Problem 19 - Prealgebra and Introductory Algebra

Question

One computer can solve a complex prime factorization problem in 75 h. A second computer can solve the same problem in 50 h. How long would it take both computers, working together, to solve the problem?

Step-by-Step Solution

Answer

When working together, both computers can solve the complex prime factorization problem in approximately 30 hours.

Step 1: Calculate the rate for the first machine

The rate at which the first machine solves the problem is calculated by finding the inverse of the time it takes for the machine to finish the task by itself. Express this metric in terms of problem per hour. The mathematical implementation of this step would be: \(Rate_1 = \frac{1}{75}\) problem/hour.

Step 2: Calculate the rate for the second machine

Follow a similar method for the second machine as well, the only difference being the time it takes for the second computer which is 50 hours. The computation is: \(Rate_2 = \frac{1}{50}\) problem/hour.

Step 3: Calculate the combined rate

The next step to solving this problem is combining the rates of the two machines. In problems regarding simultaneous work, the rates are directly additive. So, combine the two rates by summing them up as follows: \(CombinedRate = Rate_1 + Rate_2 = \frac{1}{75} + \frac{1}{50}\)

Step 4: Determine the combined time

To find how long both machines would take to finish the problem working together, find the inverse of the combined rate. This is calculated as: \(CombinedTime = \frac{1}{CombinedRate} = \frac{1}{\frac{1}{75} + \frac{1}{50}}\)

Key Concepts

Prime Factorization

Prime factorization is the process of breaking down a number into its most basic prime number components. Prime numbers are those numbers that can only be divided by 1 and themselves, like 2, 3, 5, and so on. By using prime factorization, we can simplify complex mathematical problems or express large numbers in terms of smaller, manageable prime factors. This is particularly useful in various areas of mathematics, including number theory, cryptography, and problem-solving methods like the one described in the exercise above.

Inverse Computation

Inverse computation tackles the idea of finding the reciprocal of a given number. In the context of rate work problems, it involves computing how much of a task is completed in a particular period. For instance, when we say a machine can complete a task in a specific number of hours, its rate of completion is the inverse of those hours. Therefore, if a computer takes 75 hours to solve a problem, its rate is calculated as the inverse: \(\frac{1}{75}\) of the work per hour. Similarly, a different computer completing the same task in 50 hours would have a rate of \(\frac{1}{50}\) per hour. This calculation allows analysts to precisely determine how much work is performed in any given timeframe.

Combined Rates

In combined rate problems, where multiple entities are working together to accomplish a task, it’s crucial to calculate their joint performance. To determine the combined rate of two computers solving a problem simultaneously, you sum their individual work rates. For the computers in our exercise, the rate of the first is \(\frac{1}{75}\) problems per hour, and the second is \(\frac{1}{50}\) problems per hour. Adding these rates gives the combined rate. This concept is vital in solving real-world problems, helping to calculate how long it will take multiple machines or people to finish a task when their efforts are pooled together.

Simultaneous Work

When two or more agents work together simultaneously on the same task, we deal with simultaneous work problems. These problems focus on understanding how combining efforts can reduce time required to complete a task. The principle is that when more hands are on deck, the workload is shared, thereby effectively decreasing the time it takes to finish the task. For example, in the problem provided, by looking at the combined performance rate of both computers, we are able to calculate what is known as the combined time. This is done by taking the inverse of the summed rates, giving an understanding of how long both would take working together. This method simplifies complex, time-dependent tasks into manageable calculations.