# Difference between revisions of "2010 AMC 10A Problems/Problem 8"

TheNormsta (talk | contribs) (Created page with '13') |
|||

Line 1: | Line 1: | ||

− | 13 | + | == Problem 8 == |

+ | Tony works <math>2</math> hours a day and is paid $<math>0.50</math> per hour for each full year of his age. During a six month period Tony worked <math>50</math> days and earned $<math>630</math>. How old was Tony at the end of the six month period? | ||

+ | |||

+ | <math> | ||

+ | \mathrm{(A)}\ 9 | ||

+ | \qquad | ||

+ | \mathrm{(B)}\ 11 | ||

+ | \qquad | ||

+ | \mathrm{(C)}\ 12 | ||

+ | \qquad | ||

+ | \mathrm{(D)}\ 13 | ||

+ | \qquad | ||

+ | \mathrm{(E)}\ 14 | ||

+ | </math> | ||

+ | |||

+ | ==Solution== | ||

+ | |||

+ | Tony worked <math>2</math> hours a day and is paid <math>0.50</math> 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 <math>12</math> years old, he gets <math>12</math> dollars a day. We also know that he worked <math>50</math> days and earned <math>630</math> dollars. If he was <math>12</math> years old at the beginning of his working period, he would have earned <math>12 * 50 = 600</math> dollars. If he was <math>13</math> years old at the beginning of his working period, he would have earned <math>13 * 50 = 650</math> dollars. Because he earned <math>630</math> dollars, we know that he was <math>13</math> for some period of time, but not the whole time, because then the money earned would be greater than or equal to <math>650</math>. This is why he was <math>12</math> when he began, but turned <math>13</math> sometime in the middle and earned <math>630</math> dollars in total. So the answer is <math>13</math>.The answer is <math>\boxed{D}</math>. We could find out for how long he was <math>12</math> and <math>13</math>. <math>12 \cdot x + 13 \cdot (50-x) = 630</math>. Then <math>x</math> is <math>20</math> and we know that he was <math>12</math> for <math>20</math> days, and <math>13</math> for <math>30</math> days. Thus, the answer is <math>13</math>. |

## Revision as of 16:59, 20 December 2010

## 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?

## Solution

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 .