Age word problem: Ben & William

let's do some more of these classic aged word problems so we're told that William is four times as old as men 12 years ago William was seven times as old as Ben how old is Ben now so once again it's a good idea to try to do this on on your own first and then I'll work through it so 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 let's let B equal Ben's current age Ben's age now so 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 so 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 so let's think about Ben let's think about then and then let's think about William Abdul William in blue here so let's think about William 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 so let's talk about so over here let's say today or let's call that now this will be our now column and then this will be our 12 years ago 12 years ago so let's see what we can fill in here so what is Ben's age now well we just define that is the variable B that's the UH know that's what we have to figure out so let's just stick that there so that's going to be that's just going to be B well what's Ben's age 12 years ago and so when we would maybe want to express it in terms of B well if he's B now 12 years ago he was just B minus 12 fair enough now what is Williams age today well this first sentence gave us the information William is four times as old as Ben and we can assume that they're talking about today is is four times as old as bad so if Ben is be William is going to be for B for B and so how old was William 12 years ago well if he's for B right now 12 years ago he'll just be 12 less than that so he's for B now 12 years ago it was for B minus 12 so that's kind of interesting but we haven't quite yet made use of the second statement 12 years ago William so this is William 12 years ago 12 years ago William was seven times as old as man so 12 years ago this number is going to be 7 times this number or another way to think about it take this number and multiply it by 7 7 and you're going to get this number 12 years ago then was Ben's age is 1/7 of Williams age or Williams age is seven times Ben's age so let's see if we can set that up as an equation so we can have we can have 7 times this let me write this down 7 times Ben's age 1012 years ago 7 times 10 Ben's age 12 years ago B minus 12 is going to be equal to Williams age is going to be equal to Williams 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 so the first thing we might want to do we could distribute the 7 so 7 times B 7 times essentially negative 12 or so we have 7 B minus 7 times 12 let's see 7 it's 84 84 is going to be equal to 4 B minus 12 is going to be equal to 4 B minus 12 this is this whole expression is already seven times Ben's age 12 years ago now what can we do to solve this well we can subtract 4 B from both sides so let's do that for B I can do two steps at the same time well actually let's just let's just keep it simple so I'm going to subtract 4 B from both sides that goes away on the right hand side I have a negative 12 on the left-hand side I'm left with 7 B minus 4 B is 3 B 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 let's add 84 to both sides on the left hand side I'm just left with 3 B and on the right hand side I have negative 12 plus 84 or 84 minus 12 which is 7d 272 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 a now we have our drumroll 72 divided by 3 and you might be able to do that in your head it would be 24 I believe and you could work it out on paper if you have trouble let's just do it real quick 72 3 goes into 7 2 times then you get a you get a 2 times 3 is 6 subtract you get bring down the 2 3 goes into 12 4 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 so they're telling us that William is four times as old as bed so what is Williams current age well 4 times 24 is 96 so William is a is a CD 8 we should call him mr. William so let's see is 96 years old maybe he's Ben's grandfather or great-grandfather and they say 12 years ago William well 12 years ago William was 84 years old so he was 84 years old they say that 7 times as old as Ben well 12 years ago Ben if he's 24 now Ben was Ben was 12 and indeed 84 is 7 times 12 so it all worked out