One day in a shopping mall...

A woman went to a shopping mall with a calculator. She is weak in calculation.

She took 4 items and calculate the total price by using calculator and got it as $7.11

She went to checkout queue. Unfortunately she multiply item's prices instead of sum those.

Counterman do the calculations properly (sum those) and charge the woman $ 7.11 (how strange!)

But that was true and none was mistaken.

Now what was the cost of each item she purchased?

20 comments:

sadat said...
This comment has been removed by a blog administrator.
hm83 said...

Answer is 1.7775.

juelp said...

yes the result is 1.7775

Unknown said...

Previous Answers were not correct.
A.M.=1.7775 & G.M=1.6329
I really want to know the answer.
1.775 can't be the answer because 1.7775*1.7775*1.7775*1.7775=??

Unknown said...

none of the above answers may be correct . if all the items have equal price then the answer is 1.6329

anubish said...
This comment has been removed by the author.
anubish said...

may be 1.775 is right ans but i m not sure.....

@ arina....i think 1.6329 isn't correct

earn said...

whatever as she is not expart in calculation she used simple math, she multiplied all 4 items with price and got 7.11 so ans is 1.7775

Anonymous said...

Hi all. I've read all of your answers and I got amused at your attempts to solve the puzzle. None of you except rony understood the puzzle properly. Well done rony for your understanding the puzzle! The rest of you came to sudden conclusion like 1.7775 or 1.6329 without thinking correctly. Do you guys have any idea that this puzzle involves two simaltaneous equations? If we assume that 'ALL ITEMS HAVE SAME PRICE' then let's say, x = price of each items, then according to the woman she multiplied the prices as x.x.x.x = 7.11 => x^4= 7.11. But the counterman added the prices as x+x+x+x = 7.11 => 4x=7.11. Thus, we've got two equations which are (1) x^4= 7.11 and (2)4x=7.11. Now we can interpret algebraically that we have to find the solution i.e. x and geometrically, we have to find the intersection point if we plot the graphs of those two equation. Both the algebraic or geometric conception of solution are one and same thing. Algebraically, x^4= 7.11 and 4x=7.11 have no solution because x^4= 7.11=> x= 1.6329 and 4x=7.11 => x= 1.7775. See, values of x in each equation are different! So,The results are contradictory. Geometrically, the graphs of the functions don't intersect each other, hence they don't have any common point. If you guys study higher level mathematics, then you may know that this type of situation is known as 'mathematical anomaly' or sometimes in mathematical logic we call it "reductio ad absurdum" i.e. 'reduction to absurdity'. This situation can happen if our assumption at the beginning becomes incorrect. In formal mathematics 'assumptions' are called 'axioms'. In this case our axiom was 'ALL ITEMS HAVE SAME PRICE'. Assuming this axiom to be correct, we've got a contradictory result and it means our assumption/axiom was incorrect. So, correct axiom should be ALL ITEMS CANNOT HAVE SAME PRIZE.

Anonymous said...

I've divided my entire post in several parts as the entire post will be very lenghty.
Now, I'm going to solve the puzzle using rigorous axiomatic mathematics. I'll use a little bit formal mathematical terminologies . Those who are weak at math or fear it's complexities shouldn't read my solutions because they will be completely lost in the depth of mathematics associated in my solutions. But those who are really interested in mathematics and love it truly may go on. I'm sure enough about my solution. So, if anyone including the guys studying undergraduate math, physics or engineering wants to challenge me, he can e-mail me. But those whose knowledge in mathematics is upto HSC level, do not even try to challenge me in your dreams becuase you are far far behind from the higher level mathematical concepts.
Solution of any mathematical problem is based on one or more initial axioms. This axioms are either given explicitly in the problem or we need to derive them indirectly from the data given in a problem. In this puzzle nothing is said directly whether the price of each item is same or not. All of your mistake was that you considered that each of the item has same price. Anyway, as there's no direct data given about the price of each item in the puzzle, I must go for all possibilities that can happen. For each possibility, I will assume a different axiom such that all possibilities can be covered. The possibilities are:

(1) 3 items have same price and the last one has different price( i.e. first 3 items cost s, the last one costs t )
(2)First 2 items have same price and the last two items have also same price but different than the previous two( i.e. first 2 items cost u, the remaining two costs v)
(3)2 items have same price and 3rd and 4th one have different prices( i.e. first 2 items cost w, the 3rd one costs x and the 4th one costs y)
(4) All the 4 items have different price ( i.e. they cost a, b, c and d)
All the 4 items have same price- This one I already proved as wrong. So I will not consider it again. Each of the 4 possibilities stated above may happen in this puzzle. I'm gonna solve each of the possiblity now.

Anonymous said...

Possibility 01:
Axiom :
3 items have same price and the last one has different price.

Solution:
Let,
first 3 items cost = s, the last one costs t
According to woman, s.s.s.t = 7.11 => ts^3=7.11 => t = (7.11/ s^3) .......... (1)
According to counterman s+s+s+t = 7.11 => 3s+t = 7.11 .....................(2). Replacing (1) in (2) we get, 3s+(7.11/ s^3) = 7.11 => 3s^4 -7.11 s^3+7.11 =0 ........ (3). Equation (3) is a quartic equation( degree 4 polynomial). So, it must have 4 roots. you can solve it in Ferrari's method and verify your solution by Viete's Formula. The solution steps of equation (4) is lenghty and I can't type all math statements here in so short space. Anyway, after solving the equation (3) we get four values of s. Among them two values are complex conjugate and the other two are real numbers. We consider real solutions only and they are s1= 2.121946, s2= 1.305961. Putting these values in (1) we get the values of t as t1= 0.744160 , t2= 3.192118.
So the answers are
a) three of the items cost of $ 2.122 each and the last one costs $ 0.744. or,
b)three of the items cost of $ 1.306 each and the last one costs $ 3.192.

