Number 12. Clara starts with 12 and follows the arrows using the rules shown in the picture on the right. With what number will she finish? The first thing I want to note is that multiplication is commutative, but division is not. This means that the order does not matter in multiplication, but it does for division. For example, 1 times 2 is the same as 2 times 1, and they're both 2. However, 1 divided by 2 is 1 half, and 2 divided by 1 is 2. So in division, the order does matter. And why that's important over here is because we're doing a bunch of operations. So we're going to start with 12 times 4, and then 12 times 2, and then divided by, sorry, times 4 again, and then over here, divided by 2. So there's going to be some division in this. There's also going to be some multiplication in this. But it might be easy to kind of move things around just to make the problem easier. But if there's division in it, we can't really do that. So what can we do to kind of fix this problem? Well, we can fix this problem by rewriting division by 2 as multiplication by 1 half, which of course is identical, and division by 4 as multiplication by 1 fourth. So this solves our issue about commutation and the effect on order of operations. So what operations occur in this problem? Well, now that we're addressing everything as multiplication, we can just count up how many times each operation appears, because the order, again, does not matter. So we see that we have 3 down arrows, which means multiplying by 4 3 times, 3 up arrows signifying multiplying by 1 fourth 3 times, 2 right arrows multiplying by 2 2 times, and 3 left arrows multiplying by 1 half 3 times. So right off the bat, we know that multiplying by 4 and then multiplying by 1 fourth results in the same number. And similarly, multiplying by 2 and then multiplying by 1 half will result in the exact same number, because this is equivalent to multiplying by 1, just as it is with these. So multiplying by 4 3 times and by 1 fourth 3 times, these cancel out. This is equivalent to just multiplying by 1, or not changing the number at all. Multiplying by 2 2 times and by 1 half 3 times, well, 2 of the 1 halves cancel out with these 2 twos, and we're left with 1 1 half at the end. So these cancel out to have a final effect of multiplication by 1 half. Of course, we're going to start with the number 12, so 12 times 1 half is going to be 6, and the answer will be B.

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multiplication

division

sequence

result