## What to do with the Camel Hair: A Harder Math Puzzle

A camel hair broker buys from locals and sells back to merchants. The broker never buys for himself, but upon receiving an order to buy, finds someone who wishes to sell, and charges 2 percent commission to each of them, thereby making 4 per cent on the transaction.

However, by juggling with his old fashioned balance scale that uses weights, he always manages to add to his profit by cheating!

Upon receiving a consignment of camel’s hair he placed the same upon the short arm of his scales, so as to make the goods weigh one ounce light to the pound, but when he came to sell it he reversed the scales so as to give one ounce to the pound short, and thus made $25 by cheating.

On one particular occasion, he makes $25 by cheating in the weight, as he buys a bundle of camel hair with a weight 1 ounce too heavy and sells with one 1 ounce too light.

In this case, how much does he pay for the goods?

*If you can figure it out, enter your answer into the comments section below. Will reveal submissions in a day or 2. thanks*

alederman| Profile July 8th, 2008 - 9:34 amSince his commission results will still be the same (he makes less on the seller, but more on the buyer), he will only be making his profits on 2 ounces of camel hair (gets a free extra ounce initially, then sells that plus the fake ounce).

If his cheating produced him an extra $25, it means the cost per ounce is $12.50.

daniel.leonard| Profile July 8th, 2008 - 11:10 amDo you mean that he makes $25 including his commission, or excluding it (i.e. he makes $25 solely from cheating with the adjustable scale)?

suineg| PUZZLE MASTER | Profile July 8th, 2008 - 11:30 amThis puzzle is cool. 575$ is my answer, explanation:

C= total earning= 25$

C1= earning without cheating

C2= earning just by cheating

P= price per ounce

W= weight of Hair

Equations:

1)25= C1+C2

2)C1= 4%*P*W

3)C2= 4%*P*2

–> if you sum (1) and (2)–> C1+C2= 4%*P*(W+2)–> but for (1)–> 25= 4%*P*(W+2)–>625= P*(W+2); now 625= 25*25

so P has to be 25 and W has to be 23; now how much he pays for the goods I think it is P*W—> 25*23= 575(Answer)

Verification: 575*4%= 23(C1); 50*4%= 2(C2, thats how much

he made by cheating) I think it makes sense but maybe I am wrong.

suineg| PUZZLE MASTER | Profile July 8th, 2008 - 11:47 amUps found an error in my comment, C2 should be 4%*P*1 so P still is 25 but W should be 24; 25X24=600 its the answer so the profit its 625-600=25; wow sorry about that, the rest of the explanation its ok, cool

RK| Founder | Profile July 8th, 2008 - 1:23 pmto clarify- the $25 includes the commision. thanks

michaelc| Profile July 8th, 2008 - 3:32 pmI seem to be struggling with…

“he makes $25 by cheating in the weight, as he buys a bundle of camel hair with a weight 1 ounce too heavy and sells with one 1 ounce too light.”

Unless the camel hair is a liability product (has a negative monetary value/oz), this means he gains 2 ozs of camel hair, for if the scales read 1 oz heavier buying, and 1 oz lighter selling, he loses money! So it must mean the real weight is 1 oz heavier than the scale reads when buying, and the real weight is 1 oz lighter than the scale reads when selling.

I have looked up camel hair on Wikipedia, and it appears to be a valuable commodity. (not to me, but to some folks I guess)

So I would assume that he gains 2 ozs of camel hair in the exchange.

Now, if you figure that is where the 25$ came from, then the camel hair was $12.50 per oz.

In that case, he pays $12.50 * (ro-1)

where ro is the number of real ounces.

If you figure that the 25$ was including commission, (as pointed out), I would take it as the following.

Profit from scale manipulation is 2* cc ($/oz)

where cc is cost of camel hair per oz.

Profit from commission is 4% * cc * ro.

where ro is the real number of ounces.

