Easiest way to solve equation?

Question

The equation looks like this (2*0*1*7*2*0*1*5)/(2*0*1*6*2*0*1*6) The asterisks need to be replaced by a plus or minus sign so that the result is equal to eight. What is the easiest way to do this? Is it possible to do it without manually "checking" if it is equal to eight.

Answer 1

We can use the signs in such a way that the numerator is $16$ and the denominator is $2$.

We can achieve this as follows :

$$2+0+1+7+2+0-1+5=16$$

We have $7+5+2+2=16$, so if the ones cancel out, we have the right result.

For the denominator we use

$$2+0+1+6-2-0+1-6=2$$

Since $1+1-2=0$ and the sixes cancel out, we again get the right result.

Answer 2

If you use all plus signs on the top, you get $18$. The largest multiple of $8$ less than this is $16$, so we need a minus on one of the $1$'s. Then we need the bottom to be $2$. This means we need minus signs before a total of $8$. The easiest way to do that is to put minuses before the second $2$ and one of the $6$'s. The sum on the bottom must be even, as changing a plus to a minus will always change the result by a multiple of $2$, so we cannot accept $8$ in the numerator. That would require $1$ in the denominator, which we cannot achieve.