Equations

James is twice as old as Stephanie, who, 4 years ago, was 3 times as old as Kate. If, 5 years from now, the sum of their (all 3 people) ages will be 51, how old is Stephanie?
A) 6
B) 10
C) 14
D) 20
E) 24

The fastest approach here might be to plug in the answer choices.

Having said that, here's one possible algebraic approach.

Let x = Stephanie's present age.

James is twice as old as Stephanie
So 2x = James' present age.

4 years ago, Stephanie's was 3 times as old as Kate
In other words, 4 years ago, Kate's age was 1/3 of Stephanie's age.
4 years ago, Stephanie'sage was x-4, so Kate's age 4 years ago, was (x-4)/3
So, Kate's present age = (x-4)/3 + 4

In 5 years . . .
Stephanie's age = x + 5
James' age = 2x + 5
Kate's age = (x-4)/3 + 4 + 5

5 years from now, the sum of their ages will be 51
So (x + 5) + (2x + 5) + (x-4)/3 + 4 + 5 = 51
Simplify: 3x + (x-4)/3 + 19 = 51
Subtract 19 from both sides: 3x + (x-4)/3 = 32
Multiply both sides by 3: 9x + (x-4) = 96
Solve . . . [spoiler]x = 10[/spoiler]

Answer: B

If anyone is interested, we have a free video on solving word problems involving past and future ages: https://www.gmatprepnow.com/module/gmat- ... ems?id=908

Cheers,
Brent

James is twice as old as Stephanie, who, 4 years ago, was 3 times as old as Kate. If, 5 years from now, the sum of their (all 3 people) ages will be 51, how old is Stephanie?
A) 6
B) 10
C) 14
D) 20
E) 24

Here's a super fast approach:

We'll use the fact: 4 years ago, Stephanie's was 3 times as old as Kate
In other words, 4 years ago, Kate's age was 1/3 of Stephanie's age.

Check the answer choices:
A) Stephanie is 6
4 years ago, Stephanie was 2, which means Kate was 2/3 years old.
This is no good.
Stephanie's and James' ages will be integers, so when we add Kate's fractional age, we'll get a non-integer for the sum of their ages.
So, eliminate A

B) Stephanie is 10
4 years ago, Stephanie was 6, which means Kate was 2 years old.
This might work
So, keep B

C) Stephanie is 14
4 years ago, Stephanie was 10, which means Kate was 10/3 years old.
This is no good. Eliminate C

D) Stephanie is 20
4 years ago, Stephanie was 16, which means Kate was 16/3 years old.
This is no good. Eliminate D

E) Stephanie is 24
4 years ago, Stephanie was 20, which means Kate was 20/3 years old.
This is no good. Eliminate E

So, answer choice B is the only viable option.

Cheers,
Brent

Hi vinay1983,

Both of Brent's approaches are correct.

Here's another way that you could have approached the question though. It focuses a bit more on the logic of the question and combines with TESTing the answers:

In the first sentence, we're told who's oldest and who's youngest:

James > Stephanie > Kate

We're also told that in 5 years the sum of the ages will = 51. If you subtract 5 years from each of those ages, then you'll have the sum today:

James + Stephanie + Kate = 36

Since James is 2(Stephanie), there's no way for Stephanie to be answer C, D, or E. James would end up being too old.

Furthermore, Stephanie can't be 6 because that would make James = 12 and there's no way for the sum to equal 36. Eliminate A

Final Answer: B

GMAT assassins aren't born, they're made,
Rich