From what I see in this case, ro and cc are 2 independent variables whereas…

25 = (ro*4% + 2)*cc

The price he pays for the goods is still…

(ro-1)*cc

Probably not what was in mind, but nevertheless what I see.

falwan| Profile July 9th, 2008 - 12:03 am151.5 dollars/pound

bizarette18| PUZZLE MASTER | Profile July 9th, 2008 - 7:27 amI get that there’s a whole range of values it could be depending on how much he buys or the stated cost per ounce. To get $25 profit the true weight he buys has to be (625/cost per ounce) -50. Then the amount he pays out initially is 625x(true weight -1)/(true weight + 50) (or 98% of that once he’s taken his commission). Or 625 – (51 x cost per ounce) (x or not 98%)

RK| Founder | Profile July 9th, 2008 - 12:46 pmok, sorry about this-I didn’t word the problem as I should have. Went back and edited the problem.

suineg| PUZZLE MASTER | Profile July 9th, 2008 - 11:27 pmok, let me see now: so he buys like there are 15 ounces(1 pound – 1 ounce) FOR EVERY POUND and sell like there are 17 ounces(1 pound + 1 ounce) FOR EVERY POUND, in this case the equations may be:

P= price per ounce; NP= number of pounds; NO= number of ounces; GP= goods price

C1= commision of buy

C2= commision of sell

CE= cheating earning

1)C1= 2%*(15)*P*NP

2)C2= 2%*(17)*P*NP

3)CE= 2*P*NP

4)25= C1+C2+CE

5)NO= 16*NP

6)GP= NO*P

7)25= NP*(18)*P*4% + NP*(2)*P

I am tired now RK but I think with that equations you can get the solution, tomorrow I will finish my comment, cool

—>

daniel.leonard| Profile July 9th, 2008 - 11:39 pmThanks for clarifying the question. In my previous understanding of it, it seemed impossible to solve.

There are 16 ounces in a pound. Initially, he pays 98% of the value for 15/16 of the real weight. He then sells for 102% of the value for 17/16 of the weight. The difference equals $25. Therefore:

(.98)(15/16 pound)($x/pound) + $25 = (1.02)(17/16 pound)($x/pound)

it follows:

$25 = (1.02)(17/16 lb)($x/lb) – (.98)(15/16 lb)($x/lb)

and:

$25 = 1.08375($x) – $.91875($x)

$25 = .165($x)

$x = $25/.165 = $151.52 (or $5000/33)

That number represents the actual value of the goods. He would have purchased them for 15/16 of 98% of that, or $139.21. Accordingly, he would have sold them for 17/16 of 102% of that, or $164.21. The difference between these quantities is $25, that which was to be desired. Q.E.D.

daniel.leonard| Profile July 9th, 2008 - 11:55 pmI note that I should have left any reference to weight out of the equation; I say this because I was puzzling over whether the phrasing of my equation assumed that there is one pound of camel hair present. This leaves both the cost per pound and the number of pounds unknown, which is fine.

bilbao| Profile July 10th, 2008 - 3:43 amLet’s see if I can make my explanation not too confusing:

W= weight in ounces (real)

P= price per ounce (real)

B= profit

0.98P= P/1.02= buying price per ounce (2% commission incl)

1.02P= selling price per ounce (2% commission incl)

THIS CASE:

buys: (W-1)* 0.98P

owns real: W * P

sells: (W+1) * 1.02P

Profit= B= sells – buys

Profit without cheating: B= W*1.02P – W*0.98P

Profit by cheating: B+25= (W+1)*1.02P – (W-1)*0.98P

From these two equations we get:

P= $12.5 per ounce

B= 0.04*W*P= 0.5*W (profit function)

In order to make a profit of B= $25–> W=50 ounces

IN OUR CASE:

buys: 49*0.98*12.5 = $600.25

owns real: 50*12.5 = $625

sells: 51*1.02*12.5 = $650.25

profit: 650.25 – 600.25 = $50

profit without cheating (W=50): B= 0.04*50*12.5= $25

Thus by cheating he makes an extra $25

suineg| PUZZLE MASTER | Profile July 10th, 2008 - 8:53 amcontinue—>

forget equation 7 is unneccesary,

put everything in function of GP;

P*NP=P*NO/16–>GP/16 so substitute in equation (4)putting there the values form (1), (2), (3)—–>

25= 2%*15*GP/16 + 2%*17*GP/16 + 2*GP/16—>

25= 0.01875*GP + 0.02125*GP +0.125*GP—->

25= 0.165*GP–> GP= 151,5151515151…

THE NUMBER IS UGLY BUT THAT IS WHAT THE EQUATIONS SAY ITS THE ANSWER JAJAJAJA, MAYBE I AM WRONG, THIS PROBLEM IS REALLY HARD NOT SURE IF I GOT IT RIGHT

michaelc| Profile July 10th, 2008 - 2:17 pm2 ounces he gains per pound! That’s different. Now we have 2 dependent variables.

