Coffee tea riddle

[Born2bwire]

I don't think the size of the spoon affects it either. Even if the spoon is the size of the entire cup there is no change.

100mL cups. 100mL spoon.

Coffee cup has 0, tea cup has 100mL tea, 100mL coffee - 50/50
Take another spoonful and you have 100mL of a 50/50 mix so you still end up having perfectly equal differences, and in this case, the mixtures are the same as well.

Technically, that should only prove a less than or equal to inequality. If the comparison of the mixture ratios is dependent upon the spoon size, then one would conclude that the limiting case is when the spoon is the same size as the cups. I do not think that you could conclude that the limiting case is going to be characteristic of all cases if the spoon size was a factor.

Of course it isn't here. One can easily prove algebraically that is not the case. Specifically, if x is the volume of the cup and y is the volume of the spoon, then the final ratio of coffee to tea in the tea cup and the ratio of tea to coffee in the coffee cup both become y:x and the volumes are xy/(x+y) and x^2/(x+y) where the latter is the volume for the liquid that was originally in the cup.

The final ratio of y:x is a rather tempting answer. Its simplicity suggests that one might be able to logically reason this without explicit mathematics but if that is possible it is not within the span of my patience to figure it out.

 

[TuxDave]

I don't think the size of the spoon affects it either. Even if the spoon is the size of the entire cup there is no change.

100mL cups. 100mL spoon.

Coffee cup has 0, tea cup has 100mL tea, 100mL coffee - 50/50
Take another spoonful and you have 100mL of a 50/50 mix so you still end up having perfectly equal differences, and in this case, the mixtures are the same as well.

That's what I wrote. It doesn't matter how big the spoon is and it doesn't matter how well mixed the cups are when you take a scoop. They will be equal.

 

[TridenT]

Cough... I answered this in the thread perfectly.

I had my little, "wtf, obvious" moments but I thought it through and wrote it out. I was like, "Meh... the math ends up being same. Lame."

 

[TuxDave]

Cough... I answered this in the thread perfectly.

I had my little, "wtf, obvious" moments but I thought it through and wrote it out. I was like, "Meh... the math ends up being same. Lame."

Hell, you can change the problem where the guy repeats that back and forth process as many times as he wants and on each pair of transfers decides to use a different spoon, the answer is that they still will be equal.

 

[deadlyapp]

Technically, that should only prove a less than or equal to inequality. If the comparison of the mixture ratios is dependent upon the spoon size, then one would conclude that the limiting case is when the spoon is the same size as the cups. I do not think that you could conclude that the limiting case is going to be characteristic of all cases if the spoon size was a factor.

Of course it isn't here. One can easily prove algebraically that is not the case. Specifically, if x is the volume of the cup and y is the volume of the spoon, then the final ratio of coffee to tea in the tea cup and the ratio of tea to coffee in the coffee cup both become y:x and the volumes are xy/(x+y) and x^2/(x+y) where the latter is the volume for the liquid that was originally in the cup.

The final ratio of y:x is a rather tempting answer. Its simplicity suggests that one might be able to logically reason this without explicit mathematics but if that is possible it is not within the span of my patience to figure it out.

the only time the spoon size would matter is if the spoon was BIGGER than the cups.
IE you end up taking the full amount of one cup and add it to another - although the spoon is not full, and then taking a full spoon from the other cup, which does have enough volume to fill it.

 

[zinfamous]

I've never heard this riddle...but going by molarity, there will absolutely be more coffee in the tea, being that it was added "pure," with a theoretical ~66/33 tea/coffee amount added back to coffee.

I don't even see a reason to calculate this....

 

[zinfamous]

I've got no idea what you're trying to say here...

He's talking about General Chemistry, basically.


many people have failed to educate this country.

 

[deadlyapp]

zin, unfortunately I think you're wrong if you do it based purely on volume and as long as the mixtures are perfect mixtures.

It's already been proven in the thread but I'll do it once more.

100mL cups - one coffee one tea.
25ml spoon.

Coffee cup now has 75mL of pure 100% coffee
Tea cup has 125mL 80% tea 20% coffee.

