Manhattan/Williamsburg GRE & GMAT Math Tutor

Laura is 3 times as old as Maria was when Laura was as old as Maria is now. In 2 years Laura will be twice as old as Maria was 2 years ago. How old are they now?


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Translating english to algebra:

"Laura is 3 times as old as Maria was ":
`1_a`: `Laura_(now) = 3 * Maria_(then)`

"Laura was as old as Maria is now":
`2_a`: `Laura_(then) = Maria_(now)`

"In 2 years Laura will be twice as old as Maria was 2 years ago":
`3_a`: `Laura_(now) + 2 = 2 * (Maria_(now) - 2)`

With four unknowns and three equations, it's not yet solvable, and it seems like we've used all of the information from the problem.... except:

"Maria was when Laura was", this tells us this:
`4_a`: `Maria_(now) - Maria_(then) = Laura_(now) - Laura_(then)`

Now that we have four independent equations, let's solve with substitution:

Substituting (`1_a`) into (`3_a`) and (`4_a`) to get:
`3_b`: `3*Maria_(then) + 2 = 2 * (Maria_(now) - 2)`
and:
`4_b`: `Maria_(now) - Maria_(then) = 3*Maria_(then) - Laura_(then)`

Substituting (`2_a`) into (`4_b`) :
`4_c`: `Maria_(now) - Maria_(then) = 3*Maria_(then) - Maria_(now)`

solving (`3_b`) for `Maria_(now)`:
`3_c`: `Maria_(now) = (3*Maria_(then) + 2)/2 + 2`

Simplifying (`4_c`):
`4_d`: `Maria_(now) = 2*Maria_(then)`

substituting (`3_d`) into (`4_d`):
`4_e`: `(3*Maria_(then) + 2)/2 + 2 = 2*Maria_(then)`

solving for `Maria_(then)`:
`4_f`: `Maria_(then) = 6`

and substituting (`4_f`) into (`4_d`):
`4_e`: `Maria_(now) = 12`

and (`4_e`) into (`3_a`):
`3_e`: `Laura_(now) = 18`

and (`4_e`) into (`2_a`):
`2_b`: `Laura_(then) = 12`