NationStates Jolt Archive

today I had an argument with my friend over a PSAT math question.
There are two questions I want you to answer:

heres the problem:
if x^10 = 5555 and (x^9)/y = 5, what is the value of xy?

first off, how would you aproach it? you have a calculator(ie., the TI 89)
and scratch paper available for your use?(first question)

now, this is aproach A:
intruduce variable z, where z = xy
1. (x^9)/y=5
2. (x^9)/5=y
3. x(x^9)/5=z
4. (x^10)/5=z
5. 5555/5 = z
6. z = 1111

this is aproach B:
again intruduce variable z, where z = xy
1. x^10 = 5555
2. 10√5555 = x = 2.3685
3. (2.3685)^9=2345.3932
4. (2345.3932)/5 = y = 469.0786
5. (469.0786)(2.3685) = z = 1111

looking at these two aproaches and yours (if yours is different), which one
would you use?(assuming the calculator you hace is fully capable of helping
you do the math of any equation you put in)(requirement two)

thanks for your time!(assuming you respond)

I have now officially decided to never try arguing anything with this friend,
as he is the most stubborn 455h013 in the entire world, and , I quote his
words but paraphrase) 'I live by the belief that my logic is supperior to all
others and therefore I do not need to consider the fact that they may be
right'. what he said he lived by is some quote of a phylosipher or something,
I forget his/her/its name and exact words. I could rant on about how stupid
that is and how stupid he is, but this post is getting long enough as it is

Oh, you know whats really funny? the ninth graders this year were given a
practice PSAT, or in other words, a practice practice SAT. crazy!

You don't need a calculator.

(X^9)/y=5

(X^9)=5y

X(X^9)=5Xy=(X^10)=5555

5XY=5555

Xy=1111

You'd probably still need a calculater to determine what the specific values of X and Y are unless they were intended to be exact, in which case:

x = (5555)^.1
y = 1/5*(5555)^.9

In other words the simplest way is to use HMW's way and then solve the rest.

EDIT: crap, I missed the part where you're solving for XY. HMW is right. That is the simplest way (essentially the first way you listed).

You'd probably still need a calculater to determine what the specific values of X and Y are unless they were intended to be exact, in which case:

x = (5555)^.1
y = 1/5*(5555)^.9

In other words the simplest way is to use HMW's way and then solve the rest.

I assumed the question was only asking for the value of Xy (the product of X and y). If I was wrong, then yeah, you need the calculator for the specific values.

You don't need a calculator.

(X^9)/y=5

(X^9)=5y

X(X^9)=5Xy=(X^10)=5555

5XY=5555

Xy=1111

. . . That's basically what aproach A is. . .

I assumed the question was only asking for the value of Xy (the product of X and y). If I was wrong, then yeah, you need the calculator for the specific values.

You were right. I missed that you are solving for XY. But quincys is right that you basically gave solution A.

. . . That's basically what aproach A is. . .

basically, yes. I think mine looks a little better, though.

basically, yes. I think mine looks a little better, though.

Actually, since you brought up, his is the usual notation.

Actually, since you brought up, his is the usual notation.

In what sense? the introduction of an additional variable?

Out of interest, I'd guess the OP is an American, right? And you, Jocabia?

In what sense? the introduction of an additional variable?

Out of interest, I'd guess the OP is an American, right? And you, Jocabia?

Yeah, American. The introduction of the additional variable is standard notation because it makes it clearer that you're solving for xy. I'm a little disappointed I missed it at first.

EDIT: Remember that variables always replace unknown specific values but there are two types. The ones you solve for and the ones that drop out during the process of solving. Z is the variable you're solving for and the rest drop out. It's cleaner.

Yeah, American. The introduction of the additional variable is standard notation because it makes it clearer that you're solving for xy. I'm a little disappointed I missed it at first.

Ah, ok. I was taught not to introduce additional variables in a situation like this, but I'm British so maybe we just do it differently (or maybe my teacher just did it differently. Usually my teachers were just ecstatic I'd actually used some of my potential)

So your friend was suggesting solution B? It's flawed. Using decimals requires you to round and thus introduces errors. Those errors may not be large enough to get the 'wrong' answer, but when testing mathematic abilities, process is as important as the solution. A is the correct process and B is a flawed process (HMW's process is also correct, but as I said standard notation has you introduce a variable to solve for).

