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Algebra / Systems of two linear equations in two variables Difficulty: Medium

A group of 202 people went on an overnight camping trip, taking 60 tents with them. Some of the tents held 2 people each, and the rest held 4 people each. Assuming all the tents were filled to capacity and every person got to sleep in a tent, exactly how many of the tents were 2-person tents?

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Explanation

Choice C is correct. Let x represent the number of 2-person tents and let y represent the number of 4-person tents. It is given that the total number of tents was 60 and the total number of people in the group was 202. This situation can be expressed as a system of two equations, $x plus y, equals 60$ and $2 x plus 4 y, equals 202$ . The first equation can be rewritten as $y equals, negative x plus 60$ . Substituting $negative x plus 60$ for y in the equation $2 x plus 4 y, equals 202$ yields $2 x plus, 4 times, open parenthesis, negative x plus 60, close parenthesis, equals 202$ . Distributing and combining like terms gives $negative 2 x plus 240, equals 202$ . Subtracting 240 from both sides of $negative 2 x plus 240, equals 202$ and then dividing both sides by $negative 2$ gives $x equals 19$ . Therefore, the number of 2-person tents is 19.

Alternate approach: If each of the 60 tents held 4 people, the total number of people that could be accommodated in tents would be 240. However, the actual number of people who slept in tents was 202. The difference of 38 accounts for the 2-person tents. Since each of these tents holds 2 people fewer than a 4-person tent, $thirty eight halves, equals 19$ gives the number of 2-person tents.

Choice A is incorrect. This choice may result from assuming exactly half of the tents hold 2 people. If that were true, then the total number of people who slept in tents would be $2 times 30, plus, 4 times 30, equals 180$ ; however, the total number of people who slept in tents was 202, not 180. Choice B is incorrect. If 20 tents were 2-person tents, then the remaining 40 tents would be 4-person tents. Since all the tents were filled to capacity, the total number of people who slept in tents would be $2 times 20, plus, 4 times 40, equals, 40 plus 160, which equals 200$ ; however, the total number of people who slept in tents was 202, not 200. Choice D is incorrect. If 18 tents were 2-person tents, then the remaining 42 tents would be 4-person tents. Since all the tents were filled to capacity, the total number of people who slept in tents would be $2 times 18, plus, 4 times 42, equals, 36 plus 168, which equals 204$ ; however, the total number of people who slept in tents was 202, not 204.