The Harmonic Mean

I keep coming across a certain type of problem and so I gave Google's Gemini the opportunity to explain the solution. What follows is our interaction.

Here is a problem. 

A worker A can complete a task in 50 minutes and another worker B can complete the same task in 40 minutes. If they work together, how long will it take them to complete the task. 

Can you explain how the answer is reached?

Absolutely! Here's how we can find how long it takes workers A and B to complete the task together:

1. Rate of Each Worker:

Instead of focusing on the complete time each worker takes, we can think about their rates of completing the task.

Worker A's rate: We can say A completes 1/50th of the task every minute (since they take 50 minutes to do the whole thing).

Worker B's rate: Similarly, B completes 1/40th of the task every minute.

2. Combined Rate:

When A and B work together, they essentially combine their rates. So, their combined rate is the sum of their individual rates:

Combined Rate = Rate of A + Rate of B

Combined Rate = \( \dfrac{1}{50} + \dfrac{1}{40} \)

3. Finding the Combined Time:

Imagine the combined rate represents how much of the task they complete every minute when working together. Since they combine their efforts, they'll finish the entire task (which is 1 unit of work) when the combined rate reaches 1.

Therefore:

Time taken to complete the task together:

$$ \begin{align} x &= \frac{\text{Total work (1)}}{\text{ Combined Rate} }\\ &= \frac{1}{ \dfrac{1}{50} + \dfrac{1}{40}} \end{align}$$

4. Simplifying the Math:

To simplify the above equation, we can find a common denominator for the fractions (which is the least common multiple of 50 and 40, which is 200):

$$ \begin{align} x &= \frac{1}{ \dfrac{1}{50} + \dfrac{1}{40}} \\ &= \frac{1}{\dfrac{9}{200}} \end{align}$$

5. Final Answer:

Inverting the fraction gives us the total time: $$ \begin{align} x &= \frac{200}{9} \\ &= 22 \frac{2}{9} \text{ minutes (approximately)} \end{align} $$

Explanation:

By combining their rates, we essentially calculate how much faster they complete the task together compared to working individually. This combined rate helps us determine the total time it takes for them to finish the entire task.

Has this got anything to do with means, such as the harmonic mean or the geometric mean?

You've got a good eye! In this specific case, the method we used to find the time to complete the task together is closely related to the harmonic mean.

Here's why:

The harmonic mean is ideal for finding the average rate when rates are involved.

In this problem, we weren't finding the average time to complete the task, but the average rate at which they work together (which is the reciprocal of the average time).

Mathematically, the harmonic mean of two numbers a and b is:

harmonic mean = \( \dfrac{2 a  b}{  a + b} \)

While we didn't use this exact formula, the underlying principle is the same. We took the individual rates (1/50 and 1/40), essentially representing how much of the task they complete per minute, and combined them proportionally using a common denominator. This combined rate represents the average rate at which they work together, which is what the harmonic mean helps us find.

The geometric mean, on the other hand, wouldn't be applicable here. It's used for finding the average of multiplied quantities, which isn't the case in this scenario.