AMC 10 Weekly Practice Round 2

Complete problem set with solutions and individual problem pages

Problem 26 Hard

Teams A, B, and C jointly undertake two projects, A and B. The workload of project B is \frac{4}{5} of the workload of project A. If working alone, Teams A, B, and C can complete project B in 40, 48, and 60 days respectively. At the beginning, Teams B and C work together on project A, while Team A works alone on project B. After working for a certain number of days, the arrangement changes so that Team B works alone on project A, while Teams A and C work together on project B. Both projects are completed at the same time. How many days did Team C work on project B?

  • A.

    3

  • B.

    4

  • C.

    5

  • D.

    6

  • E.

    7

Answer:D

Method 1:

Let the total number of days be x.

\left( \frac{4}{5 \times 40} + \frac{4}{5 \times 48} + \frac{4}{5 \times 60} \right) x = 1 + \frac{4}{5}

Solving gives x = 36.

Then the number of days Team C worked on project B is:

\left( \frac{4}{5} - \frac{1}{50} \times 36 \right) \div \frac{1}{75} = 6 \ \text{days}.

Method 2:

Let the workload of project B be [40, 48, 60] = 240.

Then the work rates are: Team A = 6, Team B = 5, Team C = 4.

The workload of project A is: 240 \div \frac{4}{5} = 300.

The total time for all three teams to complete both projects is: \frac{240 + 300}{6 + 5 + 4} = 36 \ \text{days}.

The number of days Team C worked on project B is: \frac{240 - 36 \times 6}{4} = 6 \ \text{days}.

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