Assuming a perfect mixture, if you take 25mL out of the mixture, you will have 20mL tea and 5mL coffee

Adding this back into the original cup you will have 80mL coffee and 20mL tea
The tea cup will have 80 mL tea and 20mL coffee.

This question is purely based on concentration.

 

[zinfamous]

zin, unfortunately I think you're wrong if you do it based purely on volume and as long as the mixtures are perfect mixtures.

It's already been proven in the thread but I'll do it once more.

100mL cups - one coffee one tea.
25ml spoon.

Coffee cup now has 75mL of pure 100% coffee
Tea cup has 125mL 80% tea 20% coffee.

Assuming a perfect mixture, if you take 25mL out of the mixture, you will have 20mL tea and 5mL coffee

Adding this back into the original cup you will have 80mL coffee and 20mL tea
The tea cup will have 80 mL tea and 20mL coffee.

This question is purely based on concentration.

cool....


...but what's the molar weight of coffee compared to the molar weight of tea?

:sneaky:

 

[Born2bwire]

the only time the spoon size would matter is if the spoon was BIGGER than the cups.
IE you end up taking the full amount of one cup and add it to another - although the spoon is not full, and then taking a full spoon from the other cup, which does have enough volume to fill it.

I didn't say that the spoon size mattered. However, I would not assume that taking the limiting case should be taken as a valid proof without first proving that the size of the spoon was not factor.

 

[sdifox]

You have two awful tasting drinks.

well, the proportions are off, but in HK they do drink coffee and tea mix (with milk and sugar). I can't stand the hot one but I love the cold version.

 

[sdifox]

My assumptions

cup contains 9 teaspoon of liquid, with room for 1 more spoonful (just to foil the spill crowd)

It's a closed system, meaning the source of the spooned liquid is the two cups in question.

This tells us the first spoonful is 100% coffee, the second is 1/10 coffee, 9/10 tea, thus the tea cup has more coffee in it than the coffee cup has tea.

The other scenario

cups contains 9 teaspoon full of liquid
1 teaspoon of coffee, not from the cup of coffee

Total liquid volume is 19 units

teaspoon of coffee is added to tea cup and mixed, making that mixture 1/10 coffee, 9/10 tea.

take one teaspoon of that mixture. Tea cup is left with 9 units of 1/10 coffee, 9/10 tea.
and add to cup of coffee then stir

the coffee cup is now 10 units, ten of which is the original coffee in the cup and the other unit is made up of 1/10 coffee, 9/10 tea.

9 C + 1/10 C = 9.1C
0.9T

Constitution of the two cups
Cup of Tea = 8.1T + 0.9C
Cup of Coffee = 9.1C+0.9T

0.9/9>0.9/10

More coffee percentage in tea than more tea percentage in coffee.

 

[Born2bwire]

My assumptions

cup contains 10 teaspoon of liquid, with room for 1 more spoonful (just to foil the spill crowd)

It's a closed system, meaning the source of the spooned liquid is the two cups in question.

This tells us the first spoonful is 100% coffee, the second is 1/11 coffee, 10/11 tea, thus the tea cup has more coffee in it than the coffee cup has tea.

Except you're adding it to 9 units of coffee, so you have 99/11+1/11 = 100/11 units of coffee in the coffee cup and 10*10/11 = 100/11 units of tea in the tea cup (which you readily know since the ratio of tea to coffee in the tea cup does not change when you spooned out a teaspoon).

 

[Fenixgoon]

ok now i'm waiting for this to turn into plane vs. treadmill epic...

 

[ShawnD1]

Think about this;
Each cup has 100ml of is respective drink.
You use a 100ml spoon to move coffee from cup 1 to cup 2 and stir.
Use the same 100ml spoon to move 100ml of that mixture back to cup 1.

lol. As ridiculous as this is, that's an interesting perspective. I like it

 

[sdifox]

Except you're adding it to 9 units of coffee, so you have 99/11+1/11 = 100/11 units of coffee in the coffee cup and 10*10/11 = 100/11 units of tea in the tea cup (which you readily know since the ratio of tea to coffee in the tea cup does not change when you spooned out a teaspoon).

Thy math sucketh

/facepalm.

