I’ve seen problems like that in math classes at multiple high schools.
“College” level is quite ambiguous, but, yes, quadratic equations and this type of one-variable algebraic manipulation are appropriate for a first course in algebra. It’s not an advanced topic. Definitely to be mastered before beginning a university course.
Systems of polynomial equations, e.g. x² + 2y² = x² + xy + y² = 2 are a more advanced topic, though.
{(sqrt(2),0) , (sqrt(2)/3,sqrt(2)/3)}
Sorry, I couldn’t resist.
You have the right idea, but have omitted negative solutions, and also the second solution is wrong (perhaps a typo), so not quite top marks 
Of course, in the general case algorithmically eliminating variables in a system of polynomial equations is a lot more complicated than in that example, and I doubt students ever learn how to do it unless they enroll in university-level algebra courses. It’s not really a general skill like solving quadratic equations in a single variable or even basic calculus.
Braino, not typo. I did it in my head, and got a couple of the steps in the wrong order.
I bought a book on doing algebra problems at a bookstore, and it explained how to do this type.
Heh. I’m the opposite: had no trouble with algebra I but algebra II was largely over my head. Only class other than English I ever had to retake.
w = 20 *.60 = 12 ounces
You said this was complicated but it is so easy that I feel like I missed something. Did I?
You missed that I said it was more complicated (than the earlier problems in the same set).
You may have also missed another point, that I didn’t explicitly mention: For some reason, many beginning students seem to have a lot of trouble with these “mixture” problems. Something about them seems to be confusing to a lot of people, and that’s on top of the problem that so many students don’t really know how to work with percentage problems.
Remember, these are “simple” problems from the earlier portion of a beginning algebra text, for new algebra students. Anyone with substantial math knowledge should find these easy. I’m trying to describe them from a beginning students’ point of view.
Even more complicated are those problems on the general form: You have a certain amount of an mixture at a certain percentage, and a certain amount of a mixture at another percentage, and you mix them together. What do you end up with?
Or those problems that ask: How much of that second mixture to you need to add to the first mixture to end up with a new mixture at a desired percentage?
I always had trouble with those. But again, the difficulty was always in setting up the right equation, not in solving the equation once it was set up.
See that’s what happens when you don’t show all your work! :smack:
For those of you who are more current in math teaching that I am:
Maybe I should ask: Is there a distinction between beginning college-level algebra at universities versus community colleges?
My observations come from community colleges, as of about 20 years ago. There, algebra classes are, by intention, addressed to students to didn’t take it in high school, or did poorly, or need a refresher. I suspect they are often calibrated for students who really aren’t expected to go on to more advanced math.
OTOH, I don’t think universities even teach beginning algebra, or other high-school level classes like the community colleges do.
I did some tutoring back then (mostly in Statistics). But I saw that various algebra texts were way too perfunctory than I thought was good. I discussed it with the chairman of the math/science department, who agreed. He agreed that the book they were using was shit, but he also said it was the best one they could find. :eek: You might say, he and I are both “old school”.
That 1948 book that I have is definitely “old school”, and is definitely a college-level course.
You don’t even need two trains to have fun stuff happen. Consider the airplane that flies across the International Date Line and arrives at its destination the day before it left. 
See, this is exactly the part of the problem that we should see explicitly laid out. You’ve described the thinking, pretty much step by step, in going from the verbal description to getting to the equation that needs to be solved.
My response is only applicable to a single community college and 4-year state university here, but virtually every math course at the community college is a straight transfer to the 4-year university. This includes everything from pre-algebra to vector calculus. This was useful to me because I could take all my math coursework at the (cheaper) community college and transfer it to the 4-year university when I moved into upper division coursework. So presumably, the 4-year university saw MTH111 (college algebra) at both schools as being equivalent.
I’m no pedagogue, but it seems that if the university student studies classics, none of this is particularly relevant, and if the student is admitted to math/science, they are already supposed to know college-level algebra.
A real word problem I found in an (advanced university-level) book on solving polynomial equations:
That looks horrifying.
I think this example requires an understanding of the subject matter. While I think most people would guess the answer to 17(a) because there are only two numbers to work with so you might as well multiply them together and see what happens.
Back in the 1980s, when I went to college, the engineering curriculum assumed that students knew algebra (up to use of matrices for simultaneous equations) when they walked in the door. First semester math for an engineering student was Calculus I (basically, simple integration and differentiation). Heaven help you if you didn’t know algebra cold.
Now when I took Econ 101, the assumed math knowledge was considerably lower (we spent a fair amount of time in one lecture covering what the sum of a convergent geometric series is).
It starts off fairly easy. Each ball must travel a distance of 10/2 -.1 = 4.9 m. If the balls are travelling at 1 m/s then I think the answer is 4.9 seconds.
“Nina next suggests to Pascal that they replace their balls by more interesting semialgebraic objects”
:eek:
Apparently kids on the playground are a lot more sophisticated than we were in my youth. I check out when semialgebraic objects are involved and wait for help from someone a lot smarter.