In a 1 pound exchange (16 ounces), I calculate the price per oz to be ~ $9.47. More exactly $9 + 31/66 per oz.

He buys for $142.05 and he sells for $160.98, thereby making $18.94 by the fudge factor.

His commission would be $6.06 for the buying and selling.

In a 2 pound exchange (32 ounces), the $/oz would be (9 + 31/66)/2. In a 3 pound exchange, ( 9 + 31/66)/3 and so forth.

He still buys for $142.05 which I believe is the question.

michaelc| Profile July 11th, 2008 - 8:50 amFor those that got 151.52 as the answer, did you remember to subtract the price of 1 ounce from the cost?

suineg| PUZZLE MASTER | Profile July 11th, 2008 - 11:22 amhello Michaelc, thats a good reasoning, I see your point, you bought 1 ounces less for each pound you purchased, however I think that if the good price is in function of the commission you are already substracting that value in order to get the initial amount, so no need of substracting that again, however I am not so sure now, you make me doubt jajaja cool, need more time to be completely sure.

suineg| PUZZLE MASTER | Profile July 11th, 2008 - 11:33 amAnd more on, because you sell 1 ounce more for every pound so your camel hair weight for the sell is 1 ounce per pound more than the real weight, so your profit also is 1 ounce per pound * price of the ounce more michaelc, but again as you put everything in function of the commission all that I mention is already considered 98,98….% sure jajajajaja cool, thats is included in the my equation for cheating earning, now 100% sure, hope I help, but now I am 99.99999% sure that the answer is 151.5151515151515151.. to infinite, $ is the unit, however falwan did not understand why your units are $/ounce, for me the units are ($) because its the price for the whole goods(hair) not the unitary price, cool

michaelc| Profile July 11th, 2008 - 2:47 pmMaybe this will help?

ro = real ozs

cc = cost camel hair per oz

bo = buying ozs = ro*(15/16)

so = selling ozs = ro*(17/16)

Tch = total profit from cheating

Tcom = total profit from commmission

25 = Tch + Tcom

Tcom = bo*cc*2%+so*cc*2%

Tcom = (15/16)*ro*cc*2% + (17/16)*ro*cc*2%

Tcom = 4%*ro*cc

Tch = so*cc – bo*cc

Tch = (1/8)*ro*cc

25= (1/8)*ro*cc +4%*ro*cc

ro*cc = 151.52 or 151 + 17/33

That’s what the cost of the goods should have been, but he cheated!

bo = (15/16)*ro

bo*(16/15)*cc =151.52

bo*cc = 151.52*(15/16)

Same as daniel.leonard except taking the commission out.

suineg| PUZZLE MASTER | Profile July 11th, 2008 - 5:47 pmNow I see my mistake, thanks, the answer is as daniel.leonard and michaelc point out, I thought that the difference in the values (commision of sell, commision of buy) regulate the value, the truth that the equation (5) and (6) of my explanation should have give me that hint, thank michaelc your last explanation clarify that point, cool

mel| Profile July 11th, 2008 - 5:51 pm{oops: I accidentally deleted my introduction}

As I read the problem, the $25 that the cheater made should be equal to profit he actually made minus the profit he would have made without cheating. [Let C = $25 cheating premium; N = net; P = amount cheater paid; G = amount cheater gets; f = fair; a = actual]

C = Na – Nf

Na = Ga – Pa

Nf = Gf – Pf

[Let x = the fair price of the goods without commission]

Pf = 0.98x

Gf = 1.02x

Nf = 1.02x – 0.98x = 0.04x

We also know that for every pound (16 oz) brought in, he only pays for 15oz, so the cheater pretends (15/16)x is the fair price without commission. Also when he sells the goods, he charges a full pound price for every 15 oz he gives, so when selling he pretends that (16/15)x is the fair price.