 

[darkxshade]

Wouldn't it be easier to think of it this way: If you had half a cup of coffee and half a cup of tea and you pour the 1/2 cup of coffee into the tea, mix it and pour half back into the empty cup, which has more coffee and which as more tea? The answer is that they are both equal.

 

[sdifox]

Wouldn't it be easier to think of it this way: If you had half a cup of coffee and half a cup of tea and you pour the 1/2 cup of coffee into the tea, mix it and pour half back into the empty cup, which has more coffee and which as more tea? The answer is that they are both equal.

Sure, change the question.

 

[Lonyo]

Sure, change the question.

That's no different to doing the 100ml of liquid and 100ml spoon method, the result is the same, you are just eliminating using the spoon (and the amount the spoon transfers as a % of the whole doesn't matter as long as the spoon does not exceed liquid volume).
So while he changed the wording, the effective meaning and result stayed the same.

 

[sdifox]

Where is the poll?

 

[sdifox]

That's no different to doing the 100ml of liquid and 100ml spoon method, the result is the same, you are just eliminating using the spoon (and the amount the spoon transfers as a % of the whole doesn't matter as long as the spoon does not exceed liquid volume).
So while he changed the wording, the effective meaning and result stayed the same.

Ratio of liquid transfer is changed, that is the important bit.

 

[actuarial]

My assumptions

cup contains 9 teaspoon of liquid, with room for 1 more spoonful (just to foil the spill crowd)

It's a closed system, meaning the source of the spooned liquid is the two cups in question.

This tells us the first spoonful is 100% coffee, the second is 1/10 coffee, 9/10 tea, thus the tea cup has more coffee in it than the coffee cup has tea.

After step 1, your volumes are:
Coffee cup: 8 tsp coffee = 8 tsp total
Teacup: 9 tsp tea, 1 tsp coffee = 10 tsp total

1 tsp from the Teacup would be .1 tsp coffee, .9 tsp tea

After step 2, your volumes are:
Coffee cup: 8.1 tsp coffee, .9 tsp tea
Teacup: 8.1 tsp tea, .9 tsp coffee

Edit: So the ratios are the same.

 

[Blackjack200]

ok now i'm waiting for this to turn into plane vs. treadmill epic...

IMO, this will be the new airplane treadmill on ATOT.

It's a good one because at first blush, it seems obvious that there is more coffee in the teacup than vice versa.

But when you pause and just think of it this way, you can tease out the truth:

Both cups begin and end with the same volume of liquid, so however much coffee is in the tea must be the same as the amount of tea in the coffee.

I like this.

 

[dank69]

IMO, this will be the new airplane treadmill on ATOT.

It's a good one because at first blush, it seems obvious that there is more coffee in the teacup than vice versa.

But when you pause and just think of it this way, you can tease out the truth:

Both cups begin and end with the same volume of liquid, so however much coffee is in the tea must be the same as the amount of tea in the coffee.

I like this.

The ratio will always be the same, but if you want to get really anal about the phrasing of the OP, you can get different volumes:

300ml cups
100ml tea in one cup
100ml coffee in one cup
150ml spoon

100ml coffee into tea cup
150ml tea/coffee mixture into coffee cup
75ml tea and 75ml coffee in coffee cup
25ml tea and 25ml coffee in tea cup

There is more tea in the coffee than there is coffee in the tea.

 

[actuarial]

General Solution (I'm really trying to distract myself waiting for exam results right now):

Cup volume: X
Spoon volume: Y

After step 1, your volumes are:
Coffee cup: X - Y coffee = X - Y total
Teacup: X tea, Y coffee = X + Y tsp total

Y from the Teacup would be Y^2 / (X + Y) coffee, XY / (X + Y) tea

After step 2, your volumes are:
Coffee cup:
Coffee: X - Y + [ Y^2 / (X + Y) ] = (X^2 - Y^2 + Y^2) / (X + Y) = X^2 / (X + Y)
Tea: XY / X + Y
Ratio Tea to Coffee: XY / X^2
Teacup:
Tea: X - [ XY / (X + Y) ] = X^2 + XY - XY / (X + Y) = X^2 / (X + Y)
Coffee: Y - [ Y^2 / (X + Y) ] = XY + Y^2 - Y^2 / (X + Y) = XY / (X + Y)
Ratio Coffee to Tea: XY / X^2