Math Is Fun Forum

How many different sandwiches can we make?

I think you want to ask "In how many ways can we make these system of 3 sandwiches while satisfying the constraints?"

Hi phrontister;

Okay, thanks for coming in. See you later.

Hi chen.aavazi;

Calling x = apples and y = bananas, we can reduce the system down to:

A)

100 - x > 3 y

3 y > 4 x

4 x > 100 - y

We can now graphically solve for the answer, I used Geogebra.

http://i.imgur.com/4A8ZlQC.png

The answer lies in the small darker triangle and must be an integer for (x,y). That reduces down the possibilities to just a few, as a matter of fact there is just one integer coordinate in there.

We can go a bit further with a trick that is used in numerical work which I invented?!

The three inequalities in A) can be changed to equations with the addition of what I call a slack variable... This is a term used in linear optimization but I give it an added meaning.

If 100 - x > 3 y  we only have to add something to the RHS to pick up the slack. I add e ( slack variable) which balances the inequality and gives us the equality of 100 - x = 3 y + e. We do that with all 3 inequalities and hope for the best.

100 - x = 3 y + e

3 y = 4 x + e

4 x = 100 - y + e

this can be solved by ordinary means

x = 19.0476, y = 26.1905, e = 2.38095

Remembering that x = apples and y equals bananas we can guess and try the closest integers and get apples = 19 and bananas = 26. We test to see that we are right and we are done!

Hi chen.aavaz,
I am glad that it fits well.
I found it by actual division of (10^n+k)/(k+1)^2
since k has to be low to meet demands of n I chose least possible value of k=6. denom=49
We do not know n.we know that with last remainder of 10000..../49 ,6 is to be appended to give a multiple of 49 . clearly it is 196.So we should look out for remainder 19. In my casio fx-825X calculator there is a button a b/c for integer division which gives dividend and remainder. Start with 100/49 = gives 2\frac {2}{49} remainder is not 19. To continue store it in memory and multiply by 10 ,look at remainder ,replace the number in memory by this number and go on multiplying by 10 until it gives decimal number.this is due to the ceiling on such operation.now recall the number in memory, write down integer part of division on paper (20408 in my calculation) and restart with remainder 8 and 0 to the right i.e 80 divide by 49 the same way from this batch I get integer as 1632 which you write on the right hand side of previous part(20408) and remainder 32 . continue the process until you get remainder of 19. Append 6 to the right and the dividend is 196/49=4 which is the last digit of number "a". i had done this much of work and posted it. I did not even calculate "b".I left it to chen.aavaz. He must have had horrible time for (a+b)^2.

If you find calculator too tiresome you can try excel.
start with a series 100. 1000.10000,100000 on a column "a"say. in the next column use=quotient(a1,49) and in column c =mod(a1,49). Copy the formula to lower cells. however after a few cells down the calculation fails. Note down the last quotient and erase all failed values. and start with new series say 80,800,8000 etc (if 8 was the last remainder) continue the process until you spot remainder=19.

hurrah!! Found a number.
a=20408163265306122448979591836734694
k=6
utilities used;scientific calculator, pen ,paper and some grey cells.
Please check whether it satisfies all the conditions.

Having trouble finding any more integer results. It seems likely that a is bigger than 10^12....

Hi, I don't understand... is the problem not just to choose three positive integers such that a=(k+10^n)/(k+1)^2 ?

Hi, chen.aavaz,
n should be the the number of digits of 'b" ,not "a".
I can only say that 'k" can not be an odd number.
So computational work is reduced by 50%