Problem Solving

Bob and John form a team together. Bob is as old as John will be when Bob is twice as old as John was when Bob was half as old as the sum of their current ages. John is as old as Bob was when John was half as old as he will become over ten years.

How old are Bob and John?

A. John is 40 years old and Bob 30
B. John is 45 years old and Bob 36
C. John is 30 years old and Bob 40
D. John is 36 years old and Bob 45
E. John is 32 years old and Bob 44

OA C

We have two statements here.

1. Bob is as old as John will be when Bob is twice as old as John was when Bob was half as old as the sum of their current ages.

Here the statement talks about 4 ages of Bob and John. Their current age and ages at 3 different times of their life. From the First line Bob is as old as John will be implies that Bob is older than John, hence we have remove choices A and B.

Now analyze the statement one by one. Last part says that Bob was half as old as the sum of their current ages.

B1 = (Bc + Jc)/2 ------- (1)
Where B1 is age of Bob at that particular time and Bc and Jc the current ages of Bob and John.
When Bob was B1, John was J1.

Second part of the statement says that 'Bob is twice as old as John when John's age was J1'

B2 = 2J1 ------ (2)
B2 is age of Bob at that particular time.

Now first part of the statement says that - 'Bob is as old as John when John will be J2'

Bc = J2 --------- (3)
i.e current age of Bob, Bc is equal to J2

2. John is as old as Bob was when John was half as old as he will become over ten years.


From second part of the above statement we have
J3 = (Jc + 10) / 2 - (4)

Where J3 is John's age at that particular time.
Bob's age at that time will be B3.

Now from the first part of the statement we get John is as old as Bob was when John was J3
i.e
Jc = B3 - (5)


Lets substitute each answers into these equations. Since the second statement gives simple equations, we will start with that

C Says Jc = 30 years and Bc = 40
Substituting in (4) we have J3 - (30 + 10)/2 = 20
When John was 20 (i.e 30 - 10), Bob should be 30 (i.e 40 - 10)
ie B3 =30
This satisfies the equation (5)

D Says Jc = 36, Bc = 45
In the same way we have J3 = (36 + 10)/2 = 23
When John was 23 , Bob should be 32, i.e B3 = 32
This doesn't satisfy eq (5). Hence ruled out

E says Jc = 32, Bc = 44
We have J3 here as (32 + 10)/2 = 21
When John was 21, Bob would be 33, i.e B3 = 33.
This doesnt satisfy eq (5).

Hence answer is C.

C will also satisfy all of the other 3 equations.

even after read the explanation, I still think this is the hardest GMAT reading comprehension question I've ever seen. :roll: :twisted:

Let us call the current age of Bob B and of John J.

The first fact is that Bob is as old as John will be when Bob is twice as old as John was when Bob was half as old as the sum of their current ages.

When Bob was half as old as the sum of their current ages, he had reached the age of (B + J) / 2 years. This is now B – (B + J) / 2 = B/2 – J/2 years ago. At that moment, John was J – (B/2 – J/2) = 3/2 J – B/2 years old. If Bob is twice as old as John at that moment, then Bob is (3/2 J – B/2) * 2 = 3 J – B years old. That moment is 3 J – 2 B years from now. Then John will be J + 3 J – 2 B = 4 J – 2 B years old. And that is exactly the age of Bob, so B = 4 J – 2 B which gives that 3 B = 4 J.

The second fact is that John is as old as Bob was when John was half as old as he will become over ten years.

When John had half the age as he will have over ten years, he was (J+10)/2 years old. This is now J – (J + 10) / 2 = J/2 – 5 years ago. At that moment Bob was B – (J/2 – 5) years old. According to the second fact, John is now as old as Bob was at that moment, so J = B – (J/2 – 5). It now follows that 3 J / 2 = B + 5.

Summarizing, we end up with two equations. 3 B = 4 J B + 5 = 3 J / 2

Multiplying the bottom equation with -3 gives after adding the top equation -15 = (4 – 9/2) J Solving this equation gives J = 30. And now it easily follows that B = 40.

So John is 30 years old and Bob 40.



explanation takes more time than action