Drinking Soda

Algebra Level 1

When I was a teenager, a friend and I stole soda from our parents. I took the soda bottle, drank half of what's there, and then passed it to my friend. My friend drank half of what was left, and passed it back to me. I again drank half of what was left, and passed it back to him.
We kept up with this until the entire soda bottle was empty.

Approximately what percentage of the soda did I drink?

80% 67% 50% 75%

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13 solutions

Michael Mendrin
Jul 5, 2015

Whatever you drank, your friend drank only half as much. Ergo, you had 2/3 of the total, your friend 1/3.

Moderator note:

So simple and elegant.

There's another approach that applies the sum of geometric progression but that's much lengthier compared to yours!

Theoretically, can we (I and my friend) drink til its empty (in a finite time)? Doesn't this resembles Zeno's Paradox?

It cannot be wholly drunk in finite time, if each drink takes a finite time, however brief. For the chalice to be emptied in this fashion, ultimately the friends must partake each drink in infinitesimal time.

This assumes an ideal soda which is infinitely divisible.

In Zeno's Paradox, this is is not a problem because the time it takes to traverse a certain distance is proportional to it, so that as distances become infinitesimal, so does the times it takes to traverse them.

Michael Mendrin - 5 years, 11 months ago

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But what if the time that it took to transfer the chalice and drink the soda also halved each time?

(It is physically unlikely though.)

Chung Kevin - 5 years, 11 months ago

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"It is physically unlikely though" is the whole point. If the time it takes for each drink is proportional to the amount drunk, including the time for transfer between the drinkers, then it can be all drunk in finite time. The big difference is that runners routinely make it to the finish line in finite time, whereas something like this is not likely to ever happen.

Michael Mendrin - 5 years, 11 months ago

See my calculation below how long it takes to finish off the bottle: 7 minutes and 10 seconds. :)

Rainer Schrom - 5 years, 3 months ago

The only problem with this is eventually you will get down to the smallest molecule (not sure of the right term) and not be able to drink half of it thereby finishing the soda in a finite amount of time.

Nadeem Merchant - 2 years, 8 months ago

This seems like a supertask to me. But it can be achieved assuming that each half of the remaining soda was drunk half as long as the previous one. For example, let's say they were to finish the task in 1minute, the first half will be drunk in half a minute, the second drink will take only 15 seconds, the next being, 7.5 and soon. They may drink from the bottle infinite times but we know that at the end of one minute, the bottle will be empty.

Billy Gene Balsamo - 5 years, 3 months ago

67% can not be the right answer. In fact a whole number is impossible to get.

Santosh Biswas - 5 years, 7 months ago

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The question states "Approximately".

Chung Kevin - 5 years, 7 months ago

This problem reminded me a lot of the trains and fly problem so I just picked 2/3.

Jared Engler - 4 years, 7 months ago

I am sorry I dont understand your solution. Can you please elaborate? Thank you

mohan nayak - 3 years, 5 months ago

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"Whatever you drank, your friend drank only half as much."

so if you drank 2x, your friend drank x thus total drink amount is 3x and your portion is 2/3

Anirban Das Deb - 3 years, 4 months ago
Jason Everetts
Jul 9, 2015

You can do the math, but for this problem, given the choices, I think process of elimination gets it done faster. It can quickly and easily be determined that you had more than 50 but less than 75 percent of the soda. You would have 50 if you stopped drinking after the first turn. You would have had 75 if you drank all of what was left on your second turn. That eliminates 50, 75, and 80, leaving only the correct answer, 66.

Yo same!! :P as mine :S

Tasin Winchester - 5 years, 3 months ago

great, man

Arian Attanazo - 3 years, 7 months ago

My reasoning, too! (Thank you, “approximately.”) But on reading other replies, I wondered just how approximate 67% was. So I played out the scene— I drank 50%, leaving 50%. My friend drank half of 50%(25%), leaving 25%. I drank half of 25%(12.5%), leaving 12.5%. (62.5% drunk so far.) My friend drank half of 12.5%(6.25%), leaving 6.25%. (31.25% drunk so far.) I drank half of 6.25% (3.125%), leaving 3.125%. (65.625% drunk so far.) My friend drank half of 3.125%(1.5625%), leaving 1.5625%. (32.8125% drunk so far.) ... (You can take it from here.) .... So in light of “approximately”, looks like 67% (66 2/3%, leaving 33 1/3% for my friend) is decidedly the best answer.

Phillip Wagoner - 2 years, 8 months ago

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Yes, this is more precise.

Vijay Simha - 11 months, 2 weeks ago

This was EXACTLY my reasoning

Kristin Davis - 5 years, 11 months ago

You and you friend drank in the form of a GP with common ratio 25℅ . sum if an infinite gp = a/1- r thus you practically drank 66.67℅ Thank god this question is not integer type

Good observation!

Chung Kevin - 5 years, 11 months ago

Wow. .That's an interesting idea

ALI AYUBA - 2 years, 5 months ago

You would drink n = 1 1 2 2 n 1 = 2 3 \sum_{n=1}^{\infty} \frac {1}{2^{2n-1}} = \boxed{\frac 23}

Uttkarsh Kohli
Jul 12, 2015

you drink50

your friend drinks 25

you drink 12.5

your friend drinks6[aprox.]

you drink 3

your friend drinks1.5

..................................

              so 50+12+3=65

Approximating the answer would be a possible way of solving this problem.