Pa = 0.98(15/16)x

Ga = 1.02(16/15)x

Na = 1.02(16/15)x – 0.98(15/16)x

Remember that C = $25, so…

$25 = [1.02(16/15) – 0.98(15/16) – 0.04]x

Divide by the mess on the right and plug into a calculator to get the fair price

x = 193.4236

But the question asked what he actually paid, so plug into the formula for Pa

Pa = 0.98(15/16)x = 135.709, call it $135.71, though this guy he probably rounded down and paid even less. What a cheater.

falwan| Profile July 11th, 2008 - 10:16 pmDoes that mean you don’t have the right answer as yet?!

mel| Profile July 12th, 2008 - 12:35 pmReading over the other comments, I see two common points of confusion. The first is whether the cheating rate on the sale is 17/16 or 16/15. The text of the problem reads “One ounce short per pound” which implies we should be using the number 15 (one ounce short of a pound). Also, it is stated that this is done by reversing the scale, so the rate should be the reciprocal of the selling cheat-rate, which everyone agrees is 15/16.

The next point of confusion is whether $25 represents the total net of the cheater or the extra amount the cheater gets by cheating. The question seems to be truly ambiguous on this point, so in my notation above, that means either

C = Na ==> x = 147.7105

C = Na – Nf ==> x = 193.4236

Which result in final answers of either

Pa = $135.71 (assuming $25 is the total earning)

Pa = $170.71 (assuming $25 is the extra earning)

Ooops. I mixed the two in my original post, having calculated it both ways. I showed the math for one and the answer for the other. My Bad.

RK| Founder | Profile July 12th, 2008 - 4:14 pmI’ll put up the answer I have shortly….

RK| Founder | Profile July 13th, 2008 - 12:33 pmok, here’s the official answer given to this very old puzzle. I believe a mistake was made when I said the $25 includes the commission….)

“In the 1st place, if the broker weighed the goods with a pound weight one ounce too heavy, he got 17 ounces for a pound. When he sold them by a weight one ounce light he gave 15 ounces for a pound, and had 2 ounces over. If these 2 ounces were sold at the same price, so as to make $25 by cheating, it is plain that the 2 ounces represent 2/15ths of what he paid for the whole and charged for the 15 ounces. One-fifteenth (1/15) being worth $12.50, fifteen-fifteenths(15/15), or the whole, would be $187.50, which, if there was no question of commission, would be what he paid for the goods.

We find, however, that he received 2 per cent from the seller, $3.75, and $4.25 from the purchaser, making $8 brokerage in addition to $25, by cheating. Now, if he had dealt honestly, he would have paid for 17 ounces, which, to be exact, would have been $199.21875. His brokerage for buying and selling would therefore only be $7.96875, so he has made an additional 3 1/8 cents by cheating. As the puzzle said he made $25 by cheating, we must reduce the $187.50 price so that his two cheatings will amount to just $25.

Now, as 3 1/8 cents is the 001th part of $25.03125, we must reduce $187.50 by its 801th part, which will bring it down to $187.27, so that he will make just $25 and the .0006 of a cent by cheating. To such as wish to be very exact and honest, I would suggest that the seller be paid $187.2659176029973125 less the 2 per cent brokerage of $3.745 plus”

ughh-my head is starting to hurt :cry:

mel| Profile July 13th, 2008 - 1:10 pmThis answer is wrong. It ignores the fact that the price per ounce given to the seller and received from the buyer differ because of the unfair scale, so material from the two sources cannot be assumed to have a consistent price per weight.

suineg| PUZZLE MASTER | Profile July 13th, 2008 - 4:40 pmI would agree with Mel in that the commison has to be taken in different scales, one for the seller and one for the buyer, and I think you have to substract the commision in order to get the original payment RK, my answer would be close to the daniel.leonard answer, so when you put 1/15= 12,50$ you are not considering commisions, when you substract both commisions, the value has to be smaller, however dont know seeing from your expalnation why the equations give a value close to 142$

michaelc| Profile July 13th, 2008 - 5:12 pmThis is actually pretty interesting. It’s always interesting when different folks interpret things differently.

