Difference between revisions of "2013 AMC 10A Problems/Problem 5"
(→Solution 2) |
(→Solution 3) |
||
| (8 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
| − | Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid | + | 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 <math>t</math> dollars, and Dorothy gave Sammy <math>d</math> dollars. What is <math>t-d</math>? |
| Line 14: | Line 14: | ||
==Solution 2== | ==Solution 2== | ||
| − | The difference in the money that | + | 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>. |
| + | |||
| + | ==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 <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. | ||
| + | |||
| + | 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. | ||
| + | |||
| + | 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> | ||
| + | |||
| + | We want to find <math>t-d=30-10=\boxed{B) 20}</math> | ||
| + | |||
| + | ~Batmanstark | ||
| + | |||
| + | ==Video Solution (CREATIVE THINKING)== | ||
| + | https://youtu.be/gPuN0eVqPUI | ||
| + | |||
| + | ~Education, the Study of Everything | ||
==See Also== | ==See Also== | ||
Latest revision as of 12:09, 1 July 2023
Contents
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 
dollars to spend. This means that after the vacation we want the amount of money, they currently have. After the trip, Tom would've 
dollars, Dorothy would've 
dollars, and Sammy had 
dollars. This gives us a total of 
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 
dollars. We know that Tom gave Sammy dollars meaning that we want to split this money first. As Tom gives money to no one else, we want him to reach dollars in this trade, meaning that as Tom has dollars and Sammy has dollars, we can do a trade of 
so 
. After this trade, we get that Tom has dollars, Sammy has 
dollars, and Dorothy has dollars.
Next trade is where Dorothy gives dollars to Sammy. Dorothy has dollars and Sammy has dollars. As both of these don't have dollars and this is the last trade, we need to make sure both have dollars at the end. This is possible if 
We want to find 
~Batmanstark
Video Solution (CREATIVE THINKING)
~Education, the Study of Everything
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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
