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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
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Since 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.
Do you mean that he makes $25 including his commission, or excluding it (i.e. he makes $25 solely from cheating with the adjustable scale)?
This puzzle is cool. 575$ 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.
Ups 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
to clarify- the $25 includes the commision. thanks
I 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.
151.5 dollars/pound
I 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%)
ok, sorry about this-I didn’t word the problem as I should have. Went back and edited the problem.
ok, 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
—>
Thanks 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.
I 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.
Let’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
continue—>
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
2 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.
For those that got 151.52 as the answer, did you remember to subtract the price of 1 ounce from the cost?
hello 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.
And 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
Maybe 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.
Now 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
{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.
Does that mean you don’t have the right answer as yet?!
Reading 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.
I’ll put up the answer I have shortly….
ok, 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 $3.745 plus”
ughh-my head is starting to hurt
This answer is wrong. It ignores the fact that the price per ounce given to the seller and received from the buyer differ because of the unfair scale, so material from the two sources cannot be assumed to have a consistent price per weight.
I would agree with Mel in that the commison has to be taken in different scales, one for the seller and one for the buyer, and I think you have to substract the commision in order to get the original payment RK, my answer would be close to the daniel.leonard answer, so when you put 1/15= 12,50$ 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$
This 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…
One 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 $199.21875.”3
Can anyone out there explain this to me? Seems to me if he would have dealt honestly, he would have paid for 16 ounces, and sold for 16 ounces, but actually he paid for 15 and sold for 17.
Or if 17 ounces were the actual real number of ounces, then by cheating, he would be paying (15/16)*17 ounces or 15.9375 ounces.
Hi RK, I think I see two errors in the official answer:
1)The commisions have to be calculated by different scales, because they are proportional to the ratio (15/16;17/16) respectively.
2)This is the more important error, suppose that there are no commisions involved, you say for every 2 ounces in a cheat pound you make 25$—> 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.
RK, 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->P=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.
Ah, 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).
maybe 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.
Ah! 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 $135.71 minus commission.
Mel, I think I concur with your answer.
Hi Michael. I just looked over your equations and they seem topologically equivalent to mine. The only difference we had we the 16/15 versus 17/16 constant and the total earning versus extra-by-cheating ambiguity.
I think we nailed this one.
now I think I found an explanation RK, I repeated “THINK” JAJAJA I am not sure yet:
if 2/15=25; 15/15= 187.5 but that is the money you make by selling the whole goods, because it includes the profit, so you have to substract the 25$ 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???
I 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.
Noble efforts, gentlemen
However, personally this is one problem where I’ve had to throw in the white towel….