Age word problem: Ben & William

Let's do some more of these classic age word problems. So we're told that William is 4 times as old as Ben. 12 years ago, William was 7 times as old as Ben. How old is Ben now? Once again, it's a good idea to try to do this on your own first, and then I'll work through it. What's the unknown here? Well, the unknown here is how old is Ben now. So let's set a variable equal to that, and we do x or y. But since Ben starts with a b, let's use b for Ben. So let's let b equal Ben's current age, Ben's age now. Let's see how all of this other information relates to Ben's current age, and then maybe we can set up some equation and then solve for things. I'll do it a little bit more structured in this one. You could have done many of the problems we've been working on in this way. Let's think about Ben, and then let's think about William. I'll do William in blue here. So let's think about William. And then there's two points in time we're talking about. We're talking about now, today, and we're going to talk about 12 years ago. Over here, let's call that now. This will be our now column, and then this will be our 12 years ago. Let's see what we can fill in here. What is Ben's age now? Well, we just defined that as the variable b. That's the unknown. That's what we have to figure out. So let's just stick that there. That's just going to be b. Well, what's Ben's age 12 years ago? So maybe we want to express it in terms of b. If he's b now, 12 years ago, he was just b minus 12. Fair enough. Now what is William's age today? Well, this first sentence gave us the information. William is 4 times as old as Ben. And we can assume that they're talking about today, is 4 times as old as Ben. So if Ben is b, William is going to be 4b. And so how old was William 12 years ago? Well, if he's 4b right now, 12 years ago, he'll just be 12 less than that. So he's 4b now. 12 years ago, he was 4b minus 12. So that's kind of interesting. But we haven't quite yet made use of the second statement. This is William 12 years ago. 12 years ago, William was 7 times as old as Ben. So 12 years ago, this number is going to be 7 times this number. Or another way to think about, take this number and multiply it by 7, and you're going to get this number. 12 years ago, Ben's age is 1/7 of William's age, or William's age is 7 times Ben's age. So let's see if we can set that up as an equation. We can have-- let me write this down-- 7 times Ben's age 12 years ago, b minus 12, is going to be equal to William's age. And it seems like we've done the hard part. We've set up the equation. Now we just use a little bit of our algebraic tools to solve for b. So let's do that. The first thing we might want to do, we could distribute the 7, so 7 times b, 7 times, essentially, a negative 12. We have 7b minus 7 times 12-- let's see. That's 84-- is going to be equal to 4b minus 12. This whole expression is literally 7 times Ben's age 12 years ago. Now what can we do to solve this? Well, we can subtract 4b from both sides, so let's do that. I could do two steps at the same time. Well, actually, let's just keep it simple. I'm going to subtract 4b from both sides. That goes away. On the right-hand side, I have a negative 12. On the left-hand side, I am left with 7b minus 4b is 3b. And then I still have a minus 84. Well, I want to get rid of this negative 84, this minus 84 on the left-hand side. So let's add 84 to both sides. On the left-hand side, I'm just left with 3b. And on the right-hand side, I have negative 12 plus 84, or 84 minus 12, which is 72. Now if I want to solve for b, I just have to divide both sides of that equation by 3. And so I am left with b is equal to-- and now we have our drum roll-- 72/3. And you might be able to do that in your head. It would be 24, I believe. You could work it out on paper if you have trouble. Let's just do it real quick. 72, 3 goes into 7 two times. You get a 2 times 3 is 6, subtract, you bring down the 2, 3 goes into 12 four times. So b is equal to 24. Going back to our question, what is Ben's age now? It is 24. And let's verify that this is actually the case. They're telling us that William is 4 times as old as Ben. So what is William's current age? Well, 4 times 24 is 96, so William is a senior. We should call him Mr. William. He is 96 years old. Maybe he's Ben's grandfather or great-grandfather. Then they say 12 years ago, William-- well, 12 years ago, William was 84 years old. So he was 84 years old. They say that's 7 times as old as Ben. Well, 12 years ago, if he's 24 now, Ben was 12. And indeed, 84 is 7 times 12, so it all worked out.