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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.


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.


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.


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.

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  • $\begingroup$ You should take a look at Proportionality. More precisely, Inverse Proportionality. Read this: en.m.wikipedia.org/wiki/… $\endgroup$ – Filburt Apr 15 '17 at 10:15
  • $\begingroup$ Oh. And this link contains references... ;-) $\endgroup$ – Filburt Apr 15 '17 at 10:20
  • $\begingroup$ Yep... super basic topic... classed under 'ratios', which is a subset of 'elementary mathematics' and 'algebra'.... these references are highly advanced and complicate/confuse things needlessly. I need a resource where I can learn/understand this and the other topics (that are useful to everyday life). This is an encyclopedia, which is largely historical. I read almost everything besides the math topics. I can't learn math like this. I need an actual learning resource. $\endgroup$ – ambw Apr 15 '17 at 10:40
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    $\begingroup$ What exactly do you classify as practical?I could list practical uses for about every precalculus topic,I also feel that Khan academy has one of the best easy to learn resources,and at least 70% of the lets say theoretical questions can be "translated" into practical problems/solutions. $\endgroup$ – kingW3 Apr 15 '17 at 10:42
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    $\begingroup$ The Khan Academy knowledge map is not just videos, by the way, it's an actual learning system where you answer math problems to prove your proficiency and progress through the knowledge map. I recommend giving it a try -- actually doing the problems and earning the badges. $\endgroup$ – littleO Apr 15 '17 at 11:10
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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.

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  • $\begingroup$ For the 3 letters, the only one that's unclear is x. I'm not sure what to put there or how I would understand what to put there. You of course did and put what's suppose to be there: 'total frames'. Looks like x is constant, but there's so many other things that are constant that I can put there. For example, a 'button on the UI' is constant but it doesn't seem like it would make sense to put that as v. But how do I figure out what to put in there for all cases and problems of this pattern? Is there a mathematical way to go about this? Or I don't understand.. What do I read or learn? $\endgroup$ – ambw Apr 15 '17 at 11:13
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    $\begingroup$ What chances in time? only the number of frames you have watched. Also, if you think about the relevant units - if we agree that time is in seconds, and the velocity is frames per second, we must get that the place is frames/sec *sec which is frames. $\endgroup$ – The way of life Apr 15 '17 at 11:15
  • $\begingroup$ I don't understand what you mean by 'chances in time'. Explaining is hard. Should read all the other comments. $\endgroup$ – ambw Apr 15 '17 at 22:20
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    $\begingroup$ *changes in time, sorry $\endgroup$ – The way of life Apr 16 '17 at 6:37
  • $\begingroup$ I meant x when I wrote v on that line: "would make sense to put that as v" -- where's a list of all 'changes in time' words so i know what to put in x? $\endgroup$ – ambw Apr 17 '17 at 1:08
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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.

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  • $\begingroup$ As for further reading, Khan Academy is absolutely fantastic. You may not see the 'real life' applications of many of his videos directly, but a great deal of them are the foundations for bigger topics that do have applications in the real world. It seems that you need to learn a lot of the fundamentals of Mathematics --- I would suggest learning everything in the Mathematics GCSE schedule (or any foreign equivalent). There is very little Mathematics taught at the level that doesn't have applications, whether you can see it or not. $\endgroup$ – Bill Wallis Apr 15 '17 at 10:26
  • $\begingroup$ So looks like the formula generally at least is [total time taken] : [number of people, speed, or whatever the variable is] -- but of course this isn't helping me understand anything. I mean, do I always put the variable (the number that is being changed) in the second slot? Saying that something is 'twice as fast' equates to 'half the time' is a HUGE jump. How in the world did we get there? Why is it that 3 people is 1/3 the time? Every single step from the starting point (2x) to the end point (50% less) needs to be explained FULLY -- IN ABSOLUTE DETAIL. That's what understanding means, $\endgroup$ – ambw Apr 15 '17 at 10:48
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    $\begingroup$ @ambw Again, I'm going to stress the fact that there is no book that goes this in depth. If you're at school or college, talk to a teacher/lecturer about it. But, I would strongly suggest just sitting down and thinking about it yourself. Try writing out some of your own examples --- attempt to figure out these relationships, and what would actually make sense. Thinking independently and trying your own examples is so vastly important for coming to a full understanding of simple laws like these. $\endgroup$ – Bill Wallis Apr 15 '17 at 13:06
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    $\begingroup$ @ambw No they don't. There are some things that are objective truths. It is also common in the community to leave some work for the reader when it comes to self explanations. Being able to piece parts of a proof together to get an understanding of the whole proof is part pf the learning experience; giving someone all the information straight away prohibits free thinking and original thought. There are some things that you are expected to learn yourself, and this is totally fair. $\endgroup$ – Bill Wallis Apr 15 '17 at 22:54
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    $\begingroup$ Stack Exchange is that community. The reason no one develops on this topic is because people don't usually struggle with it --- irrespective of ones background, my explanation or the Physics explanation are two of the explanations that are easiest to understand for this topic. We're not 'set in our ways' --- it's like asking why dividing by two is the same and multiplying by a half, there's only so many ways it can be explained without overdoing it! I stand by the comment that some things should be left to you when they are simple steps like these as it helps you to develop yourself. $\endgroup$ – Bill Wallis Apr 17 '17 at 10:00
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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.

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  • $\begingroup$ 1) How I do I know what equation to use? 2) How do I know what numbers to put into what letters? 3) I can put anything into D? - "distance D can be anything" - can I put the 'speed'? 4) In your example, [distance] = 100 m, so then... actually I have a billion other questions, but I'll check back tomorrow. What's a good source that would help give understanding for all these questions, and all the followup questions that would surely derive from the previous? $\endgroup$ – ambw Apr 15 '17 at 11:54

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