If video speed increase by 2x, total time is reduced by 50%. How do I understand this with math? Please recommend basic resources for my skill level..

Question

Problem: If you increase the speed of a video by 2x, you're reducing the total time by 50%. I have no understanding of how to calculate this, or how I got to this outcome.

Request: Please recommend exactly what specific topics I should learn and understand.

Continuation Of Problem: For example, if I increase playback speed by 2.5x, I have no idea what % the total time is reduced by. It's likely around ~66% or something, but I've no idea how to calculate this. Or more simply, I don't understand how to do this in math.

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Please recommend in comments or answers (doesn't matter; the important thing is being helpful):

• online textbooks or any other resources specifically on practical math for everyday life

Other sources like Khan Academy has a lot of math that isn't useful or needed in everyday life.

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I wish there was a math curriculum that specifically listed the top 10 or so specific topics for practical math, and the math topics that tend to be more useful relative to other topics

Whatever specific topic the question/problem I asked here should be on that top 10, 20 or whatever

I'm highly knowledgeable and understand many concepts in many academic fields/areas outside of math (that don't require math), but I don't understand whatever basic math topic this is. I'm assuming the math-orientated had given this specific topic a specific label/word -- as to make communication easier as well as a host of many other benefits.

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It's generally said and understood that math is easier to learn via programming, but I do not know of any good sources, or if someone has made this yet as of 2017

Side note: I support all the people making progress in how math is being taught at all levels, besides the most abstract/theoretical. Please do not recommend any academic or theoretical math outside of the kind of practical resources that was asked for.

Answer 1

I would actually recommend looking at this at a physics view point.

The basic equation for constant-velocity motion is $x=vt$, where $x$ stands for the place, $v$ for the velocity, and $t$ for time.

You can look at $v$ as the number of frames per second ("the speed of the video"), $t$ as the time it takes to watch the video, and $x$ as the total number of frames in the video.

In the question you are asking, we have the same $x$ for any velocity (the number of frames don't change). So we are solving $vt=const$. In case the speed is say $\times 2.5$, the time must be $\times \frac{1}{2.5}$ for us to have the same constant.

The remaining part is to understand the relation between a number such as $\frac{1}{2.5}$ and precentage. This is the definition of precentage - $ 1\%\ =\frac{1}{100}$. So we have $\frac{1}{2.5}=\frac{40}{100}=40\%\ $, and the time would be $40\%\ $ of the original time.

To end my answer, I would recommend some generic middle school algebra book, these things are usually covered quite good there.

Answer 2

This is what is called inversely proportional. Think about this example: it takes $1$ man $24$ hours to build a wall. However, if there were $2$ men, they'd get it done twice as fast, so you half how long it takes --- it would take two of them $12$ hours to build together. For $3$ people, it would take a third of the time, so it would only take $8$ hours. In general, if there are $n$ people, it would take $24/n$ hours to build this wall.

Your video is the same --- you're playing it twice as fast, so it only plays for half the time. If you play it at $2.5$ times speed, then it plays for $1/2.5$ percent of the original time.

EDIT:

The law of proportionality is evident in a lot of things. You may conjecture that sunny skies improve ice cream sales; or that better tasing food will cost more; or that more time in the sun will give you a darker tan. These are each examples of relationships that are directly proportional, since an increase in one causes an increase in the other. Conversely, an increase in rainy weather may cause a decrease in ice cream sales, or an increase in burgers eaten per week may decrease how long you live, both of which are examples of inverse proportionality. Now, these are not strict relationships since there are more factors there, but I'm trying to convey the idea of proportional relationships.

We will move to 'stricter' relationships. Say we can buy a box of five pens. If I buy one box, I get five pens. If I buy two boxes, I get ten pens. If I buy $n$ boxes, I get $5\times n$ pens. The relationship is directly proportional and follows the formula $$ y = 5 \times x $$ where $y$ is the number of pens, and $x$ is the number of boxes bought. It is nonsense to talk of a 'proof' in this sense --- the relation is obvious (I hope!). In general, the formula for direct proportionality is $$ y = k \times x $$ where $x$ and $y$ are your variables (you have to think which is which --- there's not just some 'magic formula' for every possible situation, you have to apply some logic yourself), and $k$ is the proportionality factor. In the pens example, this is five. There are many more examples of this relationship, but you just need to think which was round it goes --- that is, if I buy more boxes, does it make sense that I get more or less pens? Of course, you get more!

The other type of proportionality is inverse proportionality. This happens when an increase in one variable cause a decrease in the other. For my wall example, does it make sense that more workers means that it takes longer to build the wall? Of course not! (Unless the workers stand about chatting all day, but we're not considering that kind of situation.)

