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2013 AMC 10A Problem 5

2013 AMC 10A Problem 5AMC 10 Step by Step

Name: Date: September 13, 2026

Linear Equations & Word Problems · difficulty 1 of 5 · about 2 min · position P1-10

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 $t-d$ ?

A)
$15$
B)
$20$
C)
$25$
D)
$30$
E)
$35$

Answer

Key insight

Everyone's fair share is the total 405 divided by 3; each payment to Sammy is the gap between 135 and what that person already paid.

Solution

The total cost is $105 + 125 + 175 = 405$ dollars, so each person's fair share is $405/3 = 135$ dollars.

Tom paid $105$ , which is $135 - 105 = 30$ short, so $t = 30$ . Dorothy paid $125$ , which is $135 - 125 = 10$ short, so $d = 10$ . (Sammy overpaid by $175 - 135 = 40 = t + d$ , as expected.)

Therefore $t - d = 30 - 10 = 20$ , and the answer is $\boxed{\textbf{(B)}\ 20}$ .

Why this works

Splitting costs evenly means everyone ends up having paid the mean. Each transfer is simply "share minus amount already paid." Verifying that the transfers received equal Sammy's excess is a free consistency check.

Alternative approach

Skip the shares: $t - d = (135 - 105) - (135 - 125) = 125 - 105 = 20$ . The difference between what Tom and Dorothy owe is exactly the difference between what they paid, regardless of the total.

The trap

Computing t - d as the difference of what Tom and Dorothy paid with the wrong sign, or averaging only two of the three payments.

Common mistakes

Computing t - d as the difference of what Tom and Dorothy paid with the wrong sign, or averaging only two of the three payments.
Reporting $t = 30$ (choice D) instead of the requested difference $t - d$ .

Techniques

Set up the equation/formula and compute; no special trick needed