What Is The Missing Number?

Relevant wiki: Arithmetic Puzzles - Fill in the Blanks

The fourth equation (call it $A$ divided by $B$ equals $C$ ) is the same as $B$ times $C$ equals $A$ , which is the same form as the third equation.

Neither of these equations can use the number $1$ (or else there would be a repeated number), so in one of these two equations, the smallest number must be $2$ and in the other the smallest number must be $3$ .

If the smallest number is $3$ the only possibility is $3 \times 4 = 12$ . This leaves $2 \times 5 = 10$ as the only possibility for the other equation (since we cannot repeat the numbers $3$ or $4$ ).

So the last two equations must use the six numbers $3,4,12,2,5,10$ , and there are various ways this can happen, for example we could have used $5 \times 2 = 10$ and $12 \div 4 = 3$ .

Now the second equation (call it $X$ minus $Y$ equals $Z$ ) can be written as $Z$ plus $Y$ equals $X$ , which is the same form as the first equation. So we need to find two equations of the form $Z$ plus $Y$ equals $X$ using only numbers from $1,6,7,8,9,11,13$ .

One of these equations cannot use the number $1$ so it must be $6+7 =13$ . Then the only possibility for the other equation is $1+8=9$ . So the first two equations must use the numbers $6,7,13,1,8,9$ in some order.

The missing number must be $\boxed{11}$ .