Difference between revisions of "2013 AMC 10A Problems/Problem 5"
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The difference in the money that Tom paid and Dorothy paid is <math>20</math>. In order for them both to have paid the same amount, Tom must pay <math>20</math> more than Dorothy. The answer is <math>\boxed{{(B)20}}</math>. | The difference in the money that Tom paid and Dorothy paid is <math>20</math>. In order for them both to have paid the same amount, Tom must pay <math>20</math> more than Dorothy. The answer is <math>\boxed{{(B)20}}</math>. | ||
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+ | ==Solution 3== | ||
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+ | The meaning of sharing costs equally is meaning that, after the vacation, they are equally dividing the money in a way such that, each person would have the same amount left. As each person spends an amount of money, greater than 100, let it be that they all had <math></math>200<math> to spend. This means that after the vacation we want the amount of money, they currently have. After the trip, Tom would've </math>95<math> dollars, Dorothy would've </math>75<math> dollars, and Sammy had </math>25<math> dollars. This gives us a total of </math>95+75+25=195<math> dollars. | ||
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+ | We want to equally split this money, as that is what happens after splitting the cost equally. This means that we want Dorothy, Tom, and Sammy to each have </math>65<math> dollars. We know that Tom gave Sammy </math>t<math> dollars meaning that we want to split this money first. As Tom gives money to no one else, we want him to reach </math>65<math> dollars in this trade, meaning that as Tom has </math>95<math> dollars and Sammy has </math>25<math> dollars, we can do a trade of </math>30<math> so </math>t=30<math>. After this trade, we get that Tom has </math>65<math> dollars, Sammy has </math>55<math> dollars, and Dorothy has </math>75<math> dollars. | ||
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+ | Next trade is where Dorothy gives </math>d<math> dollars to Sammy. Dorothy has </math>75<math> dollars and Sammy has </math>55<math> dollars. As both of these don't have </math>65<math> dollars and this is the last trade, we need to make sure both have </math>65<math> dollars at the end. This is possible if </math>d=10<math> | ||
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+ | We want to find </math>t-d=30-10=20\qquad\textbf{(C)}$ | ||
==See Also== | ==See Also== |
Revision as of 22:28, 21 September 2022
Contents
[hide]Problem
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share costs equally, Tom gave Sammy dollars, and Dorothy gave Sammy dollars. What is ?
Solution 1
The total amount paid is . To get how much each should have paid, we do .
Thus, we know that Tom needs to give Sammy 30 dollars, and Dorothy 10 dollars. This means that .
Solution 2
The difference in the money that Tom paid and Dorothy paid is . In order for them both to have paid the same amount, Tom must pay more than Dorothy. The answer is .
Solution 3
The meaning of sharing costs equally is meaning that, after the vacation, they are equally dividing the money in a way such that, each person would have the same amount left. As each person spends an amount of money, greater than 100, let it be that they all had $$ (Error compiling LaTeX. Unknown error_msg)20095752595+75+25=195$dollars.
We want to equally split this money, as that is what happens after splitting the cost equally. This means that we want Dorothy, Tom, and Sammy to each have$ (Error compiling LaTeX. Unknown error_msg)65t65952530t=30655575$dollars.
Next trade is where Dorothy gives$ (Error compiling LaTeX. Unknown error_msg)d75556565d=10t-d=30-10=20\qquad\textbf{(C)}$
See Also
2013 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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 |
2013 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 4 |
Followed by Problem 6 |
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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