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How long will it take you to catch up on a new podcast?

An example of basic high school mathematics helping to answer simple questions from our everyday life
How long will it take you to catch up on a new podcast?

A friend shared with me a new podcast (for the curious, wethesalesengineers). He also told me that he was such a fan that he had listened to all the podcasts episodes, by far, 120 episodes.

I've decided to start listening to it and I started to wonder: If I listen one episode every day, how long will it take me to listen to all the podcast episodes?

What do we want to know?

It's important to define precisely what do we want to know in first place. In this case:

How many days will it take me to listen to all podcasts episodes?

At first thought, you could say: "Well, if you listen 1 episode everyday, it's obvious that it will take you 120 days"

But that's only true if the podcast is finished and there won't be more episodes. And for this one, it is not. The author is releasing one episode every week. So we will have to take that into account.

Now, if you have studied mathematics on high school this question shouldn't suppose any difficulty.

Give it a try before reading the answer below.


Gathering the relevant data

Let's summarize what we know about the problem:

  • The podcast has 120 episodes.
  • I wan to listen to one episode every day (as a detail, each episode last 1 hour but for the problem that's irrelevant)
  • The author releases a new podcast every week, or every 7 days

Translating the problem to mathematical terms

Today, the podcast has 120 episodes. In 7 days, the podcast will have 121 episodes. In 14 days, the podcast will have 122 episodes.

If we approach this from a mathematical perspective we can say that:

1 episode takes 7 days to make

Or

1/7 of the episode is made every day

Expressing that in mathematical notation:

$$ podcasts = \frac{1}{7} \cdot (days) + 120 $$

Which gives the number of podcasts episodes based on how many days have passed since today. For example: Today, day 0, the number of episodes is 120, in 7 days it will be 121, and in two weeks, 122.

Now, creating the formula for how many podcasts I've listened after some number of days is more simple. As I want to see 1 podcast every day:

$$ y = x $$

On day 1, I'll have seen one podcast, on day 2, I'll have seen two podcasts, etc.

So, back to our original question. How many days until I catch-up?

The intersection of the two functions gives the answer: After 140 days, I will have seen 140 podcasts.

In that time, 20 weeks will have passed and the author will have released 20 new episodes.