Age word problems
Ex 3 Age word problem William is 4 times as old as Ben. 12 years ago, William was 7 times as old as Ben. How old is Ben now?
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- 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?
- So, once again, it's a good idea
- to try to do this on your own first and
- then I'll work through it.
- So, what's the unknown here?
- The unknown here is: How old is Ben now?
- So let's set a variable equal Ben.
- We do x or y but since Ben starts with a B,
- let's use B for Ben.
- So let's let B = Ben's current age
- Ben's age now.
- So let's see how all these other information relates to
- Ben's current age and then maybe we could set up
- some equations and solve for this.
- I'll do a little more structure in this,
- where you could have done many of the problems
- you've been working on in this way,
- so let's about Ben, and then let's think about William
- I'll put 'William' in blue.
- So let's think about, William.
- And then there's 2 points in time we're talking about.
- We're talking about now, today and
- we're gonna talk about 12 years ago.
- Over here, let's say today, or let's call that 'Now'
- This will be our 'Now' column and this
- would be our '12 years ago'.
- So let's see what we can fill in here.
- So 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 that's just going to be 'B'.
- What's Ben's age 12 years ago?
- Well, maybe we want to express it in terms of B.
- If it's 'B' now, 12 years ago it was just 'B - 12'.
- Fair enough.
- Now, what is William's age today?
- This first sentence gave us the information.
- William is 4 times as old as Ben, and we can assume
- that they are talking about today.
- So if Ben is 'B', William is going to be '4B'.
- And so how old was William 12 years ago?
- If he's '4B' right now, 12 years ago it would be 12 less than that.
- 12 years ago it would be '4B - 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 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 it is to take
- this number and multiply it by 7
- and you're gonna get this number.
- 12 years ago, Ben's age was 1/7 of William's age, or
- William's age is 7 times Ben's age
- Let's see if we can set that up as an equation.
- We can have, 7 times this, Ben's age 12 years ago,
- is going to be equal to William's age
- And it seems like we've done the hard part.
- We set up the equation, we just have to use
- our algebraic tools to solve for 'B'
- Let's do that.
- The first thing that we want to do. We can distribute the 7.
- 7 times B, 7 times negative 12.
- So we have '7B - 84' is going to be equal to '4B - 12'
- this whole expression is 7 times Ben's age 12 years ago
- Now what we can do to solve all of these?
- Well, we can subtract '4B' from both sides.
- Actually I can do 2 steps at the same time, but let's just keep it simple.
- Let's subtract '4B' on both sides.
- That goes away, and on the right hand side I have a '-12'.
- On the left hand side, I'm left with, '7B - 4B = 3B'
- and then I still have my '-84'.
- I want to get rid of this '-84', so let's minus 84 on the left hand side.
- So let's add 84 on both sides.
- On the left hand side, I'm just left with '3B',
- and on the right hand side, I have
- '-12 + 84 = 72'
- Now if I want to solve for B, I just have to divide the whole
- equation by 3.
- And so I am left with 'B' is equal to, drum roll, '72/3'
- It would be 24.
- So 'B' is equal to 24
- Going back to our question: What is Ben's age now?
- It is 24.
- Let's verify this is actually the case.
- So they are telling us that William is 4 times as old as Ben.
- So what is William's current age?
- So '4 x 24 = 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, William was 84 years old.
- They say that he is 7 times as old as Ben.
- Well, 12 years ago, if he's 24 now, Ben was 12.
- And indeed, '84 = 7 x 12'
- And it all worked out.
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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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