Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

2013 AMC 10A Problems/Problem 5

Revision as of 22:28, 21 September 2022 by Batmanstark (talk | contribs)

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 $$ (Error compiling LaTeX. Unknown error_msg)200$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$ (Error compiling LaTeX. Unknown error_msg)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$ (Error compiling LaTeX. Unknown error_msg)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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2013_AMC_10A_Problems/Problem_5&oldid=178381"