Bunuel wrote:

Five years ago Jim was three times as old as Raoul was and Monica was six years older than Raoul was. If all three are still living in five years, which of the following must be true about their ages five years from now?

I. Monica is older than Jim.

II. Raoul is six years younger than Monica

III. The combined ages of Jim and Raoul are more than Monica's age.

A. I only

B. II only

C. I and II

D. I and III

E. II and III

We can let Jim’s age today = J, Raoul’s age today = R, and Monica’s age today = M.

Let’s set up their ages 5 years ago: Jim was (J - 5), Rauol was (R - 5), and Monica was (M - 5).

Since five years ago Jim was three times as old as Raoul was:

(J - 5) = 3(R - 5)

J - 5 = 3R - 15

J = 3R - 10

Since five years ago Monica was six years older than Raoul was:

(M - 5) = (R - 5) + 6

M - 5 = R + 1

M = R + 6

Notice that Raoul is the youngest of the three people, and Raoul must be more than 5 years old since only then can we talk about their ages 5 years ago.

Let’s now test each Roman numeral:

I. Monica is older than Jim.

We can represent Monica's age in 5 years as M + 5, or R + 6 + 5 = R + 11.

We can represent Jim’s age in 5 years as J + 5, or 3R - 10 + 5 = 3R - 5.

Is M + 5 > J + 5 ?

Is R + 11 > 3R - 5 ?

Is 16 > 2R ?

Is 8 > R ?

Is R < 8?

We know that R > 5, however, we can’t determine whether R < 8. Thus, we cannot determine whether Monica is older than Jim.

II. Raoul is six years younger than Monica

Since M = R + 6, in 5 years Raoul will still be 6 years younger than Monica. Roman numeral II is true.

III. The combined ages of Jim and Raoul are more than Monica's age.

We already see that in 5 years, Monica’s age will be R + 11, Jim’s age will be 3R - 5, and Raoul’s age will be R + 5. We can create the following inequality:

Is 3R - 5 + R + 5 > R + 11 ?

Is 4R > R + 11 ?

Is 3R > 11 ?

Is R > 11/3 ?

We know that R > 5, so R > 11/3. Thus Roman numeral III is true.

Answer: E

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Scott Woodbury-Stewart

Founder and CEO

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