Verification: Notice that both the results satisfy the given two equations. so, both are correct.
a) For woman, (2.122)*(2.122)*(2.122)*(0.744) = 7.109 =7.11. And for countermanager, (2.122+2.122+2.122+0.744) = 7.11
b) For woman, (1.306)*(1.306)*(1.306)*(3.192) = 7.109 =7.11. And for countermanager, (1.306+1.306+1.306+3.192) = 7.11

surprised?? Yeah, that's the power of math if you can think logically and put all your thoughts in a framework. Notice, equation (1) and (2) are multi-variable functions( i.e. function of s & t) but none of you thought of it. You only thought the functions as a single variable and that's why you got incorrect results like 1.7775 or 1.6329.

Anonymous said...

Ok, let's now check the possibility 02.
Possibility 02:
Axiom :
Two items have same price and the other two items have also same price but different than the previous two.

Solution:
Let,
first 2 items cost = u, The last two cost = v
According to woman, u.u.v.v = 7.11 => (u^2)(v^2)=7.11 .......... (1)
According to counterman u+u+v+v = 7.11 => 2u+2v = 7.11=> v = (7.11-2u)/2 .....................(2). Replacing (2) in (1) we get, (u^2)(7.11-2u)/2)^2=7.11 =>(u^2)(7.11-2u)^2=28.44 =>(u^2)(50.552-28.44u+4u^2)^2=28.44 => 4u^4 -28.44u^3+50.552u^2-28.44=0............(3)
Again, equation (3) is a polynomial of degree 4 and it must have 4 roots. After solving it, we find that (3) has three real solutions and one negative solution -0.6362.So we don't consider it. The remaining solutions are real number which are u1= 4.191233, u2= 2.479641 and u3= 1.075343. Putting these values in equation (2) we get, v1= - 0.636233 (as v1 becomes negative, it means we must not consider the value of v1 and u1), v2= 1.075359 and v3= 2.479657. Notice (u2,v2) and (u3,v3) are same set of solution.
So the answer is
two of the items cost $2.480 each and the other two cost $ 1.075 each.
Verification:
For woman, (2.480)*(2.480)*(1.075)*(1.075) = 7.107 =7.11. And for countermanager, (2.480+2.480+1.075+1.075) = 7.11

Amazing, isn't it?

Anonymous said...

For possibility (3) let, first 2 items cost = w, the 3rd one costs= x, the 4th one costs = y.
According to woman, w.w.x.y = 7.11 => xyw^2=7.11 .......... (1)
According to counterman w+w+x+y = 7.11 => 2w+x+y = 7.11.......(2) Now look at (1) & (2) carefully, they are functions of three variables (w,x &y). But we have only two simaltaneous equations. If you have studied Linear Algebra, you may know that when number of variables is greater than number of equations we HAVE TO express at least one variable in term of other one i.e. we can't never find a specific constant value for each variable such that each of the variable becomes independent of one another. Similar situation happens for possibility (4) where we get for the woman, abcd= 7.11 and for countermanager a+b+c+d= 7.11. Here, each of the two functions have 4 variables but we have only two equations. So we can never be able to express each of the variable independent of one another. If we solve this type of system, we would see one solution becomes a function of another. So we can ignore the possibility (3) & (4) in this puzzle.
Anyway guys, my solutions are no doubt lenghty but correct. I could solve the puzzle in 5 to 6 lines of sentences but you guys could have never understood how I solved it. So to make the solution easy I wrote every detailed steps such that you don't get confused. Even after that if anyone has question he can e-mail me nazmulht@gmail.com . Actually, proving or solving math problem becomes as simple as water if you know formal mathematics like abstract algebra( it's an intense theorem-proving subject focusing on sets, groups, rings, fields etc.) or better Axiomatic set theory which is the toppest level of understanding the foundations, mechanics and formalism of mathematics. And I know both of them.
It's my challenge to you all if you can prove me wrong. I know you can't.

Unknown said...

the items are all different costs. No matter what we have no way of knowing the price of each item. if she has four items, a, b, c, d, however you factor it, yes if they all cost the same the answer is 1.7775, but that is assuming the cost is the same which is unlikely and it does not say that she bought four of the same item (and this is unlikely and is an unknown variable). Without knowing that, it seems to me very difficult to know the solution to this equation. You need more information, I think. I'll consider it further, but on the face of it, you cannot find the cost of each individual item, only the total cost.

srp

Anonymous said...

@heleina
I think u didn't read my 5 posts that contain solutions of the puzzle before your comment. Read them. I solved the puzzle in details covering four possibilities that can happen.
You don't need more information to solve the puzzle. Data which are given is sufficient. All that u need to sit in a quiet place and think to solve it.

nur said...

What's going on. What is the real answer. No need to passing time like this way.

Unknown said...

Ya an excellent Brain storming job. I'm a newcomer on this site. Very Enjoyable site. OK. Rony has come near to problem & feel complexity of the PUZZLE. However, Analytical ability of Nazmul is Excellent. Ya I agree that, this is the actual way to analyze & solve this PUZZLE By calculating all the possibilities which is shown BY NAZMUL.
But dear Nazmul It is not so important to challenge some body. It is a game, I think no body is here no challenge somebody. Express ur self with analytical ability. OK Thaks to Nazmul for ur math.

Masud said...

Yes, the ans is 1.7777

Unknown said...

question is not correct.

Sayef said...

hey guys why r u spending so much time thinking on this puzzle
"BOTH" of the answer r right
***easier way***to think...

as "She is weak in calculation" may be she didn't grab the calculator straight. therefore, she thought/used "x" sign as "+" sign...