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PS: Ages of Ebo and Atu

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by saadishah » Fri Jan 01, 2016 4:25 pm
I am stuck and unable to solve this using the single variable method. Please help.

5 years ago, Ebo was 3 times as old as Atu. In 3 years, Ebo will be twice as old as Atu. What is the sum o their ages now?

A) 18
B) 24
C) 28
D) 35
E) 42
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by Brent@GMATPrepNow » Fri Jan 01, 2016 4:46 pm
saadishah wrote:I am stuck and unable to solve this using the single variable method. Please help.

5 years ago, Ebo was 3 times as old as Atu. In 3 years, Ebo will be twice as old as Atu. What is the sum o their ages now?

A) 18
B) 24
C) 28
D) 35
E) 42
Here's one approach:

5 years in the PAST
Let A = Atu's age 5 years ago
So, 3A = Ebo's age 5 years ago

3 years in the FUTURE
A + 8 = Atu's age 3 years from now
3A + 8 = Ebo's age 3 years from now


In 3 years Ebo will be twice as old as Atu
This means that 3A + 8 is TWICE AS BIG AS A + 8
To make these values EQUAL, we'll multiply the smaller value by 2
We get: 3A + 8 = 2(A + 8)
Expand: 3A + 8 = 2A + 16
Solve, A = 8

5 years in the PAST
Atu's age 5 years ago = 8
So, Ebo's age 5 years ago = 24

So, their PRESENT ages are 13 and 29
SUM of present ages = 13 + 29 = 42

Aside: Here's a video solution that uses 2 variables- https://www.gmatprepnow.com/module/gmat- ... /video/909

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by [email protected] » Sat Jan 02, 2016 10:20 am
Hi saadishah,

While you can treat this as a one-variable question (as Brent has shown), this prompt is pretty easy to deal with as a 2-variable 'system.'

(E-5) = 3(A-5)
(E+3) = 2(A+3)

With 2 variables and 2 unique equations, you can solve for A and E.

Are you comfortable with the 2-variable approach (and were you just looking for an alternative?)?

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by GMATGuruNY » Tue Jan 05, 2016 4:49 am
5 years ago, Ebo was 3 times as old as Atu. In 3 years, Ebo will be twice as old as Atu. What is the sum of their ages now?

A) 18
B) 24
C) 28
D) 35
E) 42
An alternate approach is to TEST CASES.

5 years ago, Ebo was 3 times as old as Atu.
What is the sum of their ages now?

Case 1: Five years ago A=1 and E = 3*1 = 3
Sum of the their ages now = (1+5) + (3+5) = 14.
14 is not among the answer choices.

Case 2: Five years ago A=2 and E = 3*2 = 6
Sum of the their ages now = (2+5) + (6+5) = 18.
In 3 years, A = 7+3 = 10 and E = 11+3 = 14.
Since Ebo is not twice as old as Atu, eliminate A.

Case 3: Five years ago A=3 and E = 3*3 = 9
Sum of the their ages now = (3+5) + (9+5) = 22.
22 is not among the answer choices.

Notice the PATTERN:
The resulting sums -- 14, 18, 22 -- keep increasing by 4.
Thus, the correct sum must be in the following list:
14, 18, 22, 26, 30, 34, 38, 42...

Only the value in red is among the remaining answer choices.

The correct answer is E.
Last edited by GMATGuruNY on Sun Jul 24, 2016 7:54 pm, edited 1 time in total.
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by Matt@VeritasPrep » Fri Jan 08, 2016 1:16 pm
You won't be able to solve it with a single variable since you have two, but you could think of it this way.

Five years ago, Ebo was (E - 5) and Atu was (A - 5). Back then, Ebo = 3 * Atu, so (E - 5) = 3 * (A - 5).

Three years from now, Ebo will be (E + 3) and Atu will be (A + 3). We'll then have Ebo = 2 * Atu, so (E + 3) = 2 * (A + 3).

The two equations simplify as

E = 3A - 10
E = 2A + 3

Both equations = E, so they equal each other: 3A - 10 = 2A + 3, or A = 13. From there, E = 2*13 + 3, or 29, and the sum of their ages is 42.
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