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## Algebra 1

### Course: Algebra 1 > Unit 6

Lesson 6: Systems of equations word problems- Age word problem: Imran
- Age word problem: Ben & William
- Age word problem: Arman & Diya
- Age word problems
- System of equations word problem: walk & ride
- Systems of equations word problems
- System of equations word problem: no solution
- System of equations word problem: infinite solutions
- Systems of equations word problems (with zero and infinite solutions)
- Systems of equations with elimination: TV & DVD
- Systems of equations with elimination: apples and oranges
- Systems of equations with substitution: coins
- Systems of equations with elimination: coffee and croissants
- Systems of equations: FAQ

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# Age word problem: Ben & William

Sal solves the following 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? Created by Sal Khan.

## Want to join the conversation?

- i don't understand age word problems please if you have any advice ,recommendations answer this question because I've tried many different websites but i don't seem to get how to solve age word problems .(29 votes)
- what do you not get about it?(0 votes)

- I feel the crux to understanding Sal in this is that 4b-12 is already 7 times as old as Ben's b-12.

That is why he multiplied Ben's (b-12) by 7, so it can equal to William.

I guess I'm sorta saying the 7 times is already applied to the equation right from the beginning. So you have to compensate by multiplying Ben's past age by 7 to equal William's.

Therefore 7*(b-12) = 4b-12(21 votes) - Can someone help explain this to me? I need help with these kinds of things.(8 votes)
- Does anyone have a quick way to calculate division with big numbers like 256/4? (I know its 64 so don't JUST give me the answer) I could do the 72/3 in my head but if there is a good way to do those types of division mentally I'd be glad to hear it.(3 votes)
- There are many tips and tricks to dividing numbers in your head.

In your example, 256/4 is the same as 256/2/2 (you can plug this into your calculator to check). So, in reality all you have to do is halve the number twice. Ex: 256->128->64, which is easier to do in your head than doing brute force dividing. There is no quick way to divide by 3 other than just doing it the long way in your head, but there is a divisibility rule (a rule that tells you if 3 can evenly go into a number) for it. If you add up the digits of the number you are trying to divide by 3, and the sum of the digits is divisible by 3, the entire number is divisible by 3. With your number, 72 you can add the digits. 7+2=9. 9, we know, is divisible by 3, so 72 is divisible by 3. This divisibility rule works for all numbers.(11 votes)

- Couldn't you make an equation differently and end up with the same answer? Like I got the equations w=4b and w-12=7b and solved using substitution but didn't get the correct answer? Why?(0 votes)
- You have a mistake in your 2nd equation.

"b" represents Ben's current age. When you create the 2nd equation, both Williams and Ben's ages need to be 12 years younger. So, your 2nd equation need to be:`w-12 = 7(b-12)`

Hope this helps.(17 votes)

- At2:32; I don't understand why for William (12 years ago column) it's 4b-12 and not just 7b?(5 votes)
- Think of it this way: Pretend William is 6 years old and Ben is 2, so Will is 3 times the age as Ben, so the equation is (3)2=6, the 2 is Ben's age, the 3 is how many times Ben's age goes into Wills, and the 6 is Will's age. Now, since the real question says that William is 7 times as old as ben, we can set up the equation as this: 7(b-12) = 4b-12. B-12 is Ben's age 12 years ago and 4b-12 is Will's age 4 years ago.(1 vote)

- I still have
**ZERO**idea on how to do this. Can u suggest other videos that might help and include the link?(5 votes) - Wait a sec...if 12 yrs ago, William was 7x as old as Ben, and they both have a birthday every year, shouldn't he stay 7 times older? And, i would understand if the 7 was instead an even number. But 7? I don't really get that.(1 vote)
- If I am 12 now, and my younger brother is 4 years old, then next year, I will be 13 (at the end of the year), and my brother will be 5. Before, I was 3 times as old as my brother, now, I am only 2.6 times as old as him. So the ratio of the ages will change over time; however, the difference between 2 peoples ages will stay the same.(10 votes)

- I let William age w and Ben age b

And make these equations

W= 4b therefor w-4b=0

W-12= 7(b-12) solved this and got w-7b=-72

Then eliminated w by subtracting both w- 4b = 0 and w-7b=-72.

I got the same answer. Is it correct?(5 votes)- yuh youre right(1 vote)

- how was william 7 times older then ben but now he is 4 times older then ben(3 votes)
- Because people age at the same rate. Look at a simple example: William is 1 and Ben is 3, Ben is three times older. In a year, William will be 2 and Ben will be 4, so Ben will be only twice as old. We always divide Ben's age/Williams age. In 20 years, William will be 21 and Ben will be 23, so Ben would be 23/21 older than William.(4 votes)

## Video transcript

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.