2465.11 – The Luxury Yacht Gemini

The Gemini is twice as old as its boiler was when the ship was as old as the boiler is now. If the sum of their current ages is 49, how old is the Gemini? How old is the boiler?

Solution

Let ss be the ship's age; bb be the boiler's age; and xx be the number of years to subtract to get back to the time when "the ship was as old as the boiler is now." There are three equations. One gives the sum of the ages of ship and boiler:

s+b=49. (1)s + b = 49. \text{ (1)}

Another is some years back, say xx years back, when "the ship was as old as the boiler is now:"

s−x=b⇝s−b=x. (2)s - x = b \leadsto s - b = x.\text{ (2)}

Finally the Gemini is now twice as old as the boiler was back then:

s=2(b−x)=2b−2x. (3) s = 2 (b - x) = 2b - 2x. \text{ (3)}

Combining (2) and (3) we can get a second equation that, like (1), only has s and x in it:

s=2b−2x=2b−2(s−b)=4b−2s⇝3s=4b. \begin{aligned} s = 2b - 2x = 2b - 2(s - b) = 4b - 2s \\ \leadsto 3s = 4b. \end{aligned}

So let us next take (1) and multiply it by 3.

3s=3(49−b)=147−3b⇝4b=3s=147−3b⇝7b=145⇝b=21. \begin{aligned} 3s = 3 (49 - b) = 147 - 3 b &\leadsto& 4b = 3s = 147 - 3b \\ &\leadsto& 7b = 145 \\ &\leadsto& b = 21. \end{aligned}

Knowing bb, we find that ss = 49 - b = 28. Incidentally, x=7.x = 7.

By the way, there is (or was) a wonderful yacht named Gemini, that once belonged to the Duke of Windsor. Do Google images for "yacht Gemini 1941" and you can find a picture of it. Stella sailed on that yacht twice in 1969, the date in which this problem is set.