how does dividing by d also cancel all the ds on the bottom two fractions instead of just one? And what is Harmonic Mean?

[Originally posted in the Discussions]
Module 1 Week 4 Day 15 Challenge Explanation Part 3
Ok so how does dividing by d also cancel all the ds on the bottom two fractions instead of just one? And how is 2 over 4/12 the same as 2 times 12 over 4?
If we had just divided the top only by d, then our expression would not have kept the same value. You see, this expression isn't an equation. When you have an equation, it's okay to divide the left and right sides by the same amount, because they "balance" each other. But when you just have an expression like this, you don't want to change the value of it. That's why we have to divide d from not only the top, but the bottom as well, because this is effectively the same as multiplying by d/d which is the same as multiplying by 1.
Also, for the dividing by a fraction, you can think of comparing 2/4 with
Whenever you have a divide, you can think of the numbers as expressing the sharing of something, like cookies:
In the first fraction, there are 4 people. In the second fraction, there are 4/12, which is a little weird, but you can think of it as that there are 12times fewer people.
If there are fewer people to share cookies with, then each person gets more. This means that 2/(4/12) is bigger than 2/4.
(Remember that the answer to the fraction means how much cookie each person gets.) How many times as much cookie? They should get 12 times as much cookie, because there are 12 times fewer people.This is why 2/(4/12) is equal to 12 times 2/4, or
I hope this helps! I'm glad that you're continuing with the course and making such great progress! I hope that you've learned a lot in this course, and will continue to learn with us.
Happy Learning,
The Daily Challenge Team
[Originally posted by a student in the Discussions.]
Also what exactly is the harmonic mean? Why does it work and why the divide by 2 in the formula?
And what I mean with the ds is that when they canceled the d from the top, why did they remove the d for both fractions at the bottom? Why not just remove d for one fraction?
And lastly, how do you make fractions like 1/a and 1/b into fractions with common denominators because I saw that in an explanation but don't understand it
[Reply:]
Are you asking about this:
You are asking why we can't cancel d's to get something like this:
Let's think about a divide like trying to figure out how much cookie each person gets if we are sharing cookies over some number of people:
We just want to find out how much cookie each person gets, so we don't need to look at all the people to figure this out. We could put the cookies and people evenly into some number of rooms. Suppose the number of rooms is d. Then you would divide the cookies by d. You have to divide the people evenly into the d rooms as well. This is why you can't have
because only some of the people (the number on the bottom) are divided by d.
Happy Learning,
The Daily Challenge Team
Hi Xiao,
Thank you for your question.
For the purpose of this question, it's not super important that you know why we use the word "harmonic." What's important is that you know what the harmonic mean is.
A mean, normally, is when we add up all the numbers, and divide them by how many numbers there are. For example, if we have the numbers 1,2, and 4, we add them up, and divide them by 3:
When we talk about the harmonic mean we do something a little different. First, we take one over all the numbers (we "flip" them upside down). Then, we take that mean:
Finally, we flip the whole thing again. A good way to think about this is that because we flipped them to start with, we have to flip the answer back, after we take the mean. This give us:
This may look a little bit weird, because it is strange compared to the mean you may be used to. But as Po describes, this is actually really helpful for many problems in math, including this one!
About your question on why this works, try picking two letters instead of numbers, like "a" and "b," as speeds up and down the hill. Then try the problem again. You'll see you get the same formula! In the problem, we did this with a specific example (4 mph and 12 mph), but it didn't have to be those speeds, it could have been anything!
I hope this helps! You're asking great questions. Feel free to reach out if you ever get stuck, and keep doing a great job learning!
Happy Learning!
The Daily Challenge Team

This post is deleted! 
Thank you for your question.
For the purpose of this question, it's not super important that you know why we use the word "harmonic." What's important is that you know what the harmonic mean is.
A mean, normally, is when we add up all the numbers, and divide them by how many numbers there are. For example, if we have the numbers 1,2, and 4, we add them up, and divide them by 3:
$$ \frac{1 + 2 + 4}{3} = \frac{7}{3} $$
When we talk about the harmonic mean we do something a little different. First, we take one over all the numbers (we "flip" them upside down). Then, we take that mean:$$ \frac{\frac{1}{1} + \frac{1}{2} + \frac{1}{3}}{3} $$
Finally, we flip the whole thing again. A good way to think about this is that because we flipped them to start with, we have to flip the answer back, after we take the mean. This give us:
$$ \frac{1}{\frac{\frac{1}{1} + \frac{1}{2} + \frac{1}{3}}{3}} = \frac{3}{\frac{1}{1} + \frac{1}{2} + \frac{1}{3}} $$
This may look a little bit weird, because it is strange compared to the mean you may be used to. But as Po describes, this is actually really helpful for many problems in math, including this one!
About your question on why this works, try picking two letters instead of numbers, like "a" and "b," as speeds up and down the hill. Then try the problem again. You'll see you get the same formula! In the problem, we did this with a specific example (4 mph and 12 mph), but it didn't have to be those speeds, it could have been anything!
I hope this helps! You're asking great questions. Feel free to reach out if you ever get stuck, and keep doing a great job learning!
Happy Learning!
The Daily Challenge Team

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