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I'm a simple man living my every day life and have not much understanding of math or science.

Today I read an article where someone claimed they can charge a battery $200\%$ faster. This got me thinking. What do they mean? How can something be faster than $100\%$.

Let's say I charge battery in $5$ minutes. $100\%$ faster would then be $5$ minutes faster - $0$ minutes $0$ seconds - instant charging (impossible).

This guy claims $200\%$ faster charging, so what does he mean? He charges batteries in $-5$ minutes?

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    $\begingroup$ This shirt is 100% means 100% discount - free give away? No? Let's say it cost $10. 90% off would be $9 off - it would cost $1. 99% off would be it cost 10 cents. 99.9% off is 1 cent. 100% off is $0. 200% off is they pay me $10? No? $\endgroup$ – bodacydo Aug 20 '15 at 19:26
  • $\begingroup$ If George is driving at 20 mph and Ralph is driving at 60 mph then Ralph is driving 200% faster than Ralph. If one battery charges in 15 minutes then the speed of that charge is 1/15 batteries per minute. Another battery charging in 5 minutes is charging at a speed of 1/5 batteries per minute. That's three times faster, or 200 % faster. $\endgroup$ – David C. Ullrich Aug 20 '15 at 19:27
  • $\begingroup$ Ooooh - does it mean it's all relative? $\endgroup$ – bodacydo Aug 20 '15 at 19:28
  • $\begingroup$ So if one battery charged in 60 minutes, and the new battery charged in 20 minutes, then the new battery is charging 200% faster, right? $\endgroup$ – bodacydo Aug 20 '15 at 19:28
  • $\begingroup$ Seems that way to me, yes. $\endgroup$ – David C. Ullrich Aug 20 '15 at 19:29
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"Faster" refers to speed (units of something per unit of time) or velocity (units of something per unit of time, with a directional component).

In this case you are talking about charging a battery in $5$ minutes. The speed of that operation is $(1.0\,\text{battery})/(5.0\,\text{minutes})$, or $0.2$ batteries per minute.

$100\%$ faster than $0.2$ batteries per minute would be $0.4$ batteries per minute, or charging a single battery in $2.5$ minutes.

$200\%$ faster than $0.2$ batteries per minute would be $0.6$ batteries per minute, or charging a single battery in $1\frac{2}{3}$ minutes, or $1$ minute and $40$ seconds.

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  • $\begingroup$ This is easy to understand now. Thanks. $\endgroup$ – bodacydo Aug 20 '15 at 22:32
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You're applying the percentage to the wrong magnitude. 100% faster means speed is increased by 100%, i.e., double speed. So, you don't reduce time by 100%, but by 50%. If it's 200% faster, you get 3x speed.

In your example, you'd charge the battery in 2.5 and 1.6 minutes, respectively.

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Interesting question. When people say that, i don't think it is strictly mathematical. As far as i'm concerned what is implied is the following.

$100\%$ faster charging = half charging time required = $\frac{5}{2} = 2.5$ minutes to charge (twice as fast).

$125\%$ faster charging = 2.25 times less charging time = $\frac{5}{2.25} = 2.22$ min. to charge.

$200\%$ faster charging = 3 times less charging time = $\frac{5}{3} \approx 1.67$ min. to charge.

You can't say you can charge something in less than $100\%$ faster, because that would be slower, not faster.

On the other hand you can say that something took $x\%$ times slower to occur. The speed at which it happened is less, thus the time is greater than the way it originally happened. So the extra time is added to the original time.

$50\%$ slower charging = 50% more charging time required = $5 + \frac{1}{2}\times 5= 7.5$ minutes to charge.

$75\%$ slower charging = 75% more charging time required = $5 + \frac{75}{100}\times 5 = 5 + \frac{3}{4}\times 5 = 8.75$ min. to charge.

$166\%$ slower charging = 166% more charging time required = $5 + \frac{166}{100}\times 5 = 5 + 1.66\times 5 = 13.3$ min. to charge.

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"350% faster" is promotional vocabulary for "3.5 times faster". A ratio of 3.5 between speeds or times.

This is inconsistent with the usage for smaller percentages. 10% faster would mean subtraction; the action happens in 0.9 times the original duration.

Where is the boundary between the two modes of speech? I'd guess the descriptions switch to the type in the question before the improvement reaches 50% faster (subtractively) = 200% faster (multiplicatively).

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Percentage is nothing but a number: $100\% = 1$. So, $200\% = 2 \cdot 100\% = 2$. Saying that something is $200\%$ faster is same as saying it is $2$ times faster, which essentially translates to your battery being charged in half the time.

EDIT: It seems that "faster" means $+200\%$ and not $\times 200\%$. Oh, well... luckily enough, it could just as well say $2134321543\%$ faster and probably be equally true.

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    $\begingroup$ 200% faster is not the same as twice as fast; 200% faster than is three times as fast. $\endgroup$ – shoover Aug 20 '15 at 19:29
  • $\begingroup$ Yes but there is also a problem when using the word faster. When i hear $200$% faster, i think its $200$% in addition to what it was before so $100 + 200$% If you want to be for that procentage to have the meaning you explained, I think you should say that something is working at $200$% capacity $\endgroup$ – JamMaster Aug 20 '15 at 19:30
  • $\begingroup$ Yes, it seems I misinterpreted use of the term "faster", but yeah, I can see how "faster = more fast" translates to $+200\%$ instead of $\times 200\%$. $\endgroup$ – Ennar Aug 20 '15 at 19:35
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I think you're being too literal. 100% faster, in this context, means you cut the time in half. Definitely not the correct usage though. For instance, saying "That shirt is 100% off" when you mean "that shirt is half off" is really bad form.

If you see that battery $x$ is charging $b$% faster, which as a proportion is $p=1+{{b} \over {100}}$, what is meant is that the new battery charging time $x_1$ multiplied by the derived proportion $p$ yields the original batteries charge time $x_0$. In math symbols,

$$x_0=x_1 \cdot p$$ $$\Rightarrow x_1={{x_0} \over p}$$

The original battery charges in 5 minutes, and you get a method to charge the battery 100% faster, you get,

$$x_1={{5} \over {p}}={{5} \over {1+{{100} \over {100}}}}={5 \over 2}=2.5$$

Which gives a charge time of two minutes and thirty seconds.

Basically, when you're in the "wild", the real world, you have to give people some leeway in how they use mathematical jargon. Don't even get me started on the use of the words integral and integrated in common usage...

Fun Fact: $-5$ minutes means you travelled back in time...

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    $\begingroup$ Yea so this guy invented a time machine to charge his battery 200% faster. What a douchebag. $\endgroup$ – bodacydo Aug 20 '15 at 19:30

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