The standard formula for inverse proportionality is $$ y = \frac{k}{x} $$ where the variables are the same as before. You have to look at your situation and figure out which variable matches up with which letter in the formula. For your video example, you actually set $k = 1$ because you are considering only one video, but we follow this thought process:

"My video plays for some amount of time. If I speed up the video, do I expect it to play for a longer or for a shorter amount of time? Since it's playing faster, it plays more frames in each second. If it's playing more frames in each second, and it's playing the same amount of frames, logic dictates that it must play for a shorter amount of time (again, I stress that there is no formal proof for this in the sense that you are talking, you just need to think about what makes sense). If I play it at $2\times$ the speed, then it can either double or half because of the $2$. If it can't double since that doesn't make sense, it has to half (this is shown Mathematically in the other two answers). Now, we know that the relationship is $(2\times)$ speed $\implies$ $\frac{1}{2}$ the play time, we can jump to the formula $$ (s\times)\text{ sped up} \implies \frac{1}{s}\text{ times the original play time.}" $$ Again, this is 'proved' Mathematically in the other two answers, but it's such an obvious relationship that the above steps certainly suffice.

Answer 3

Let's think about this in terms of something a little more concrete. Think of a sprinter running a 100-meter dash. If they run $5$ meters every second, how long will it take them to complete the race? Well, they must cover a the distance $D = 100~\text{m}$ at a speed (or rate) of $r = 5~\text{m}/\text{s}$. This will take them a time $t = \frac{D}{v} = \frac{100~\text{m}}{5~\text{m}/\text{s}} = 20~\text{s}$. I have derived this from the equation $D = r\times t$, which says that one travels a distance $D$ when moving at a speed $r$ for a length of time $t$. Notice how the units ($\text{m}$ and $\text{s}$) are treated algebraically, just as if they were numbers or variables. In particular, the unit of meters cancels as a common factor in the division, and the unit of seconds ends up in the numerator.

In terms of your video example, the distance $D$ is the length of the video. So, say we have a video that is three minutes long: $D = 3~\text{min} = 3~\text{min}\times\frac{60~\text{s}}{1~\text{min}} = 180~\text{s}$.

We can multiply by the conversion factor of $\frac{60~\text{s}}{1~\text{min}}$ since it equals $1$, and since multiplying by $1$ leaves a number unchanged.

If we play the video at normal speed ($1$x), then we have playback rate $r = 1~\text{s}/\text{s}$. Of course, in this case, we have that the playback time $t = \frac{D}{v} = \frac{180~\text{s}}{1~\text{s}/\text{s}} = 180~\text{s} = D$ is the same as the length of the video.

What happens when we consider playback rate of 2.5x? In this case, we have $r = 2.5~\text{s}/\text{s}$, so that $t = \frac{D}{r} = \frac{180~\text{s}}{2.5~\text{s}/\text{s}} = 72~\text{s}$. So, at 2.5x playback, a $3$-minute ($D=180~\text{s})$ video only takes $1$ minute and $12$ seconds ($t=72~\text{s}$) to watch. This is a reduction of $1$ minute and $48$ seconds ($D-t=108~\text{s}$), which is a reduction by $\frac{D-t}{D} = \frac{108~\text{s}}{180~\text{s}} = 0.6 = 60\%$. (Thus your guess of $66\%$ was rather close.)

Takeaway: The unit of the distance $D$ can be anything and the equation will still apply, assuming the rate $r$ remains constant for the duration $t$. In the sprinter example, $D$ was traditional distance, length. In the video example, it was (video playback) time.

But it could be pizzas. Final example: If I have to fill an order of 15 pizzas and can make one pizza in 7 minutes, how long will it take to finish all of the pizzas? Well, our distance is $D = 15~\text{pizzas}$ and our rate is $r = \frac{1~\text{pizza}}{7~\text{min}}$. Thus it would take $t = \frac{D}{r} = \frac{15~\text{pizzas}}{\frac{1~\text{pizza}}{7~\text{min}}} = 15\times 7~\text{min} = 105~\text{min}$ to complete this order.

Answer 4

After reading these answers, I was left personally more confused, as the given examples used everything but video examples. So I made my own answer. I hope this helps you (and others) understand.

Watch speed multiplier total time taken equation

t=Y/nZ t=seconds Y=total frames Z=frames per second (fps) n=watch speed multiplier

In this, we are assuming the video was filmed at 60fps and plays for 300 seconds at x1 speed. This gives us 18,000 frames. How many frames are in a given video won’t change, but the fps rate will change with your multiplier. So for the example video, the equation looks like this:

0.5x watch speed multiplier (half speed) t=18,000/0.5*60 t=18,000/30 t=600 seconds (10 minutes)

1x watch speed multiplier (normal speed) t=18,000/1*60 t=18,000/60 t=300 seconds (5 minutes)

2x watch speed multiplier (double speed) t=18,000/2*60 t=18,000/120 t=150 seconds (2 minutes 30 seconds)

2.5x watch speed multiplier t=18,000/2.5*60 t=18,000/150 t=120 seconds (2 minutes)

2.75x watch speed multiplier t=18,000/2.75*60 t=18,000/165 t=109.09(repeating) seconds (1 minute 49.09(repeating) seconds)

As you can see, the overall time and the fps both change, but the total frames never do. This is because video is made of frames. A typical frame rate is 60fps. So to determine how long your video will last given a specific watch speed multiplier, you need to know what fps your video has at normal (1x) watch speed. Then you can take the seconds it lasts and multiply the two together for total frames. Once you solve for total frames, you can use the simple equation above to determine how long any video will play for, for any given watch speed multiplier.