Hey guys! Let's dive into a math problem that involves a charity call center, workers, calls, and time. It's a classic scenario that combines rates, proportions, and a little bit of algebraic manipulation. So, buckle up, and let's figure out how long it would take a certain number of workers to make a specific number of calls, all while keeping the spirit of giving alive!
The Charity Call Center Scenario
Imagine a bustling charity call center where 50 dedicated workers are making calls to solicit donations. These 50 workers can collectively make c calls in h hours. Now, the big question is: if we have w workers, how long would it take them to make 2,295 calls? This is where our mathematical skills come into play. We need to find a way to relate the number of workers, the number of calls, and the time it takes to make those calls. The key to unraveling this problem lies in understanding the rate at which calls are made. Let’s break this down step-by-step to make sure we nail it!
Understanding the Call Rate
To solve this problem, the call rate concept is very important. Think of it this way: if 50 workers make c calls in h hours, we can determine how many calls one worker makes in one hour. This is crucial because it gives us a baseline to compare different scenarios. To find the number of calls one worker makes in one hour, we can divide the total number of calls ( c) by the number of workers (50) and then divide by the number of hours ( h). Mathematically, this looks like:
Call rate per worker per hour = c / (50 h)
This fraction tells us the efficiency of each worker in making calls. It's like the engine of our problem, powering the solution. Grasping this rate is the cornerstone to solving our problem effectively.
Scaling Up to w Workers
Now that we know the call rate for one worker in one hour, let's scale this up to w workers. If one worker makes c / (50 h) calls in an hour, then w workers would make w times that amount. So, the combined call rate for w workers in one hour is:
Combined call rate = w * [c / (50 h)] = (w c) / (50 h)
This new rate represents the collective power of our w workers. It's how many calls they can make together in a single hour. Keep this number in mind, because it’s our stepping stone to finding the total time it takes to make 2,295 calls.
Calculating the Time to Make 2,295 Calls
Okay, guys, we're in the home stretch! We know the rate at which w workers make calls, and we know the total number of calls we need to make (2,295). To find the time it takes to make these calls, we'll use a simple formula:
Time = Total calls / Call rate
In our case:
Time = 2,295 / [(w c) / (50 h)]
To divide by a fraction, we multiply by its reciprocal. So, we flip the fraction and multiply:
Time = 2,295 * [(50 h) / (w c)]
Time = (2,295 * 50 * h) / (w c)
So, the time it takes for w workers to make 2,295 calls is (2,295 * 50 * h) / (w c) hours. Looking at the answer choices, we need to rearrange our expression to match one of them. Let’s do that now!
Matching the Answer Choices
Let’s take a look at the options given. Our calculated time is:
Time = (2,295 * 50 * h) / (w c)
We can rearrange this to match the format of the answer choices. Notice that we can keep the 2,295 separate:
Time = (50 h * 2,295) / (w c)
Now, let’s look at the options again. Option A looks promising:
A) (w c * 2,295) / (50 h)
Wait a second! Our equation doesn't quite match option A. We have the inverse of that. We have the total number of calls (2,295) times 50h divided by wc, but Option A has wc times 2,295 divided by 50h. So, it seems like we might need to rethink our approach, or perhaps we made a small mistake along the way. Let’s retrace our steps to make sure everything is in order.
Retracing Our Steps and Correcting the Inversion
Alright, team, let’s rewind a bit and double-check our calculations. It’s super easy to make a small slip-up, so no worries! We had:
Time = 2,295 / [(w c) / (50 h)]
When we divide by a fraction, we multiply by its reciprocal. So, we should have:
Time = 2,295 * [(50 h) / (w c)]
Time = (2,295 * 50 * h) / (w c)
Okay, so far so good. But here’s where we need to be extra careful. We are trying to find the time it takes, so our formula should indeed be:
Time = Total calls / Call rate
We calculated the call rate for w workers as (w c) / (50 h) calls per hour. So, if we plug that into our formula:
Time = 2,295 / [(w c) / (50 h)]
Time = 2,295 * (50 h) / (w c)
Time = (2,295 * 50 h) / (w c) hours
Now, let's rearrange this to match the answer choices. We have:
Time = (2,295 * 50 h) / (w c)
This is the same as:
Time = (2,295 * 50 * h) / (w * c)
Let's look at Option A again:
A) (w c * 2,295) / (50 h)
Nope, that's still not it. We have the reciprocal of this. What about the other options?
B) (w c) / (50 h * 2,295)
This is definitely not it. We need the 2,295 and 50h in the numerator.
The Correct Approach
Hey, sometimes math problems throw us a curveball, and that's totally okay! Let's step back and make sure we're thinking about this logically. We know that 50 workers can make c calls in h hours. That's our starting point. We need to figure out how long it takes w workers to make 2,295 calls.
Let’s find the rate at which one worker makes calls. If 50 workers make c calls in h hours, then one worker makes c/50 calls in h hours. To find the rate per hour, we divide by h:
Rate per worker per hour = (c/50) / h = c / (50h) calls per hour.
Now, we have w workers, so their combined rate is:
Combined rate = w * (c / (50h)) = (w c) / (50h) calls per hour.
We need to make 2,295 calls. To find the time it takes, we divide the total calls by the combined rate:
Time = 2,295 / [(w c) / (50h)]
Time = 2,295 * (50h) / (w c)
Time = (2,295 * 50 * h) / (w * c)
Okay, we're back to this. Let's rearrange it to match the answer choices. We can write it as:
Time = (2,295 * 50 * h) / (w * c) hours
Now, let's look at the options again:
A) (w c * 2,295) / (50 h)
This is still the inverse of what we have. It looks like there might be a mistake in the options provided, or we need to express our answer differently.
Expressing the Answer
Since our calculation seems correct, let's express our answer in the simplest form and see if it clarifies anything:
Time = (2,295 * 50 * h) / (w * c)
Time = (114,750 * h) / (w * c) hours
This is the most simplified form we can get without additional information. If we had to choose the closest answer, we would look for an option that has a similar structure, but it seems none of the options match our calculation perfectly.
Conclusion
So, guys, we've taken a deep dive into this charity call center problem. We've calculated the call rates, scaled up the workers, and determined the time it takes to make 2,295 calls. Our final expression for the time is:
Time = (114,750 * h) / (w * c) hours
It looks like none of the provided options match our result exactly, which could indicate an error in the options or a need for further simplification. But hey, we tackled the problem head-on, and that's what matters! Keep those mathematical gears turning, and who knows what we'll solve next!