I'm portuguese and I wouldn't introduce extra variables. I would also use <=> between each line.

What is wrong with this method ?

x^10 = 5555
(X^10) /XY = 5555 /XY
5 = 5555 /XY
XY = 1111

EDIT: nvm; basicly the same as that of Heron.

Ah, ok. I was taught not to introduce additional variables in a situation like this, but I'm British so maybe we just do it differently (or maybe my teacher just did it differently. Usually my teachers were just ecstatic I'd actually used some of my potential)

Yeah, I was a prick in school. My paper would have said only '1111'. I didn't become more careful about noting my processes until I started tutoring and teaching children and adults. Eventually, I had to admit they were right, showing the process is essence of showing you understand.

I had one teacher tell me that if I only wrote the answer on tests they could only be marked all right or all wrong, but if I showed my work that he would give me most of the credit if I did it essentially right but made a simple arithmetic error or something. I turned in every test after that with no work shown. The teacher and I got along great, but I swear he wanted to stab me every time I handed in a test and every time he handed back a 96.

Side-note: 96 was my magic number when I was kid. I was in school in the military for nearly two years and I got a 96 every single week except two. One week I got a 92 and one I got a 100. 25 question test once a week and I missed one every time.

What is wrong with this method ?

x^10 = 5555
(X^10) /XY = 5555 /XY
5 = 5555 /XY
XY = 1111

EDIT: nvm; basicly the same as that of Heron.

No, Heron covered all the steps. You skipped at least one. If you goal is just to do it in as few as steps as possible, then just write 1111, the problem can easily be done in your head. However, if you're going to show your work you should show all of it.

I'm definitely with Heron on this one. His solution looks much clearer.

What's all this stuff about introducing extra variables anyway? If you're solving for XY, why not just solve for XY and leave zed at home where he belongs? Having Z in the equation just muddies the waters as far as I'm concerned.

I'm definitely with Heron on this one. His solution looks much clearer.

What's all this stuff about introducing extra variables anyway? If you're solving for XY, why not just solve for XY and leave zed at home where he belongs? Having Z in the equation just muddies the waters as far as I'm concerned.

A solution should stand alone. It should not be required to read the original problem in order to understand the solution.

What's the difference between saying let z = xy and solving for z or just solving for xy? Solving for z makes it clear that this is the intent. Solving for XY (particularly when you can get solutions for both X and Y) looks like you didn't complete the problem. Most people don't notice the notation in math books (and they don't explain it) and most teachers either don't know or don't care that it's standard to note it as such.

You don't need a calculator.

(X^9)/y=5

(X^9)=5y

X(X^9)=5Xy=(X^10)=5555

5XY=5555

Xy=1111
I also did it algebraically.

In my opinion, if something can be solved algebraically, then that's generally preferable.

hey all, thanks for responding and stuff. I hope you don't mind if I show my friend things you said, cause I will. (could that even be considered offensive/ rejectable?) also, I noticed after writing down his process of thinking that his aproach is quite similar to mine:

((10√5555)^9/5)(10√5555)=xy (his equation)
(x^9/y)(x) = xy (my equation)

actually, come to think of it, the main point made that I would show him:

So your friend was suggesting solution B? It's flawed. Using decimals requires you to round and thus introduces errors. Those errors may not be large enough to get the 'wrong' answer, but when testing mathematic abilities, process is as important as the solution. A is the correct process and B is a flawed process (HMW's process is also correct, but as I said standard notation has you introduce a variable to solve for).

wouldn't actually wont work, since he was arguing that the calculator is man's most advanced tools and would then say in responce that he puts the equation in the calculator as one mass and thus avoids needing to use decimals and such.

he also said that we should use the tools provided, aka, the calculator (I had just finished telling him I hardly ever use my calculator on my tests) and I argued that the brain is just as good as the calculator; in fact, the calculator would not be here if not for the brain.

although his main point was that the calculator is far faster than any human brain.
which is a good point, although it would take more time to think up aproach B and plug it into the calculator than to think up aproach A and figure it out in a jiffy.
plus, aproach B is a whole lot more confusing than simple substatution, so aproach A is more useful by that sense.

come to think of it, bringing up this argument in the first place is far more insulting [to him] than using your points to trump him is [to you], no?

I guess I failed at keeping you guys from knowing who had which oppinion, as a few of you guessed aproach A was mine. oh well.

