Car Talk Puzzler: A...Question about a Ship and Its Boiler

[MarleysGh0st]

Car Talk has asked another mathematical question for this week's puzzler. Another $26 gift certificate is up for grabs for one person out of all those to send in the right answer!

An Age-Old Question about a Ship and Its Boiler

RAY: This is from my shipping series and it was sent in by Fred Gluck. He writes: "Back in the old days ships of yesteryear were driven by steam and it was often the case that the ship would outlive the engine and the boiler and they'd have to replace them.

"Now, take one ship for example. When you add the age of the ship and the age of its boiler, it totals 42 years. So S + B = 42. Now pay attention! The ship is twice as old as the boiler was when the ship was as old as the boiler is now."

The question is how old are they?

Answers can be sent to Car Talk using this link:
http://www.cartalk.com/ct/3500.jsp

For me, the hardest part was properly composing the second equation to solve for. Good luck, everyone!

[Bob Juch]

Spoiler

The ship is 24 and the boiler is 18.

[Jeemie]

Spoiler

...three equations. It's been a while since I did this stuff, so I'm guessing I missed something where I could have gotten it in two equations.

Anyhow:

Ship Age = S, Boiler Age = B, Years ago that Ship's Age was Boiler Age today = Y

The equations:

B = 42 - S

Y = 2S - 42

S = 2B - 2Y

Substituting through in sequence, you get

S = 24

B = 18

Y = 6 (years ago that the ship's age was the boiler's age today)

[Jeemie]

Spoiler

Actually, while I was writing it all out, I figured out how to get to two equations!

Years ago ship's age was boiler's age now = S - B

So, we have B= 42 - S and

S = 2 * (B - (S - B)) or S = 2 (2B - S) or S = 4B - 2S or

S = (4B)/3

[MarleysGh0st]

Spoiler

...three equations. It's been a while since I did this stuff, so I'm guessing I missed something where I could have gotten it in two equations.

Anyhow:

Ship Age = S, Boiler Age = B, Years ago that Ship's Age was Boiler Age today = Y

The equations:

B = 42 - S

Y = 2S - 42

S = 2B - 2Y

Substituting through in sequence, you get

S = 24

B = 18

Y = 6 (years ago that the ship's age was the boiler's age today)

Actually, while I was writing it all out, I figured out how to get to two equations!

Years ago ship's age was boiler's age now = S - B

So, we have B= 42 - S and

S = 2 * (B - (S - B)) or S = 2 (2B - S) or S = 4B - 2S or

S = (4B)/3

Spoiler

Funny that you should mention that about two equations! That's the approach I took, althought I called it X, the "difference in ages between S and B" instead of Y, the "'Years ago that Ship's Age was Boiler Age today" and plugged that into that longer equation you have. Same thing.

Then I started wondering about the old rule that you need one more equation than independent variables, so two variables need three equations. Where was my third equation??? I figured it must have been rolled into that big one without my noticing somehow.

And it's still not clear to me that Y = 2S - 42 is one of the three equations. Where did you get that, without starting with Y = S - B and then plugging in the first equation for B?

[MarleysGh0st]

Spoiler

And I'm completely wrong about that, aren't I?

Two unknowns require two equations.

Duh.

[Jeemie]

Spoiler

Funny that you should mention that about two equations! That's the approach I took, althought I called it X, the "difference in ages between S and B" instead of Y, the "'Years ago that Ship's Age was Boiler Age today" and plugged that into that longer equation you have. Same thing.

Then I started wondering about the old rule that you need one more equation than independent variables, so two variables need three equations. Where was my third equation??? I figured it must have been rolled into that big one without my noticing somehow.

And it's still not clear to me that Y = 2S - 42 is one of the three equations. Where did you get that, without starting with Y = S - B and then plugging in the first equation for B?

Y= 2S - 42 comes from

S - Y = 42 - S (42 - S is the current age of the boiler, so Y years ago, the ship was the current age of the boiler).

All becomes simpler when you just remember that Y = S - B. Then you don't have to have three equations, just two: S+B=42 and S=2*(B-(S-B)) (the second equation comes from the fact that the age of the ship now is twice the age the boiler was when the ship was the current age of the boiler! Whew! try saying that 10 times fast!)

And I'm completely wrong about that, aren't I?

Two unknowns require two equations.

Duh.

Yes- if you have n unknowns, you only need n equations to solve for all the unknowns.