I’m trying to understand the question from RK’s answer. The way I see it, if you figure the 25$ comes completely from cheating, I think the commision falls out of the equations. Here goes the explanation with my old equations above.

ro = real ozs

cc = cost camel hair per oz

bo = buying ozs = ro*(15/16)

so = selling ozs = ro*(17/16)

Tch = total profit from cheating

Tcom = total profit from commmission

Tp = profit he should have made without cheating

25 = Tch + Tcom – Tp

Tp = ro*cc*2% + ro*cc*2% = 4%*ro*cc

Tcom = bo*cc*2%+so*cc*2%

Tcom = (15/16)*ro*cc*2% + (17/16)*ro*cc*2%

Tcom = 4%*ro*cc

Tch = so*cc – bo*cc

Tch = (1/8)*ro*cc

25= (1/8)*ro*cc +4%*ro*cc – 4%*ro*cc

ro*cc = 25*8

ro = bo*(16/15)

bo*(16/15) *cc = 200

bo*cc = $187.50

commission …

bo*cc*2% = $187.50*2% =$3.75

Take it out or leave it. Depending on how you look at it (he pays for the goods then the seller gives him his commision (187.5), or he pays for the goods and takes his commission out at the same time 183.75).

From the looks of this puzzle, there are several other ways to look at it…

michaelc| Profile July 13th, 2008 - 5:23 pmOne thing I have a hard time following from RK’s answer…

“Now, if he had dealt honestly, he would have paid for 17 ounces, which, to be exact, would have been $199.21875.”3

Can anyone out there explain this to me? Seems to me if he would have dealt honestly, he would have paid for 16 ounces, and sold for 16 ounces, but actually he paid for 15 and sold for 17.

michaelc| Profile July 13th, 2008 - 5:32 pmOr if 17 ounces were the actual real number of ounces, then by cheating, he would be paying (15/16)*17 ounces or 15.9375 ounces.

suineg| PUZZLE MASTER | Profile July 13th, 2008 - 7:54 pmHi RK, I think I see two errors in the official answer:

1)The commisions have to be calculated by different scales, because they are proportional to the ratio (15/16;17/16) respectively.

2)This is the more important error, suppose that there are no commisions involved, you say for every 2 ounces in a cheat pound you make 25$—> 2/15 = 25$ -> 1/15= 12.5

so you say 15/15= 187.5$–> price if there were no commision, but that is assuming the number of pound weighted is 1; the equation if there were no commision should be: NP*2/15 = 25$, so needing more equations to solve, because thay value can have a extreme range and has to be narrowed by more equations to determine the number of pounds weighted, the less the number of pounds the greater the value paid for the goods, the commisions conditions regulate the number of pounds, cool.

suineg| PUZZLE MASTER | Profile July 13th, 2008 - 11:32 pmRK, to clarify my last comment go to comment #10; equations 3 & 4; no commisions so C1 & C2 are 0; you get:

CE= 2*P*NP; 25= 0 + 0 + CE; if NP=1; CE=25; 25=2P->=12.5$, that would be the price per ounce with no commisions and only 1 pound of hair weighted; but he cheated so he pays for 15 ounces only –> 15*12.5= 187.5$, but that is if your hair only weights 1 pound.

mel| Profile July 14th, 2008 - 1:03 amAh, I’m looking at the crossed-out text, and I see where everyone is getting the 17/16 number, because it used to read “using a weight once too heavy” (which means 17/16) instead “giving an ouch short per pound” (16/15).

suineg| PUZZLE MASTER | Profile July 14th, 2008 - 9:20 ammaybe I am wrong, but why the equations did get that answer, dont know yet, cause the official answer is great too, no answer to that but cool.

michaelc| Profile July 14th, 2008 - 11:30 amAh! Mel Now I see the error of my ways.

I guess I didn’t re-read the new post carefully enough. Live and learn so they say.

Let’s start over.

ro = real ozs

bo = buying ozs = (15/16)ro

so = selling ozs = (16/15)ro

cc = cost of camel hair per oz

Thp = total profit if it had been honest

Thp = 2%rocc + 2%rocc = 4%rocc

Tch = profit by scale cheating alone

Tch = socc – bocc = (31/240)*rocc

Tcom = cheating commission profit

Tcom = socc2% + bocc2%

Tcom = (962/24,000)rocc

Now if 25$ is solely from cheating…

25 = Tch + Tcom – Thp

Then

rocc = $193.42

bocc = $181.33

or 177.70 minus commission

If 25$ includes commission…

25 = Tch + Tcom

25 = (3100/24000)rocc+(962/24000)rocc

Then

rocc = $147.71

bocc = $138.48

or $135.71 minus commission.

