Okay, let's see. Frank took a 100-question test. He answered 80 questions, so that means he left 20 unanswered. Each correct answer gives 5 points, incorrect ones deduct 2 points, and unanswered ones give nothing. His total score is 232 points. The question is asking how many questions he answered correctly.

Hmm, so first, let's break this down. Let me think. Let me set up some variables. Let me call the number of correct answers as 'c' and the number of incorrect answers as 'i'. Since he answered 80 questions, the sum of correct and incorrect answers should be 80. So, c + i = 80. That's one equation.

Now, the scoring system: each correct answer is +5, each incorrect is -2. So the total points would be 5c - 2i. And his total score is 232. So the equation is 5c - 2i = 232. Now we have two equations:

1. c + i = 80
2. 5c - 2i = 232

We need to solve these two equations to find the value of c. Let's see. Maybe we can use substitution or elimination. Let me try substitution. From the first equation, we can express i in terms of c. So i = 80 - c. Then substitute this into the second equation.

So replacing i in the second equation: 5c - 2*(80 - c) = 232. Let's compute that. 5c - 160 + 2c = 232. Combine like terms: 5c + 2c is 7c, so 7c - 160 = 232. Then add 160 to both sides: 7c = 232 + 160. 232 + 160 is 392. So 7c = 392. Then c = 392 / 7. Let's compute that. 7*56 is 392, so c = 56. So Frank answered 56 questions correctly. Let me check if this makes sense.

If he answered 56 correctly, then the number of incorrect answers would be 80 - 56 = 24. Then total score would be 5*56 - 2*24. 5*56 is 280, and 2*24 is 48. So 280 - 48 = 232. Which matches the given score. So that checks out. Therefore, the answer should be 56.

Wait, but let me make sure I didn't make any calculation errors. Let me verify again. 56 correct answers: 56*5 = 280. 24 incorrect: 24*2 = 48. 280 - 48 = 232. Yep, that's correct. So the answer is 56.

Alternatively, maybe there's another way to approach this problem. Let's see. Suppose instead of using variables, we can think in terms of total points. If all 80 answered questions were correct, he would have 80*5 = 400 points. But he only scored 232, which is less. The difference is 400 - 232 = 168 points. Each incorrect answer instead of a correct one would result in a loss of 5 + 2 = 7 points (since you lose 5 for not getting it right and lose another 2 for getting it wrong). So the number of incorrect answers would be 168 / 7 = 24. Therefore, incorrect answers are 24, so correct answers are 80 - 24 = 56. That's another way to get to the same answer. So that confirms it again.

So both methods lead to the same result, so I think that's solid. Therefore, Frank answered 56 questions correctly.
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