I'm assuming all of you are math geniuses, right? cause it will hurt him more... I mean, it will convince him better when he is being told wrong by math wizes much more experienced than he or I.;)

First off, if your friend thinks that a calculator will do everything for him, I pity him once he get's to Multivariable Calculus. Hell, Linear and Differential might require the 89 for some parts, but it'll ruin you if you input the wrong notations and such.

Also, working out the problem is better for you mentally. It prepares you for other questions on the test. Also, working with decimals, as I'm sure someone has mentioned before, is very risky. Every math teacher I know has said to NEVER convert to decimals until the end. And even then, most math classes accept the exact (non-decimal) answer.

blah blah blah

What does it matter if you arrived at the conclusion in a different way? As long as you both understand how to get the right answer and do get it, who cares if one way is easier/morally superior/stupider than another?

What does it matter if you arrived at the conclusion in a different way? As long as you both understand how to get the right answer and do get it, who cares if one way is easier/morally superior/stupider than another?

This is true if you don't have a job where an error due to decimals can have devastating results. Stuff like aeronautics, thermodynamics, structural integrities, and a few others depend on ranges of error. If your error is higher than necessary because you used decimals halfway through and had to round (if say you used 5/3, which is 1.66666666666666666666666666666666666666666666666666666666blahblah, you're gonna eventually need to round to ...67. Usually you'd do it as 1.67 just to make life simple. However, if the calculations are long and complex, rounding that early may disrupt your answers later on. You might say, so I'll round later on, and make it 1.66667. 1.66667 is not a nice number to work with for many calculations. Also, 1.66667 is a nice infinite repetition. There are other decimals that aren't as pretty and apply more in the real world.

What does it matter if you arrived at the conclusion in a different way? As long as you both understand how to get the right answer and do get it, who cares if one way is easier/morally superior/stupider than another?
Both my math and calculus teachers said that last week.

Also, working with decimals, as I'm sure someone has mentioned before, is very risky. Every math teacher I know has said to NEVER convert to decimals until the end. And even then, most math classes accept the exact (non-decimal) answer.
In my math and calculus we are told to give it in the exact form unless told otherwise.

A solution should stand alone. It should not be required to read the original problem in order to understand the solution.

What's the difference between saying let z = xy and solving for z or just solving for xy? Solving for z makes it clear that this is the intent. Solving for XY (particularly when you can get solutions for both X and Y) looks like you didn't complete the problem. Most people don't notice the notation in math books (and they don't explain it) and most teachers either don't know or don't care that it's standard to note it as such.

I must admit introducing a temp variable would have been a Point for solution but a fail on method. I persnally would have just written solution with no method as I am an end justifies the means type of guy.

but introductions of additional variables was very much a no-no (hey things change apparantley people are getting taught imperial units again Arrrrgh!)

A solution should stand alone. It should not be required to read the original problem in order to understand the solution.

What's the difference between saying let z = xy and solving for z or just solving for xy? Solving for z makes it clear that this is the intent. Solving for XY (particularly when you can get solutions for both X and Y) looks like you didn't complete the problem. Most people don't notice the notation in math books (and they don't explain it) and most teachers either don't know or don't care that it's standard to note it as such.

introducing z simply adds another step that isn't required. I especially hate this because my american books tend to do rather lengthy calculations on something that can be done in one or two lines.

In exercises you also make it a bit easier: you take more steps, but each step is easier than doing it in one go. As a result, you won't get as many 'training' out of the exercise. You simply get used to doing more difficult things if you do them in one step, you'll do it faster and if you say 'hey, this looks too difficult', it is very easy to just do it in more steps for once. I remember that my math teachers usually had troubles 'following me' when i made exercises because it was natural for me to take big steps.

to work this out without out a calculator first you will need to use ethens formula and figure out the value for pq, the you will be able to figure out the value of xy

A more systematic approach:

x^10 can be seen as the vector [10,0]
x^9/y can be seen as the vector [9,-1]
these are our 'base vectors', v1 and v2
xy is the vector [1,1]

we want to represent [1,1] as a linear combination of v1 and v2 and get:
[1,1] = 1*[10,0] -1*[9,-1]

we 'translate' this by saying that xy = (x^10)^1 * (x^9/y)^-1 and simply devide 5555 by 5

This way you get an 'easy' result for all x^a * y^b