Mel, I think I concur with your answer.

mel| Profile July 14th, 2008 - 2:06 pmHi Michael. I just looked over your equations and they seem topologically equivalent to mine. The only difference we had we the 16/15 versus 17/16 constant and the total earning versus extra-by-cheating ambiguity.

I think we nailed this one.

suineg| PUZZLE MASTER | Profile July 16th, 2008 - 8:45 amnow I think I found an explanation RK, I repeated “THINK” JAJAJA I am not sure yet:

if 2/15=25; 15/15= 187.5 but that is the money you make by selling the whole goods, because it includes the profit, so you have to substract the 25$ in order to get how much did you pay for the goods if there were no commision; 187.5-25=162.5$ but if there were commsions involve this price would be lower, lets suppose that the 4% is charged only in the buy process for simplicity: 162.5 – 162.5*0.04= 156$ but you now that a 2% of that 4% is over a bigger amount so the original payment is even lower, because the number of ounces sold are more than the ounces bought; teh amount get in 140$ range, what do you or anybody think about it ?, is my reasoning is wrong???

michaelc| Profile July 16th, 2008 - 11:17 amI finally understand where the “official” answer came from!

It’s all in how you define the buying ozs and the selling ozs.

The official answer defines bo (buying ozs) and so (selling ozs) as follows…

ro = real ozs

bo = (16/17)ro

so = (16/15)ro

cc =cost camel hair per oz

Tch = profit by cheating on scales

Tcom = profit by cheating on commision

Thp = honest profit

Tch = (32/255)rocc

Tcom = (256/6375) rocc

Thp = 4%rocc

25= Tch + Tcom – Thp

so solving you get..

rocc = 198.97$

so bocc = 187.26$

Then you take out the commision out of that which is $3.745.

How in the heck they calculate that from the explanation is beyond my grasp!

So what this is saying is when there is 17 ozs on the scale, the scale reads 16 ozs during purchasing from the seller by the broker.

It also says that when the broker is selling the goods to the buyer and there is 15 ozs on the scale, the scale actually reads 16 ozs.

We are given…

“Upon receiving a consignment of camel’s hair he placed the same upon the short arm of his scales, so as to make the goods weigh one ounce light to the pound, but when he came to sell it he reversed the scales so as to give one ounce to the pound short, and thus made $25 by cheating.”

I interpret “One ounce to the pound light” as the scales reading 15 ounces when there is actually 16 ounces on the scales. So there is the difference in the answers.

Whew.

RK| Founder | Profile July 17th, 2008 - 7:45 amNoble efforts, gentlemen

However, personally this is one problem where I’ve had to throw in the white towel….

hex| PUZZLE MASTER | Profile October 16th, 2009 - 5:05 pmbilbao has just brought to my attention to this old puzzle.

The wording of the puzzle is very inaccurate and misleading.

To start with, the balance is an “old fashioned balance scale that uses weights”, so the balance ratio k is fixed either to 15:16 or 16:17 (one of these is to be used both for buying and selling). Since we are talking “one ounce light” and “one ounce short”, then i chose k=15/16.

@RK, the man cannot “get 17 ounces for a pound” and “gave 15 ounces for a pound” using this scale.

He can get 17 ounces for a pound and earn 17 ounces for a pound, or pay 15 ounces for a pound and give 15 ounces for a pound.

Now the deal starts as a buyer approaches the man and asks for a certain weight of camel hair (W pounds), so W is the weight that the buyer believes he is buying.

P = price of camel hair per pound

If the dealer does not cheat, he will actually buy W pounds of camel hair:

buy for 0.98 x W x p

sell for 1.02 x W x p

profit = 0.04 x W x p

Since the dealer cheats twice, he will pay the seller for (15/16)^2 x W pounds of camel hair:

buy for 0.98 x (15/16)^2 x W x p

sell for 1.02 x W x p

profit = 1.02 x W x p – 0.98 x (15/16)^2 x W x p

=~ 0.1587 x W x p

The difference in profit between cheating and being honest would be:

0.1587 x W x p – 0.04 x W x p

= 0.1187 x W x p

This is equal to 25, so W x p = 210.6649

And he buys at 0.98 x (15/16)^2 x W x p

= 0.98 x (15/16)^2 x 210.6649

= 181.45 net

I hope i did not create a mess 8-)