# Word Problems and Division

Here we have a list of some words that indicate division. So, if you’re trying to solve a word problem, and you come across the word quotient, per, out of, or ratio, you’ll know that you need to divide something.

Let’s look at an example. If the quotient – there’s one of our keywords – of a number and 10 is 3/5, what is the number?

The transcript is provided for your convenience
So, again, we saw that keyword “quotient”, which tells us we’re going to be dividing something. We’re going to be dividing two numbers, and those numbers we’re dividing are a number, and 10, and they must be divided in that order.

So, when we write our division problem, we can write it as x÷10, or we could write it as x/10, but it must be in that order.

If we reverse it to do 10/x, then that is not what is written here in this problem. That would be if the quotient of 10 and a number. So, order is very important, since it was written, “If the quotient of a number and 10”, then we must have that number first, divided by 10. And you’ll see here that I put x, and I put in x because we don’t know that number. It just said “a number”. So, when we don’t know a number, we use a variable in place of that.

Let’s finish this sentence. If the quotient of a number and 10 is 3/5. “Is” is equals, 3/5.

Now, since I chose to write it this way, x/10 equals 3/5, it looks like a proportion, and that’s because it is one. All a proportion is, is a ratio equal to a ratio, and that’s what I have here.

So, to solve this proportion, we’re going to cross multiply. x times 5 is 5x, and that’s equal to 3 times 10, which is 30.

And now, we need to solve for x, this is 5 times x, and the opposite of multiplying is dividing. So,  we divide both sides by 5, 5/5 is 1, 1 times x is x, and that equals 30/5, which is 6.

So, the quotient of 6 and 10 is 3/5, and that’s true. 6/10 does simplify to be 3/5.