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

#### falcios

##### New member

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)

#### Denis

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

c = Casey, a = Aaron

c = a + 4 

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

Substitute  in , then do the rock n' roll #### falcios

##### New member
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 

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

Substitute  in , then do the rock n' roll #### Denis

##### Senior Member
"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

##### Elite Member

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)

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

##### New member
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
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

##### Senior Member
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
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".