2014 AMC 12A Problem 2

Below is the professionally curated solution for Problem 2 of the 2014 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2014 AMC 12A solutions, or check the answer key.

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Concepts:ratio and proportion

Difficulty rating: 1020

At the theater children get in for half price. The price for $5$ adult tickets and $4$ child tickets is $\$24.50.$ How much would $8$ adult tickets and $6$ child tickets cost?

$\$35$

$\$38.50$

$\$40$

$\$42$

$\$42.50$

Solution:

Since a child ticket is half an adult ticket, $5$ adult and $4$ child tickets equal $5+\tfrac12\cdot4=7$ adult tickets, so one adult ticket costs $\dfrac{24.50}{7}=\$3.50.$

The second purchase equals $8+\tfrac12\cdot6=11$ adult tickets, costing $11\cdot 3.50=\$38.50.$

Thus, the correct answer is B.

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