Hard math problem?

Question

A man has bought 36 dozens of eggs, but he has spilled his coffee on his receipt. Thus, he can only read two numbers of the total price: _8,5_$ (underscores represent numbers he can't read). He remembers that a dozen costs less than 2,00$. Can you find the total price back?

That's the only thing I've been able to do:
36p = 10x+8.5+0.01y
10x+0.01y < 63.5
x has to be an integer, so x E [1,6]
Please help!

Answer 1

Use cents as the unit, not dollars. Then we get
36p = 1000x + 850 + y

The obvious constraints are that x and y must be single digit integers.
In fact, 36p <= 36*199 = 7164, so 1 <= x <= 7.

Rewrite the equation as
36p - 1000x = 850 + y
The left side of the equation is divisible by 4, so the right side also must be divisible by 4. The only single digit values for y that make this happen are y = 2 and y = 6.

Then rewrite the equation as
36p - 999x = x + 850 + y
The left side is divisible by 9, so the right side must also be divisible by 9.
Try both possible values for y
For y = 2, we get 36p - 999x = x + 852
x = 3 is the only single digit that makes the right side divisible by 9.
For y = 6, we get 36p - 999x = x + 856
x = 8 is the only single digit that makes the right side divisible by 9, and we already know x <= 7. So y cannot be 6.

So the required solution is
x = 3
y = 2
which gives a total cost of $38.52, and a price per dozen of $1.07

Answer 2

first $2.00 x 36 = $72.00 which means that the first number is less than 7. Then if you start dividing all the number combinations by 36 you will find that all but $38.52 do not divide evenly. I have no idea how to do it without just trying them all out. You can limit what you test by knowing that the last digit has to be even and the sum of the digits has to be divisible by 9. So that gives you 5 combinations to test.
38.52
18.54
88.56 <- Not Valid too High
68.58
58.50
Testing these yield 38.52 as the only valid answer. Which yields $1.07 as the price for each.

Answer 3

x price of one egg

12 x < 2
36 * 12 * x = A8.5B

combine
36 * 12 * x < 72
so A is 1,2,3,4,5 or 6
36*12 = 3^2*2^2*3*2^2 so
A8.5B is divisible by 9 and 27

also divisible by 4*4
thus B = 0, not ok ( .50 -> .25 ->.125 boom )
B=2 almost ok ( not divisble by 8 )
B=4 not ok
B=6 : 0.56 ->0.28->0.14->0.07 yessss
B=8 :0.58 ->0.29 no.

so only B=6 is ok.

A8.56
price divisible by 9 and 27
( sum of numbers divible by 9 )
none
there is no solution.

Answer 4

No. of answers can be generated using numerical analysis
1) 36*1.905 = 68.58
2)36*1.625=58.50
3)36*1.3475=48.51
4)36*1.07=38.52
5)36*0.795=28.62
6)36*0.515=18.54

Answer 5

it is 107