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Difference between revisions of "2013 AMC 10A Problems/Problem 5"
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==Solution 3==
 
==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.  
+
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>200</math> dollars 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.
+
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>
+
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)}\$
+
We want to find $t-d=30-10=20\qquad\textbf{(C)}\$
  
 
==See Also==
 
==See Also==

Revision as of 23:29, 21 September 2022

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 $t$ dollars, and Dorothy gave Sammy $d$ dollars. What is $t-d$?


$\textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35$

Solution 1

The total amount paid is $105 + 125 + 175 = 405$. To get how much each should have paid, we do $405/3 = 135$.

Thus, we know that Tom needs to give Sammy 30 dollars, and Dorothy 10 dollars. This means that $t-d = 30 - 10 = \boxed{\textbf{(B) }20}$.

Solution 2

The difference in the money that Tom paid and Dorothy paid is $20$. In order for them both to have paid the same amount, Tom must pay $20$ more than Dorothy. The answer is $\boxed{{(B)20}}$.

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 $200$ dollars to spend. This means that after the vacation we want the amount of money, they currently have. After the trip, Tom would've $95$ dollars, Dorothy would've $75$ dollars, and Sammy had $25$ dollars. This gives us a total of $95+75+25=195$ 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 $65$ dollars. We know that Tom gave Sammy $t$ dollars meaning that we want to split this money first. As Tom gives money to no one else, we want him to reach $65$ dollars in this trade, meaning that as Tom has $95$ dollars and Sammy has $25$ dollars, we can do a trade of $30$ so $t=30$. After this trade, we get that Tom has $65$ dollars, Sammy has $55$ dollars, and Dorothy has $75$ dollars.

Next trade is where Dorothy gives $d$ dollars to Sammy. Dorothy has $75$ dollars and Sammy has $55$ dollars. As both of these don't have $65$ dollars and this is the last trade, we need to make sure both have $65$ dollars at the end. This is possible if $d=10$

We want to find $t-d=30-10=20\qquad\textbf{(C)}$

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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