I helped my brother with his Math homework tonight. He's in Grade 9.*An aircraft travels 5432 km from Montreal to Paris in 7 h and returns in 8 h. The wind speed is constant. Determine the wind speed and the speed in still air.*

---> wind Let x = speed in still air M ---------------------> P y = wind speed v = (x + y) d = 5432 km t = 7h

d ---> wind v = --- , d = vt M <--------------------- P t v = (x - y) km/h t = 8h

Since the distance from A to B is the same as the distance from B to A, we have:

7x + 7y = 5432 8x - 8y = 5432

Add the two lines to get:

15x - y = 10864 15x = 10864 + y

Subtract the two lines to get:

-x + 15y = 0 x = 15y

Plug in '15y' for 'x':

225y = 10864 + y 224y = 10864 y = 48.5

Plug in 'y':

x = 15(48.5) x = 727.5Wind speed: 48.5 km/h

Speed of aircraft in still air: 727.5 km/h

Q.E.D.

## Comments

ntangFirst off, I figured the average velocity both ways - 776 kph w/ the wind, 679 kph into the wind, by dividing the distance by the time. Easy enough.

From there, I then subtracted the difference in velocities and divided by two, since it made the trip twice. That got me 48.5 kph. That's basically the crux of the question, right?

Then you just take that and add it to 679 or subtract it from 776 to get the still air velocity.

I'm sure there's some complex formulaic way of doing it, but to me math has always either been just a case of "getting it" or having no clue. I did well (really well) in the courses where I could logic my way through the questions, and really poorly in the courses that required memorizing a lot of complex formulas and numbers.

It was sort of funny, I was in the advanced calculus class in my high school and doing really well until we got to the point where we had to start plugging in trig. Since trig is mostly memorization of the various angles and formulas, I tripped up on that. I went from getting straight a's on everything to suddenly dropping to c's and d's, and my teacher was baffled.

I went to him for after-school tutoring, and he was flat out amazed when he realized my utter lack of clue at trig, considering I had been acing everything that was considered much much harder. He couldn't believe I had trouble with trig. Oh well. :)

(I did study up and managed to rough it through that section, and finished the course with a low a or high b, I forget.)

bridegot me 48.5 kph. That's basically the crux of the question, right?Yup. You could do it that way too. But one of the problems with elementary Algebra concepts is that the problems are so simple that you can just think it through. It makes it _really_difficult_ to justify the learning of Math and Algebra to kids that way.

When you get into more the University Maths, you can't just _think_it_through_, you have to set up the equations, use the formulas and derive your answer from the principles. Getting kids to set it up with symbols and formulas is preparing them to think about it in a logical way, IMHO.

just a case of "getting it" or having no clueI totally agree with that one. And I think North American Math curriculae are seriously lacking in foundation skills - they fail to teach kids how to understand the most basic things. When it doesn't work, they tell kids to memorize.

Part of it, I think, is the lack of respect that the culture, on the whole, gives to Math. Math and Science are not considered "cool" here. English, Arts, Literature and Social Sciences are by far prefered subjects here.

I did well (really well) in the courses where I could logic my way through the questions, and really poorly in the courses that required memorizing a lot of complex formulas and numbers.There's a bit of memorization to everything in life though. Memorizing formulas isn't a matter of memory, so much as it is a matter of experience. If you use it often enough, you'll know it. So, I blame "not enough homework" for inability to remember formulas.

ntangMy point is they aren't teaching valuable skills, such as problem solving, or the actual understanding of mathematical concepts. I did well in those areas. I just had a lot of problem with memorizing formulas that I knew I'd never need to remember in real life NO MATTER WHAT MY JOB WAS.

To give a real life example: I don't remember every command and every config file syntax there is to know in unix, but to be blunt I'm still one of the best sys admins I know. Why? Because I understand how things work, and I can troubleshoot and problem solve like few people can, and when I can't remember a command a 30 second hit on the man page gives me the syntax details I need. When you understand how a system works, the rest is trivia, and trivia is for board games, not real life work.

That's my problem with all of this. Schools should teach kids how to think, not how to memorize crap. I could out-think my classmates, and so was successful even though my memorization skills were pretty shitty. (They still are to this day, but it's never hindered me except during a few interviews, and even then only in those interviews that didn't take the time to see how much I really knew and understood. Most of those I wasn't interested in anyways, because if they hire people based on what they've memorized it's almost always a poor work environment anyways.)

Having some memory and knowledge of facts is of course important, but education (and most people) put way too much emphasis on that aspect of learning, which I think should be one of the lower priorities, not the highest one.

Incidentally, in the US anyways, advanced calculus courses -are- university level, and I did just fine in them. Maybe they're too easy in the US, but I seemed to do fine. I went to a good private school, not public school, and so it's not even a case of just taking advanced calculus classes at a public school. But anyways... :)

brideBy calculating the velocity first and then half of the difference, you're introducing inaccuracy. In this case, the text book picked a nice number that was divisible by both 7 and 8. But in real life problems, you'd want to leave the exact number manipulation for the very end, if at all possible. You avoid rounding and preserve accuracy that way.

=)

ntangIn real life I'd never have to deal with it because I don't give a crap how fast the wind is blowing, and my job doesn't require that I know how fast it's blowing, or anything else similar. ;)