2013 AMC 10A Problem 5

P1-10~2 minPrint

Linear Equations & Word Problems Ratios, Percents & Averages Set up the equation/formula and compute; no special trick needed

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 $t-d$ ?

Show solution without answering

Key insight

Everyone should pay the same amount, one third of 405, which is 135. Each person gives Sammy the difference between 135 and what they already paid.

Solution

Total cost. $105 + 125 + 175 = 405$ dollars.
Fair share. Split evenly among $3$ people, each should pay $405 \div 3 = 135$ dollars.
What Tom owes. Tom paid $105$ , which is $135 - 105 = 30$ too little. He gives Sammy $30$ , so $t = 30$ .
What Dorothy owes. Dorothy paid $125$ , which is $135 - 125 = 10$ too little. She gives Sammy $10$ , so $d = 10$ .
Check with Sammy. Sammy paid $175$ , which is $175 - 135 = 40$ too much. He gets back $30 + 10 = 40$ . It matches.
Answer. $t - d = 30 - 10 = 20$ .

The answer is $\boxed{\textbf{(B)}\ 20}$ .

Why this works

"Split the cost evenly" means everyone ends up paying the average. Each person who paid too little makes up the gap between the average and what they paid. Checking that the person who overpaid gets back exactly the extra is a free way to catch mistakes.

Alternative approach

You do not even need the fair share. Tom owes $135 - 105$ and Dorothy owes $135 - 125$ . When you subtract, the $135$ cancels:
$t - d = (135 - 105) - (135 - 125) = 125 - 105 = 20.$
So $t - d$ is just how much more Dorothy paid than Tom.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

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 $\boxed{\textbf{(B)}\ 20}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 6: Ratios and Percents
Introduction to Algebra, Chapter 7: Proportion (§7.4)
Prealgebra, Chapter 13: Data and Statistics (§13.1)