Difference between revisions of "2013 AMC 12A Problems/Problem 5"
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| − | + | ==Problem== | |
| + | Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $<math>105</math>, Dorothy paid $<math>125</math>, and Sammy paid $<math>175</math>. In order to share the costs equally, Tom gave Sammy <math>t</math> dollars, and Dorothy gave Sammy <math>d</math> dollars. What is <math>t-d</math>? | ||
| − | + | <math> \textbf{(A)}\ 15\qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 25\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 35 </math> | |
| − | Thus, <math>t - d = 30 - 10 = 20</math> | + | ==Solution 1== |
| + | |||
| + | 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> | ||
| + | |||
| + | == See also == | ||
| + | {{AMC12 box|year=2013|ab=A|num-b=4|num-a=6}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 22:42, 9 March 2018
Contents
Problem
Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $
, Dorothy paid $
, and Sammy paid $
. In order to share the costs equally, Tom gave Sammy 
dollars, and Dorothy gave Sammy 
dollars. What is 
?
Solution 1
Simply write down two algebraic equations. We know that Tom gave dollars and Dorothy gave dollars. In addition, Tom originally paid dollars and Dorothy paid dollars originally. Since they all pay the same amount, we have: ![\[105 + t = 125 + d.\]](http://latex.artofproblemsolving.com/a/6/f/a6fe9782e94c6b81531413f701ac7f149a57878e.png)
Rearranging, we have ![\[t-d = \boxed{\textbf{(B)} 20}.\]](http://latex.artofproblemsolving.com/6/b/5/6b5b9202249b931c46ff886e2ea004aded3c478b.png)
Solution 
RandomPieKevin. All Rights Reserved. skrrt...
Solution 2
Add up the amounts that Tom, Dorothy, and Sammy paid to get $
, and divide by 3 to get $
, the amount that each should have paid.
Tom, having paid $, owes Sammy $
, and Dorothy, having paid $, owes Sammy $
.
Thus, 
, which is 
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. 
