How to solve age problem: Casey is 4 years older than Aaron....

falcios

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Can you please help me solve this age problem?


Casey is 4 years older than Aaron. Three times Casey’s age in 3 years will equal 2 less than 5 times Aaron’s age in 3 years. How old are they now?


The answer is Casey is 8, Aaron is 4.


This is how I setup the equation but it doesn’t turn out the right answer. I am not sure what numbers are inside or outside the parentheses.

Casey x+4

Aaron x

In 3 years
Casey 3(x+4+3)

Aaron 5(x-2+3)

Thanks in advance.
 

Denis

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Aaron 5(x-2+3)
That's not quite correct.

c = Casey, a = Aaron

c = a + 4 [1]

3(c + 3) = 5(a + 3) - 2 [2] ......OK?

Substitute [1] in [2], then do the rock n' roll :rolleyes:
 

falcios

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Sep 28, 2016
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Parenthesis question

Thanks Denis

How did you know to include the 3 inside the parenthesis and -2 outside? That's where I'm confused.
Does the question give clues to do that?



That's not quite correct.

c = Casey, a = Aaron

c = a + 4 [1]

3(c + 3) = 5(a + 3) - 2 [2] ......OK?

Substitute [1] in [2], then do the rock n' roll :rolleyes:
 

Denis

Senior Member
Joined
Feb 17, 2004
Messages
1,480
"Three times Casey’s age in 3 years will equal 2 less than 5 times Aaron’s age in 3 years."

That means: (5 times Aaron’s age in 3 years) - 2
 

Dr.Peterson

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Nov 12, 2017
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Can you please help me solve this age problem?


Casey is 4 years older than Aaron. Three times Casey’s age in 3 years will equal 2 less than 5 times Aaron’s age in 3 years. How old are they now?


The answer is Casey is 8, Aaron is 4.


This is how I setup the equation but it doesn’t turn out the right answer. I am not sure what numbers are inside or outside the parentheses.

Casey x+4

Aaron x

In 3 years
Casey 3(x+4+3)

Aaron 5(x-2+3)

Thanks in advance.
The grammar of a problem like this can be more complicated than the algebra -- especially since English doesn't have as precise an order of operations as algebra. So it can be helpful to take it in small steps, and write down as much as possible so you can see what you're doing.

You made a great start, defining a variable and some expressions:

Aaron's age now = x
Casey's age now = x+4
Aaron's age in 3 years = x+3
Casey's age in 3 years = (x+4)+3 = x+7

Note that there is no reason not to simplify that last bit now, so there's less to copy wrong later.

Now put those into the problem:

Three times Casey’s age in 3 years = 2 less than 5 times Aaron’s age in 3 years.
Three times (x + 3) = 2 less than 5 times (x + 7).

I used parentheses to make sure each expression is treated as a single quantity.

Now continue:

Three times (x + 3) = 3(x + 3)
2 less than 5 times (x + 7) = 5(x + 7) - 2

And the equation is

3(x + 3)= 5(x + 7) - 2

Are you okay with this last part?
 

falcios

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Sep 28, 2016
Messages
17
Parenthesis Question

So the three years is included in the brackets for Aaron because it follows the same pattern as for Casey?

I can solve the equation. I am not clear why -2 is outside of the brackets?


The grammar of a problem like this can be more complicated than the algebra -- especially since English doesn't have as precise an order of operations as algebra. So it can be helpful to take it in small steps, and write down as much as possible so you can see what you're doing.

You made a great start, defining a variable and some expressions:
Aaron's age now = x
Casey's age now = x+4
Aaron's age in 3 years = x+3
Casey's age in 3 years = (x+4)+3 = x+7

Note that there is no reason not to simplify that last bit now, so there's less to copy wrong later.

Now put those into the problem:
Three times Casey’s age in 3 years = 2 less than 5 times Aaron’s age in 3 years.
Three times (x + 3) = 2 less than 5 times (x + 7).

I used parentheses to make sure each expression is treated as a single quantity.

Now continue:
Three times (x + 3) = 3(x + 3)
2 less than 5 times (x + 7) = 5(x + 7) - 2

And the equation is
3(x + 3)= 5(x + 7) - 2

Are you okay with this last part?
 

Dr.Peterson

Elite Member
Joined
Nov 12, 2017
Messages
3,078
So the three years is included in the brackets for Aaron because it follows the same pattern as for Casey?

I can solve the equation. I am not clear why -2 is outside of the brackets?
Think of it this way: We have the phrase "2 less than 5 times Aaron’s age in 3 years". Read it bit by bit:

"2 less than [something]" means "[something] - 2".

The [something] is "5 times (x+7)", that is, 5(x+7).

Putting it together, we have 5(x+7) - 2.

So why is the -2 outside the parentheses? Because the parentheses have been there from the start, holding together the expression that means "Aaron's age in 3 years". Nothing is going to open up those parentheses until you start simplifying! That is a single quantity, which we are first multiplying by 5 to get "5 times Aaron's age ...", and then decreasing by 2 to get "2 less than" that.

The key is to always think about the order of the things being done in the phrase:

"2 less than ... what?" - the last thing we'll be doing is subtracting 2 from something.
"... 5 times ... what?" - before that, we will be multiplying something by 5.
"... Aaron’s age in 3 years" - we'll be starting with the phrase we already wrote for this, and multiplying by 5, then subtracting.

Another thing worth doing is writing an expression, then checking it against what the problem said. As you evaluate the expression, will you be doing what it says? If not, figure out why.

So, why would the -2 be inside the parentheses, in your opinion?
 

Denis

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Joined
Feb 17, 2004
Messages
1,480
Three times Casey’s age in 3 years will equal 2 less than 5 times Aaron’s age "in 3 years".

Aaron 5(x-2+3)
5(x-2+3) = 5(x+1)

So that means you're changing the given condition to:
Three times Casey’s age in 3 years will equal 2 less than 5 times Aaron’s age "in 1 year".
 

falcios

New member
Joined
Sep 28, 2016
Messages
17
Thanks so much Denis.

I think I got it now.

5(x-2+3) = 5(x+1)

So that means you're changing the given condition to:
Three times Casey’s age in 3 years will equal 2 less than 5 times Aaron’s age "in 1 year".
 
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