The Brightvale Institute of Technology needed new Maths books, so they ordered a new set from Brightvale Books.

When they got the books back from the printers, however, they discovered that there had been a horrible mistake. Apparently the printer scrambled up all the digits in the printing press, so all the numbers reflected wrong digits. Yet somehow, these mistakes were consistent throughout the book. (For example, all instances of the digit "2" would instead use the digit "7"... but that's just an example! Don't assume that that is a clue for this puzzle!)

Some examples that were printed in the book were:

518 x 70 = 6270

6481 + 3294 = 32847

7221 x ( 3334 + 3666 ) = ?

Using this numbering system, what would this book list as the answer to the problem 7221 x ( 3334 + 3666 )

This is an incredibly hard problem I can't get around, I'm pretty sure that a 3 is supposed to be a 0 but that's about as far as I can get any one out there got any ideas?

well let's split the questions up.

(100e + 10a + h) x (10g + z) = 1000f + 100b + 10g + z

1000f + 100d + 10h + a + 1000c + 100b + 10i + d = 10000c + 1000b + 100h + 10d + g

(1000g + 100b + 10b + a) x (1000c + 100c + 10c + d + 1000c + 100f + 10f + f) = ?

Here I have taken the numerical values of 0-9 into variables z-i, then rewritten the equations you are given using them.

Does this help? I'll work on it a little more, but This looks like a promising route to take.

Simplifying:

1000eg + 100ag + 10hg + 100ez + 10az + h = 1000f + 100b + 10g + z

a + 10i + 1000f = 900b + 9d + g + 90h

211000cg + 232100bc + 2110ca + 111000fg + 12210fb + 111fa + 1000dg + 110db + ad = ?

Not sure how you'll figure that one out capuchin. 2 equations with 9 unknowns...
I applied brute force to get the following answer. I stopped at the first one that worked so I'm not sure this is the only answer.
2-> 0
3-> 1
7-> 2
5-> 3
0-> 4
4-> 5
8-> 6
1-> 7
9-> 8
6-> 9

518 * 70 = 6270 -> 376 * 24 = 9024
6481 + 3294 = 32847 -> 9567 + 1085 = 10652
721 * (3334 + 3666) = ? -> 2007 * (1115 + 1999) = 6249798

So the book's answer would be 8706169.

If you have any questions please ask.