Data Sufficiency

Most Manhattan GMAT students are trying to break the 700 barrier. As a result, we've developed our own math problems written at the 700+ level; these are the types of questions you'll WANT to see, when you are working at that level. Try to solve this 700+ level problem (I'll post the solution next Monday).

Now and Then:
Bobby and his younger brother Johnny have the same birthday. Johnny's age now is the same as Bobby's age was when Johnny was half as old as Bobby is now. What is Bobby's age now?

(1) Bobby is currently four times as old as he was when Johnny was born.
(2) Bobby was six years old when Johnny was born.

(A) Statement (1) ALONE is sufficient to answer the question, but statement (2) alone is not.
(B) Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.
(C) Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient to answer the question.
(E) Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question.

I think the answer is (B)

Here is my solution -

Johnny's age now is the same as Bobby's age was when Johnny was half as old as Bobby is now means
J = B/2 + (B-J)

J = Johnny's present age
B/2 = Johnny's age when he was half the age of Bobby's present age
B/2 + (J-B) = Bobby's age was when Johnny was half as old as Bobby's present age

Solves to 4J=3B

Using statement (1), Bobby is currently four times as old as he was when Johnny was born.
B = 4*(B-J) => 4J = 3B. Hence not sufficient.

Using statement (2), Bobby was six years old when Johnny was born.
B-J=6. Which is sufficient to answer the ques about Bobby's present age.

Although we dont need to calcuate but still ... Bobby's present age is 24 and Johnny's present age is 18.

Kevin, I would say tough one! Really enjoyed doing it.

Answer

In questions like this, it helps to record the given information in a table. Upon initial reading, the second sentence is probably very confusing but what is clear is that it discusses the ages of the two boys at two different points in time: let's refer to them as “now”, and “then”. So, let's construct a table such as the one below. Let x and y denote the boys' ages “now”:

Johnny's age Bobby's age
now x y
then


Now, re-read the first few words of the second sentence: “Johnny's age now is the same as Bobby's age . . . ‘then’”. We can fill in one more entry of the table as shown:

Johnny's age Bobby's age
now x y
then x



Finally, the rest of the second sentence tells us that “then” was the time when Johnny's age was half Bobby's current age; i.e., Johnny's age “then” was (1/2)y. We can complete the table as follows:

Johnny's age Bobby's age
now x y
then (1/2)y x



One way to solve this problem is to realize that, as two people age, the ratio of their ages changes but the difference in their ages remains constant. In particular, the difference in the boys ages “now'” must be the same as the difference in their ages “then”. This leads to the equation: y - x = x - (1/2)y, which reduces to x = (3/4)y; Johnny is currently three-fourths as old as Bobby.

Without another equation, however, we can't solve for the values of either x or y. (Alternatively, we could compute the elapsed time between “then” and “now” for each boy and set the two equal; this leads to the same equation as above.)

(1) INSUFFICIENT: Bobby's age at the time of Johnny's birth is the same as the difference between their ages, y - x. So statement (1) tells us that y = 4(y - x), which reduces to x = (3/4)y. This adds no more information to what we already knew! Statement (1) is insufficient.

(2) SUFFICIENT: This tells us that Bobby is 6 years older than Johnny; i.e., y = x + 6. This gives us a second equations in the two unknowns so, except in some rare cases, we should be able to solve for both x and y -- statement (2) is sufficient. Just to verify, substitute x = (3/4)y into the second equation to obtain y = (3/4)y + 6 , which implies y = 24. Bobby is currently 24 and Johnny is currently 18.

The correct answer is B, Statement (2) ALONE is sufficient to answer the question, but statement (1) alone is not.