組織解題所需的計算：练习题

Five students (Anna, Ben, Carla, David, Emma) are ranked by their scores in a competition, with no ties. The following information is known:

• Anna scored higher than Ben but lower than Carla.
• David scored lower than Emma but higher than Anna.
• Carla did not achieve the highest score.
• Emma did not achieve the lowest score.
• David scored lower than Carla.

What is the order of students from the lowest score to the highest score?

A small bakery produces custom cakes. Each cake requires $$ 2 \text{ kg} $$ of flour and $$ 1 \text{ dozen} $$ eggs. The bakery has $$ 100 \text{ kg} $$ of flour and $$ 48 \text{ dozen} $$ eggs in stock at the beginning of the week. They receive a delivery of $$ 30 \text{ kg} $$ of flour and $$ 10 \text{ dozen} $$ eggs every Tuesday morning.

The demand for custom cakes is constant at $$ 15 $$ cakes per day, Monday through Saturday. The bakery is closed on Sundays. If they start production on Monday morning, what is the maximum number of cakes they can produce by the end of Saturday?

A project requires three tasks: Task A, Task B, and Task C. The dependencies and durations are as follows:

• Task A: Duration 3 days. Must be completed before Task B can start.
• Task B: Duration 5 days. Must be completed before Task C can start.
• Task C: Duration 4 days.

There are two workers, John and Mary. Each worker can only work on one task at a time. Tasks can only start on a full day. What is the minimum number of days to complete the entire project if John starts Task A on Day 1?

A chemist needs to create $$ 200 \text{ ml} $$ of a solution that is $$ 40\% $$ acid. They have two stock solutions: Solution X (which is $$ 25\% $$ acid) and Solution Y (which is $$ 50\% $$ acid). How much of Solution X is needed to create the target solution?

In a group of 100 students, 60 study Mathematics, 40 study Physics, and 20 study both. A student is chosen at random from this group.

What is the probability that the chosen student studies Physics, given that they do not study Mathematics?

A machine processes batches using three sequential stages: Heating (30 mins), Mixing (20 mins), and Cooling (40 mins). There is only one Heating unit and one Cooling unit available, but Mixing can be done for multiple batches simultaneously. A new batch can only start Heating after the previous one finishes Heating, and its Cooling can only start after both its own Mixing is done and the previous batch has finished Cooling. What is the minimum time required to complete two batches?