Solve This Pzzlr: Filling the Tank

Long enough to have a few beers at lunch....

Close the drain valve, fool.

1-3/5 hours.

1/(1/2 + 1/4 - 1/8) hours.

Same equation and answer as #4, packaged more neatly. Both are right, and together (it would take both parts), they would satisfy the math teacher in me.

A+

Maybe so. But I understand #4.

So did I, Lyn. It was well laid out, and very straightforward. But JohnDG's addition was the "pocket kit" version of the knowledge, as opposed to Rixter's "take the parts out of the scrap box and lay them out for use" version. Both have their place. I would not REMEMBER JohnDG's answer, if the problem were real-life, and came up when I didn't have a computer handy. But I could remember Rixter's version, and reapply it, tweaked for my situation, remembering the logic JohnDG supplied.

I am reminded of a use I made of an obscure fact when I was answering a question on CTM-2 (PO2, but with a specialty, in the USN, NavSecGru) test.

The question was multiple choice, and had to do with the wavelength of a signal in the Lower Megahertz band. I don't remember exactly where it hit, but it was something that was multiple of 60 Hz, like 15 MHz, or 30Mhz, or 22.5 Mhz (yep, that one is, too!), and I couldn't for the life of me remember the formula for calculating it from the capacitive reactance given in the problem. But the resonant frequency was given.

But I DID remember that my High School Electronics teacher had once mentioned casually that 60 Hz had about a 3300 mile wavelength.

Since I also remembered how to work ratios, and ("60hz/3300mi" = "the giiven frequency/ one of the wavelengths given") I was able to apply algebraic manipulation, and come up with the right answer.

I'd like to say that after I finished the test I ran right out and brushed up on my reactance equations, but I haven't lied yet, and I don't want to start now.

Being able to remember the "pocket kit" version of the info pulled my fat out of the fire. After I passed the appropriate Mil-Lead test that week, I made 2nd Class Petty Officer in no time, and my wife and I were very pleased with the results.

So, both pieces of info are necessary from a dedicated teacher's point of view. I want my students to have full toolboxes, AND a well supplied pocket "emergency kit", when I get done with them.

For a full-bore algebraic equation, x/4 + x/2 - x/8 = 1 gives the results already obtained.

Here's a pedantic version to push one of my favorite tools: Dimensional Analysis

In High School Chemistry I learned Dimensional Analysis, which is almost
all I took away from the class, but perhaps one of the most useful and
reused lesson I learned. It is (arguably) one of the most valuable tools in any
engineering discipline.

By keeping the units of measure in the equation, and matching or cancelling
the units in the numerators and denominators, you ensure your answer is in
the correct units. Many an error has crept in when doing distance, temperature,
time calculations that would be eliminated using Dimensional Analysis.

Using it, I'd present the solution this way:

(multiplying by dimensionless unities (2/2) and (4/4) to get common denominators)

Flow 1: (1 tank / 4 hours) * (2/2)
= 2 tanks/8 hours
Flow 2: 1 (tank) / 2 (hours) * (4/4)
= 4 tanks /8 hours
Flow 3: 1 (tank) / 8 (hours)

flow 1 + flow 2 - flow 3 = (2 tanks/8 hours + 4 tanks/8 hours - 1 tank/8 hours)
= (2 + 4 - 1) tanks / 8 hours = 5 tanks / 8 hours

Since you want one tank full you have the equation

5 tanks/8 hours = 1 tank/ x hours

cross multiplying results in:

(5x)tank-hours = 8 tank-hours

5x = 8

x = 8/5

Q.E.D.

Gorgeous!!!

Prior to this, with all the math of studied, I've never heard of Dimensional Analysis. I'll have to take back all the kind things I've said through the years about my great math teachers. THIS is a TOOL. It looks especially helpful to new Algebra Students, trying to make the jump from simple numerical values, to variables with no intrinsic, recognizable place in the scheme of things. Tanks, in place of X, would greatly simplify that understanding, and looks like it would be a neat few steps to put into the intro to Algebra classes.

It looks simple enough, but to avoid missing anything, are there any books you know of that teach it?

Links (or references to books I might find via Amazon) would be most helpful.

"I've never heard of Dimensional Analysis" - where you bin, man? Big up and GA to RufusVS for bringing it up. I do it so automatically that I forget it has a name! Especially handy IMHO for stuff like energy calculations (Joules, Watts, time, mass, distance etc.).Even more important when mixed units are used (metric/imperial etc.).

Product of California Public High Schools, Class of 72. Learned a lot of technical stuff, took all the Science they offered (Physics in my Junior Year, Chem in my Sophomore Year, Bio in Freshman), finished all the available Math by the end of my Senior Year, took three years of hands on Electronics Design. Never one time heard the term Dimensional Analysis, nor saw a presentation that looked like this one. Took both Algebra and Algebra II, plus Geometry and Trigonometry. Calculus wasn't available. Joined the Navy as a Crypto Tech, when that required the highest scores possible on all entry testing (a 114 combination on Gen and Math testing). Spent the first 9 months out of boot camp in ET-A school (before moving on to the classified stuff), with the first 3 weeks supposed to be used for a Vector Analysis course, self-paced (finished it in one week, had to sit around for two weeks till my lock-step ET-A school course started). In ET-A school, the Navy started us on Nuclear Particle Theory with an emphasis on what moves electrons in an Atom, and why. Got into advance circuit theory in about week 5, but I'd already studied all of it in High School.

I say all that to say I'm neither stupid, nor ignorant (you didn't say I am, but I wanted it on the record), but I NEVER saw that presentation, nor heard that term. And I suspect that my math teachers would have been apoplectic had they seen anyone teach something that clearly could so simplify polynomials, and probably even matrix math, while eliminating the confusion of every first-term Algebra student I've ever met or taught, about the value of something so nebulous as "X".

So, I want to learn it. Now tell me where I can. And quit acting like the blow-hard bully in the school-yard, lording your knowledge over the guy who only asked how he could find answers to his questions.

Put up, or SHUT UP!

There's no magic to it (sorry if I sounded snotty). All yer do is make sure that on both sides of any equation, the dimensions are the same. Trivial example:

distance = speed x time

Say distance in miles, time in hours,

miles = (miles/hours) x hours

Hours "cancels out" on the right, and you're left with miles on both sides. Pretty good indication that it's correct.

You could use it to find the formula for speed, just from the units:

speed(mph ≡ miles/hours) =? Oh, yeah, miles divided by hours, so that's distance over time.

Thanks, got that one. I'd like to see how the process deals with a lot of other mathematical issues. I WANT to learn it.

And I apologize for being so prickly. I meant that to start, AND END, with "product of California Public Schools", but it hit a hot spot, because I grow ever so tired of the instructor who tells the students "the only silly question is the question not asked" and then berates the student for asking the question to which the student truly needs (and does not have/understand the answers already given). If it is asked, it needs to be asked. And most of the time, the one who is willing to ask it is not the only one who needs it answered, just the bravest (or the best target for the bullying instructor. One wonders WHY those take up teaching.).