2022 AMC 8

Complete problem set with solutions and individual problem pages

Problem 11 Medium

Henry the donkey has a very long piece of pasta. He takes a number of bites of pasta, each time eating 3 inches of pasta from the middle of one piece. In the end, he has 10 pieces of pasta whose total length is 17 inches. How long, in inches, was the piece of pasta he started with?

  • A.

    34

  • B.

    38

  • C.

    41

  • D.

    44

  • E.

    47

Answer:D

Solution 1

Let's say that the first strand of pasta he had is x. The first time he takes a bite of this strand will make it 2 pieces. This is x - 3. The second time he takes the bites of EACH strand will become (x - 3) * 3(2). The 2 is for the two bites he took, 1 for each strand. This equation is showing that for every bite that he is taking, he is taking off 3 inches. This is being showed in the 3(2). He took 2 bites, each taking off 3 inches. Now, the problem is stating that their are 10 pieces of pasta and the TOTAL length is 17 inches. We have to plug in the total number of bites to the equation (x - 3) * 3（ ）. To find this out, we can see that for the number of strands there are, he took 1 less bite than that number. For example, if there are 2 pieces, he took 1 bite to get those. Since there are 10 pieces, he took 9 bites. The equation then becomes, (x - 3) * 3(9) = 17. To solve this, x = 44. Therefore, the answer is \boxed{\textbf{(D) } 44}.

 

Solution 2

If there are 10 pieces of pasta, Henry took 10-1=9 bites. Each of these 9 bites took 3 inches of pasta out, so all his bites in total took away 9\cdot 3 = 27 inches of pasta. Thus, the original piece of pasta was 27+17=\boxed{\textbf{(D) } 44} inches long.

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