2010 AMC 12A Problems/Problem 1
Problem
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?
Solution
Solution 1
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
.
Solution 2
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
See Also
2010 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
2010 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by First Problem |
Followed by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
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