 An important idea in mathematics is known as order of operations. This is really an agreement about how we're going to perform the various arithmetical operations. And it really is an agreement, just like driving on the right side of the street. So the key rules of the order of operations, when working with multiple operations, and parentheses must be done first, then multiplication and division from left to right, then addition and subtraction from left to right. And for future reference, it'll be very useful to keep the following idea in mind. The type of expression is determined by the last operation performed. For example, consider the expression 12 minus 3 plus 5. So going through our order of operations, there's nothing in parentheses. There are no multiplications or divisions, and so we have to perform addition and subtraction from left to right. And so we first evaluate 12 minus 3. That gives us 9 plus 5 we haven't done anything with, so that gets carried along for the ride. And now we evaluate 9 plus 5. Since the last operation performed was the addition of 9 and 5, we say that this expression 12 minus 3 plus 5 is a sum. How about this expression? So here, again, no parentheses. So now we want to perform multiplication and division from left to right. So first we deal with 36 divided by 12, that's 3, and then multiply this by 3. And since the last operation performed was a multiplication 3 times 3, then our expression, 36 divided by 12 times 3, is going to be called a product. What if we have parentheses? Our rules of the road say that we have to deal with the operations and parentheses first, so this 12 times 3 must be dealt with first, so we'll evaluate 12 times 3. And then we've taken care of the parentheses so we don't need them anymore. Then we have 36 divided by 36 is equal to 1. And since the last operation performed was the division, 36 divided by 36, this expression, 36 divided by quantity 12 times 3, is going to be called a quotient. So if we mix in a few more operations, 8 minus 12 divided by 3 plus 3. There are no parentheses, so now we go to multiplication and division from left to right. The only division we have is 12 divided by 3, which gives us 4. And so now we have 8 minus 4 plus 3, addition and subtraction are going to be performed from left to right. So we do 8 minus 4 plus 3 gives us 7. And since the last operation we performed was an addition, the expression 8 minus 12 divided by 3 plus 3 is a sum. What if we throw down some parentheses? So we deal with the addition 3 plus 3 first. We have subtraction and division, so we do the division 12 divided by 6 first. And now we're left with a subtraction 8 minus 2 giving us 6. And since the last operation performed was a subtraction, the expression is a difference.