is ending. Since it is a multiplier for that number of years, we can use an exponent as a way of expressing this. We can therefore write the equation as:
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S%5Bn%5D+=+S%281.05%29%5En
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Where n represents the number of the year end.
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Therefore at the beginning of the 10th year the teacher's pay will be what he was making at the end of the ninth year. So the pay through the 10th year the teacher will be paid:
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S%5B9%5D+=+S%281.05%29%5E9
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and if you raise 1.05 to the 9th power, you will find that throughout the tenth year the teacher is making 1.55 times what he made during the first year in which the pay was an amount S. This means that in the tenth year the teacher is paid 55% more than in the first year. But the problem asks what was the increase in salary after 10 years. When the teacher completes the 10th year the pay will become:
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S%5B10%5D+=+S%281.05%29%5E10
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Using a calculator to raise 1.05 to the 10th power, you find that after 10 years are completed the teacher's pay will be approximately:
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S%5B10%5D+=+S%2A1.629
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which represents a 62.9% (or rounded to 63%) increase from the salary at the start of the first year.
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Answer a is the correct answer.
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Hope this helps you understand the problem. One thing that you have to be careful of in this problem is determining what the "after 10 years" really means. It means AFTER the tenth year is completed, not during the tenth year.
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