Aric F. answered • 03/21/19
Stanford-educated Active-Learning Tutor for Math, Spanish, SAT + More
Multiplying one number by another is pretty straightforward, but it can get confusing when we are asked to multiply something by an entire expression in parentheses - like the expression (5+n) that is being multiplied by 10 in this example - instead of just a single number.
Luckily, there is a property we can use to break down problems like this so that we end up multiplying things one-by-one without any parentheses to worry about. It's called the distributive property (in regular English, to distribute means to hand out individually; so you could, for example, distribute a snack to every member a soccer team). The distributive property tells us that when we see a number being multiplied by an expression in parentheses, we should distribute a copy of that number to every individual thing inside of those parentheses. In other words, every individual thing inside the parentheses will get its own "private" multiplication by our number, and then at the very end we will add up all of our results to get one final answer.
If that sounds confusing, don't worry! It is much easier to understand with a few examples:
6(5+3) = ?
Here, the expression in parentheses is (5+3), and the number we are multiplying that expression by is 6. The distributive property says we can distribute (in other words, hand out) a copy of our 6 to each of the two numbers (5 and 3) inside our parentheses. Here's what that looks like:
6(5+3) = (6*5 + 6*3)
You can think of the 6 as hopping from outside the parentheses to inside, making enough copies of itself so that it can pair up with each term inside the parentheses. From there, with nothing outside the parentheses to worry about anymore, we just follow the order of operations and do all of that math inside the parentheses. That means we first do the multiplication, and then we do the addition:
(6*5 + 6*3) = (30 + 18) = 48
You might be thinking that we could have solved 6(5+3) just as easily by adding (5+3) first, which gives us 8, and then multiplying 6(8) to get 48. And you're right! In fact, it's always great in math to solve the same problem in different ways, especially when you learn something new, so you can see for yourself that your answers are correct and the new way works. But we can't do addition inside the parentheses first like that when there are variables instead of just numbers inside, which is when the distributive property really comes in handy:
8(5+n) = ?
We can't get to a single number like 48 as our answer on this one, because that variable n can represent any number at all, so our goal here is just to simplify the expression. We do that by using the distributive property in the exact same way as above, with both 5 and n getting their own "private" copy of the 8 we are multiplying by:
8(5+n) = 8*5 + 8*n = 40 + 8n
Notice that one of our "private" multiplication problems, 8*5, did not have any variables in it, so we were able to do that calculation and write the number 40 in its place. The other multiplication, 8*n, has a variable, so we can't simplify it any further, and we keep it as 8n in our final answer.
Now that you know how to use the distributive property, see if you can use the same steps to solve the original question you asked, 10(5+n). Feel free to come back here to ask any further questions, or to see if your answer is correct!
