Track / Stage 5 / 244 of 400 #844 of 1964

Problem 844

AIME late

Combinatorics Difficulty 5.5 Find the answer

In the mess of one of our fishing fleet's ships, the crew members are talking about their ages.

The helmsman says: "I am twice as old as the youngest sailor and 6 years older than the engineer."

The 1st sailor says: "I am 4 years older than the 2nd sailor and just as many years older than the youngest sailor as I am younger than the engineer."

The 2nd sailor says: "I celebrated my 20th birthday yesterday."

The crew consists of 6 members, and the average age is exactly 28 years.

How old is the captain?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

It is advisable to introduce designations for the ages of the participants:

Let $k, s, m, m_{1}, m_{2}$ , and $m_{j}$ be the ages of the captain, the helmsman, the engineer, as well as the 1st, 2nd, and the youngest sailor. The statements then yield the following system of equations:

Helmsman:	$s=2 m_{j}=m+6$ ,
1st Sailor:	$m_{1}=m_{2}+4$ ,
1st Sailor:	$m_{1}-m_{j}=m-m_{1}$ ,
2nd Sailor:	$m_{2}=20$ ,
Average:	$k+s+m+m_{1}+m_{2}+m_{j}=6 \cdot 28=168$ .

From (4) and (2) it follows that

m_{1}=24

, and from this and (3) it follows that

m_{j}=2 m_{1}-m=48-m

. Substituting the latter into (1) yields

m=30

and

s=36

. Substituting these results into (5) leads to

$k+36+30+24+20+18=168 \quad \Longrightarrow \quad k=40$

Thus, the captain is 40 years old.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.