I am having a problem with a problem

[dieseltaylor]

A fruit vendor bought 100 pounds of berries for £2.00 per pound and expected to double his investment by selling the berries for £4.00 per pound at an open-air market. The vendor only managed to sell 50 pounds of berries the first day and he sold the remainder on the second day. The fresh berries had a content of 99% water, but because of the hot weather, the berries dehydrated and contained only 98% water on the second day. How much profit did the vendor make?

Now I am sure some bright spark can explain the answer waaay below. But before you look give it a go.

The vendor had a hundred pound profit.

Explanation:

The decrease of water content from 99% to 98% does not reduce the weight by 1%, but by a significant 50%. The first day sale of berries with 99% water is: 50 pounds × £4.00/pound = £200.00. The vendor covers his cost on the first day, but does not make a profit.

The second day sale of berries with 98% water is: 25 pounds × £4.00/pound = £100.00. The profit is the total sales (£300.00) minus the cost (£200.00).

[hcrof]

First day sales = £200 (covers costs)

1% of 50lbs = 0.01*50 = 0.5lbs

so that means we have 0.5lbs solid, 49.5lbs water.

49.5/0.5= 99 so ratio = 99:1

if ratio is now 98:2 (49:1) and the mass of the solid doesn't change, we can reverse that equation:

w/0.5= 49 so w= 24.5lbs (of water)

24.5 + 0.5 = 25lbs of berries

[check: 24.5/25= 0.98 (98%, correct)]

25*4= £100 profit

That took me a little while but there is no harm in keeping the maths skills sharp! It seems pretty counter-intuitive though doesn't it?

Where did you find the question?

[dieseltaylor]

http://www.theengineer.co.uk/?cmpid=TE01&cmptype=newsletter

However this is appears to me to be mathematical idiocy. As the berries are indivisable - that is the dry and water content are as one - you cannot divorce them.

In any event if 1% loss of water is equal to 25lbs then presumably if another 1% of the original water lost means complete dehydation. That is not possible so it must be a question of syntax misuse.

After all can a 1% reduction of a component part really mean a 50% reduction in total weight.

[hcrof]

I don't understand it either - I was trying so hard to prove the question wrong and went through it a couple of times but the maths is there and proven.

I suspect that because the solids are so insignificant, there needs to be a really dramatic loss in the water in order for the solids to begin to take up another percentage point. In other words, the solids have to double in relation to the water to take that 1%. The next 1% would take much less water loss until eventually the roles are reversed and a tiny change in water makes a huge change in the percentages.

Or to put it another way, we are working at the very edge of an asymptote on a graph of ratios.

Like I said - not intuitive at all, but that is what made it challenging

[Grunt0302]

Let's carry the logic a step further...let's say the berries have only 80% water the next day.

The ratio is now 80:20 or 4:1.

w/0.5 = 4 so w = 2lbs of water.

2lbs + .5lbs = 2.5lbs

check: 2/2.5 = .8 (80%, Correct)

So 1% water weighs 24.5lbs and 19% weighs 2lbs. That doesn't make sense.

The fact of the matter is that if you have 50lbs of something that has lost 1% of it's weight, you have lost .5lbs. It's really that simple.

[Elmar Bijlsma]

But it hasn't lost 1% of the weight.

The amount of water in the product has been reduced to the point it no longer outnumbers the fruit 99 to 1 but rather 98 to 2. In other words, nearly half the water is gone.

[Affentitten]

Except that in reality the good old British barra boy would say "Two parns? Just unda. Awight?" So with the rounding up, the profit would be like a thousand quid.

[dieseltaylor]

The ratio thing is the ballocks part. If the weight is 99% water and 1% matter than each per cent is 1lb. I agree with Grunt. The ratio is the distractor here as it is establishing a false relationship.

Somehwere there must be a proof on how not to use ratios!!.

[hcrof]

Percentages are ratios - it is a ratio of x:100

http://www.mathgoodies.com/lessons/vol4/meaning_percent.html

As soon as you get out of the extreme edge of the percentages it becomes quite intuitive

[Skepticist]

Bored lurker here.

These are very wet berries.

Talking about percent water content makes sense if the value is low - soil containing 5% water by weight - but I think what's confusing people is that when there is very high water content we usually think in terms of concentration of the other stuff in the water, i.e. mg/L or ppm.

99% water content by weight is a 1% concentration of berries. 98% water content is a concentration of 2% berries. To double the concentration of a solution you can imagine that you would have to evaporate off half the water.

Does that help?

Edited to add: It would also help if the units of weight and units of currency didn't have the same name. Not that you have to adopt the Euro. At first I thought it was some sort of trick question with the price.

[Elmar Bijlsma]

Hiya, and welcome to the board/bored.

It definitely helps if you think less in numbers and more on what is really going on.

[Grunt0302]

But if you've boiled off half the water, you've lost 50%, not 1%. That's why I agree with dieseltaylor, the ratio of water to solid berry is not relevant, it's a red herring. Well, that and because he agreed with me.

There's a name for a mathmatical equation that can logically lead to a silly conclusion. But it's not coming to me at the moment.

Here's another example; Three men check into a hotel. The front desk clerk tells them it costs $30 per night. They each pay $10 and go to their room. Later the cleark realizes he made a mistake and should have charged only $25. So he gives five $1's to the bell boy and tells him to take it to the men. On the way the bell boy realizes he can't split this evenly among the men. So he decides to give the men $3 and to keep the remaining $2 for himself. So in effect each man has paid $9. 3 x $9 = $27. Plus the $2 the bell boy kept is $29. What happened to the other $1? The answer of course is that (3 x 9) + 2 = 29 is the wrong equation. The correct equation is 25 + 3 + 2 = 30.

