Understanding and Solving Speed Problems with Copier A, B, and C
When working with multiple copiers, it's often necessary to understand their combined efficiency to meet production goals. This article will explore a practical scenario involving three copiers—A, B, and C—with varying speeds and an average efficiency. We will dissect the problem and provide a step-by-step solution, ensuring clarity on how to derive the necessary speed of copier C.
Background and Problem Statement
We have three copiers: Copier A, which makes 18 copies per minute, Copier B, which makes 9 copies per minute, and Copier C, with an unknown rate of copy production. The average copy speed across all three copiers is 15 copies per minute.
Step-by-Step Solution
To find the rate at which Copier C makes copies, let's start by formulating the given information into mathematical expressions and equations:
We know the average speed of the copiers is 15 copies per minute. We know the speeds of Copier A and Copier B, which is 18 and 9 copies per minute, respectively. The average speed equation can be expressed as:I. (text{Average speed} frac{text{Total copies produced per minute}}{text{Number of copiers}})
Given that the average speed is 15 copies per minute and there are 3 copiers, we have:
A B C / 3 15
Substituting the known values for A and B:
18 9 C / 3 15
27 C / 3 15
To isolate C, first multiply both sides by 3:
27 C 45
Subtract 27 from both sides:
C 18
Thus, Copier C makes 18 copies per minute.
Verification and Alternative Approaches
Let's verify our solution using a different method. We can also solve this problem using the concept of weighted averages.
Method 2: Weighted Averages
Using the formula for the weighted average, we can rephrase the problem as follows:
(frac{18n 9n v_Cn}{3n} 15)
Multiplying through by 3n:
(18n 9n v_Cn 45n)
Combining like terms and solving for (v_C):
(27n v_Cn 45n)
(v_Cn 45n - 27n)
(v_C 18)
Again, we find that Copier C makes 18 copies per minute.
Conclusion and Final Answer
Using both direct and alternative methods, we have determined that Copier C, which had an unknown rate, produces 18 copies per minute. This ensures that the average speed of all three copiers is 15 copies per minute.
Understanding how to solve such problems can be crucial in various business scenarios, especially in ensuring that production targets are met efficiently. The key takeaway is to use the principles of averages and weighted averages to derive the unknown variables in your problem.