GRE & GMAT Mathy Tutor: Try this word problem

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 (

$2_a$ ):

$2_b$ :

$Laura_(then) = 12$

GRE & GMAT Mathy Tutor

Tuesday, July 3, 2012

Try this word problem

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