I think something like this is happening to our berries.

[Blackhorse]

It's fantasy.

There were no berries that were 99% water and that lost that much money, etc etc. It's a fictional problem created to highlight a math problem.

It's like asking, "How many miles per hour did John run, if it took him 30 minutes to run 2,000 miles?"

Sure, it's a simple ratio...

It's a fantasy ratio, and that is where the problem lies. You are accepting at face value that this can really happen, when it cannot.

Some of the most moist berries come in at 65% moisture.

Blueberries when dried to 2% moisture, weigh 10% of their fresh weight..that is the reality...not some made up crap designed to highlight a math problem.

[Grunt0302]

At the risk of beating a dead horse...no offense intended Blackhorse...consider this:

You have a box with 49 oranges and 1 apple. If you are asked to remove 1% of the oranges, how many do you take out? Note there is no relationship between apples and oranges. You aren't being asked to make the apples a certain ratio of the oranges or vice versa. So, as in the blueberry problem, injecting the ratio of water to blueberry solid introduces a false relationship and that's why the math doesn't work out over the other percentages.

[Michael Emrys]

I'm glad I didn't even try to figure this one out. My remaining brain power is too valuable to waste on non-productive problems like this. Oh to be young again...

Michael

[dieseltaylor]

Understanding statistics ...

The media failed to mention the statistical trickery at work. Cardiovascular events were reduced by an absolute, paltry 0.9% with Crestor use. Using a few tricks of the statistics trade, this bland number was converted into the more lucrative, "relative risk reduction" of 53%. This trickery happens so fast, it's like watching magician David Blaine pull his heart out of his chest. You don't know if it's real or just a cheap magic trick. Dr. Mark Hlatky of Stanford shows that it's just a trick.

The drug maker would make 500000$ per patient if the lifetime regime was adopted.

http://www.newswithviews.com/Ellison/shane144.htm

[Pak40]

You're all wrong. Any open air market salesman would have divided the berries by the pound on the first day into separate crates because he would have hoped to sell them all. He wouldn't have re-weighed them on the second day. What would be the purpose? Therefore he would have made the a 200 profit. Also, wouldn't a wise seller keep his berries wet, increasing his profits further?

[Lt Belenko]

Initial cost: 100 X 2.00 = 200

Day 1: sell 50 @ 4.00 = 200.

Day 2: Start with 50 pounds lose 1% for evaporation 50-1% = 49.5

Sell 49.5 lbs @ 4.00 = 198

Profit is (Day 1 200) + (Day 2 198) - (Cost 200) = $198 not really but dollars British money thingys.

[Elmar Bijlsma]

But you aren't losing 1%. It doesn't say that.

What it does say is that proportionally there is twice as much fruit in the berry on the second day in the berry than there was on the first, when compared to the amount of water. But the amount of fruit in real terms has remained constant, we are told. Thus this significant change in ratio must have come at the expense of the water.

[Elmar Bijlsma]

Okay, let's explain this with matter that is more easily understood.

The Rat Patrol bounce a convoy of Germans. They are badly outnumbered, only 1 of a hundred combatants is a Rat Patrol Member. Combat occurs. As in their habit, the Rat Patrol doesn't lose any members. Hurray! The Germans suffer such casualties that we are told at the end 2 out of every 100 surviving combatants are members of the Rat Patrol. Or 98 per hundred are German.

Hang on! 2 out of 100 seems to suggest that The Rat Patrol got bigger in combat. That obviously can't be true. So let's make that a mathematically cleaner and easier on the brain 1 out of 50.

So post combat it's really 1 out of every 50 combatant that is Rat Patrol. To recap: At start there were 99 Germans per Rat Patrol member, post combat there's only 49 Germans per Rat Patrol member.

At start: 99 Germans and 1 Rat Patrol member= 100

Post combat 49 surviving Germans and 1 Rat Patrol member=50

Thus you can see, half the participants in the combat got killed.

[hoolaman]

It took me about 5 minutes to work out the answer as stated, but most of that time was spent wondering what was wrong.

I think the unintuitiveness of the problem is because the wording leads you to assume the more believable proposition that the berries lost 1% of their water weight. You want to think that the berries have gone from 99:1 to 98:1 and the other 1% water is gone.

Instead the definition of the berries on day two is not related at all to the berries on day one.

If you want to be picky you could also argue that it isn't made clear that it is a mass/mass ratio.

99% water content by weight is a 1% concentration of berries. 98% water content is a concentration of 2% berries. To double the concentration of a solution you can imagine that you would have to evaporate off half the water.

Does that help?

Yes I find that a very good explanation, though I am a chemist, so maybe thats just me.

[Elmar Bijlsma]

Does that help?

Yes I find that a very good explanation, though I am a chemist, so maybe thats just me.

Stand by for mister picky...

It's not half the water that evaporates. It's slightly more then half. As can be seen in the result, half the weight of the berry (in water) evaporates . Thus it can't be half the weight of water, seeing as the amount of fruit doesn't change.

[hoolaman]

Well I took it to be an approximation for the sake of illustration, but since you asked:

If you are talking about a chemical solution by volume which is a slightly different real world situation, for dilute solutions the solute is considered completely integral within the volume of water, so to double the concentration you'd go from 1g/1000mL to 1g/500mL and evaporate half the water volume.

[Michael Emrys]

This very nearly the most stupid puzzle that has ever appeared in these pages.

Michael

[dieseltaylor]

There is stupider! ?

Do tell : )

Conceptually the puzzle challenges us to ignore what we know about dehydration in 24 hours and also what condition berries must be in to be saleable at $4. So your mindset is already framed when they ask a stupid question! The berries would be virtually unsaleable.