Bunuel wrote:

Tough and Tricky questions: Word Problems.

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.

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Let Bobby's current age = B and Johnny's current age = J.

Let Bobby's earlier age = B-x and Johnny's earlier age = J-x.

Here, x = the number of years between the CURRENT year and the year WHEN JOHNNY WAS HALF AS OLD AS BOBBY IS NOW.

Johnny was half as old as Bobby is now.Since Johnny's EARLIER age is equal to half Bobby's CURRENT age, we get:

J - x = B/2

J - B/2 = x

Johnny's age now is the same as Bobby's age was.Since Johnny's CURRENT age is equal to Bobby's EARLIER age, we get:

J = B - x

x =

B - J.

The expressions in blue are both equal to \(x\) and thus are equal to EACH OTHER:

B - J = J - B/2

2B - 2J = 2J - B

3B = 4J

B = (4/3)JStatement 1: Bobby is currently four times as old as he was when Johnny was bornCase 1: J=3, implying that B = (4/3)(3) = 4

Since J=3, Johnny was born 3 years ago.

Since B=4, at Johnny's birth 3 years ago Bobby must have been 1 year old, satisfying the condition that Bobby is currently four times as old as he was when Johnny was born.

Case 2: J=6, implying that B = (4/3)(6) = 8

Since J=6, Johnny was born 6 years ago.

Since B=8, at Johnny's birth 6 years ago Bobby must have been 2 years old, satisfying the condition that Bobby is currently four times as old as he was when Johnny was born.

Since B can be different values, INSUFFICIENT.

Statement 2: Bobby was six years old when Johnny was bornIn other words, Bobby is 6 years older than Johnny:

B = J+6Since we have two distinct linear equations (the green equations above) and two variables (B and J), we can solve for the two variables.

Thus, the value of B can be determined.

SUFFICIENT.

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