Chung Kevin - 5 years, 11 months ago

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atleast upvote it

uttkarsh kohli - 5 years, 11 months ago

What if the options in the question were 65%, 66%, 67%, 68%?

Vaibhav Priyadarshi - 3 years ago

By the time you have gotten to the point where you drink say, 3%, you will not be able to consume another 3%, so in your example, you won't ever reach 68%.

Spencer May - 2 years, 9 months ago

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It is true that we "won't ever reach 68%". The exact answer is 66 2 3 % 66 \frac{2}{3} \% , and not 68 % 68 \% .

The first 3 sips gives us 50 + 12.5 + 3.125 = 65.625 50 + 12.5 + 3.125 = 65.625 , and so we only need just over 1% more to go.

Chung Kevin - 2 years, 9 months ago
Marc Pfaff
Mar 6, 2018

Every round after the first one, you will have 1/4 of what you just had the round before (since your friend will have had 1/2 of what you had and you will have 1/2 of that next, i.e. 1/2*1/2=1/4). So in the second round, you will have 1/4 of what you had in the first round, in the third round, you will have 1/4 of what you had in the second round etc. Sum all those up (2nd round until infinity) and factor out 1/4 and you get 1/4 of what you will have had total after infinite rounds, including the first one. But what will you have had total after infinite rounds, including the first one? In the first round you have 1/2 of what's there total. Therefore, the total sum (x) of what fraction of the initial amount of beverage you will have had after infinite rounds equals 1/2 plus 1/4 of that very same total sum, put in algebraic terms: x = 1/2 + 1/4x. Solving for x you get x = 2/3.

Sterling Klein
Jul 11, 2015

It's a simple limit problem. At what number is it impossible to consume any more? I see it as you start with 50%. Then skip 25 and directly add the 12.5% making it 62.5% gone. Now, your friend has 6.25% and you consume 3.125% of the total compared to the beginning. Add 3.125 to 62.5 and you get 65.625. The numbers will get smaller and smaller so the limit won't let you go too far past 66%. It should theoretically pass 66% but that's a far closer number than 75% would've been.

That's a good observation. Because the answer options are far apart, we can make an estimation of the final answer.

Chung Kevin - 5 years, 11 months ago
Mohan K Basnet
Mar 7, 2016

Approximately 67%

Usman Saeed
Feb 18, 2016

Suppose Total amount in bottle is 100 ml. You drink 50 ml, your friend 50% of 50 ml i.e 25. Again you drink 50% of 25 and up to so on. So, you drink=50+12.5+3.125+..=67

Achille 'Gilles'
Nov 4, 2015

Considering that every time both have a drink, the share of the first drinker will be twice of the second so it is a 2/3 : 1/3 ratio or 66,67%.

Kenny Duran
Nov 3, 2015

I guess I'm kind of a cheater. Your friend drinks 1/4 + 1/16 + 1/64 +... You drink 1/2 + 1/8 + 1/32 +... 1/4 + 1/16 is already around .31. So your friend drank more than 30%. 100 - 30 leaves you with 70% left to drink yourself, so only 67% and 50% are feasible choices. 1/2 + 1/8 is already more than .5. So you drank more than 50%. So 67% is the answer.

A simple geometric progression with r being 1/4. A/(1-r) with a being 1/2 gets you 2/3 or 66%

Joey Titus
Jul 19, 2015

You can't finish it cause u only drink half of what's left, which means you can never finish it

That's really an interesting observation! :)

However, practically, it's akin to lim x -> 0..... and, like the question asks, it's only about "approximation".

Sure, if you start dividing 1 by 2 and keep on going, you won't get to 0. Though at some point, the figure would be so small that it would barely make an arithmetic difference to the calculation (read: 0.0000000000582076609134674072265625), let alone sipping that amount of milliliters from the soda bottle!

Hope that makes sense...

Dhruv Saxena - 5 years, 7 months ago

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Ok- I think one can actually compute very exactly how long it takes for the soda bottle to become empty. And it does not even really matter much what size soda bottle they start with, e.g. whether it is 1 liter or half a gallon, or anything of that sort. The only assumption would be the time it takes them on average to physically hand over the bottle and drink their share. Say that time is 5 Seconds. How long does it take until they are done?

Rainer Schrom - 5 years, 3 months ago

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Water has a molecular weight of 18 (H2O; 2x1 for the Hydrogen atoms and 1x16 for one Oxygen atom), and so one mole of water is 18 grams. One Liter of water has 1000 grams. The number of moles is 1000/18 = 55.55 moles. There are 6.022 x 10^23 molecules in a mole. The number of molecules is therefore 6.022 x 10 ^23 x 55.55 = 3.34 x 10^25 molecules. 2^85 = 3.87 x 10^25 So it takes (85 + 1) x 5 Seconds = 430 s = 7 Minutes and 10 Seconds until the bottle is empty. And only 5 Seconds more if it's a 2-Liter bottle.

Rainer Schrom - 5 years, 3 months ago

U r exactly right

mohammad amir - 5 years, 2 months ago

Atoms and molecules are finite

Scott Tillinghast - 2 years, 3 months ago

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