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

Difference between revisions of "2013 AMC 12A Problems/Problem 5"

(→‎See also)
m (→‎Solution 1)
 
(One intermediate revision by one other user not shown)
Line 4: Line 4:
 
<math> \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 </math>
 
<math> \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 </math>
  
−
==Solution==
+
==Solution 1==
−
Add up the amounts that Tom, Dorothy, and Sammy paid to get &#036;<math>405</math>, and divide by 3 to get &#036;<math>135</math>, the amount that each should have paid.
 
  
−
Tom, having paid &#036;<math>105</math>, owes Sammy &#036;<math>30</math>, and Dorothy, having paid &#036;<math>125</math>, owes Sammy &#036;<math>10</math>.
+
Simply write down two algebraic equations. We know that Tom gave <math>t</math> dollars and Dorothy gave <math>d</math> dollars. In addition, Tom originally paid <math>105</math> dollars and Dorothy paid <math>125</math> dollars originally. Since they all pay the same amount, we have: <cmath>105 + t = 125 + d.</cmath> Rearranging, we have <cmath>t-d = \boxed{\textbf{(B)} 20}.</cmath>
 +
 
 +
Solution <math>\textcopyright 2018</math> RandomPieKevin. All Rights Reserved. skrrt...
 +
 
 +
==Solution 2==
 +
Add up the amounts that Tom, Dorothy, and Sammy paid to get \$<math>405</math>, and divide by 3 to get \$<math>135</math>, the amount that each should have paid.
 +
 
 +
Tom, having paid \$<math>105</math>, owes Sammy \$<math>30</math>, and Dorothy, having paid \$<math>125</math>, owes Sammy \$<math>10</math>.
  
 
Thus, <math>t - d = 30 - 10 = 20</math>, which is <math>\boxed{\textbf{(B)}}</math>
 
Thus, <math>t - d = 30 - 10 = 20</math>, which is <math>\boxed{\textbf{(B)}}</math>

Latest revision as of 22:42, 9 March 2018

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 the 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

Simply write down two algebraic equations. We know that Tom gave $t$ dollars and Dorothy gave $d$ dollars. In addition, Tom originally paid $105$ dollars and Dorothy paid $125$ dollars originally. Since they all pay the same amount, we have: \[105 + t = 125 + d.\] Rearranging, we have \[t-d = \boxed{\textbf{(B)} 20}.\]

Solution $\textcopyright 2018$ RandomPieKevin. All Rights Reserved. skrrt...

Solution 2

Add up the amounts that Tom, Dorothy, and Sammy paid to get $$405$, and divide by 3 to get $$135$, the amount that each should have paid.

Tom, having paid $$105$, owes Sammy $$30$, and Dorothy, having paid $$125$, owes Sammy $$10$.

Thus, $t - d = 30 - 10 = 20$, which is $\boxed{\textbf{(B)}}$

See also

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_12A_Problems/Problem_5&oldid=93105"