Ultimate Brian Teaser

[dimava]

In the year 1660 a beaver skin brought a trader 300 purple beads and 60 white beads. Suppose that this was the same as $50 today.
how much US dollars was 1 white bead?
how much US dollars was 1 purple bead?

Also, remember that a prurple bead is worth about twice as much as a white bead

[Iain17]

Approximiatley

1 white bead = $0.238
1 purple bead = $0.119

[dimava]

correct, what did you do to find that number out?

[HarryW]

Simultaneous equations, solved by substitution:

300p + 60w = 50
p = 2w

substitute in:

300(2w) + 60w = 50
660w = 50
w = 50/660

now use value of w to find p:

p = 2w (or 100/660)

[dimava]

ok, so

p = $0.156
w = $0.078

is that correct?

BTW this is my homework, but I went totally blank when I saw it


Also if something costs 10 US Dollars it would be 65 purple beads, or 205 white beads?

[HarryW]

p should be rounded to $1.152, but I am confused: both sets of figures seem to work on my calculator, yet it is a first-degree polynomial and should have only one root. Errr... *Confused*

[Iain17]

Well i dont remember my simultaneous equations that well, so i cheated.

If we take the white beads to be worth twice as much as a purple bead then.

2 * 60 = 120
       + 300

       = 420
'work out the percentage of beads
120 / 420 = 0.28571428571428571428571428571429
300 / 420 = 0.71428571428571428571428571428571

'work out the cost using the percentage
50 * 0.28571428571428571428571428571429 = 14.2857142857142857142857142857
50 * 0.71428571428571428571428571428571 = 35.71428571428571428571428571425

'divide the cost by the number of beads
14.2857142857142857142857142857 / 60 = 0.238095238095238095238095238095
35.71428571428571428571428571425 / 300 = 0.1190476190476190476190476190475

[dimava]

ok... this is exactly what it said on my work sheet:

In the year 1660, a beaver skin brought a trader 300 purple beads and 60 white beads. Suppose that this was the same as $50 today. Write a price list in wampum(the bead currency) for five items that you might buy this week. Remember that a purple bead was worth about twice as much as a white bead.

so here are my 5 things:

1. CD------------$16
2. Video Game----$50
3. Book----------$20
4. Pack of Pens--$3
5. Head Phones---$10


so what would I put for the wampum currency?

[Iain17]

We are not here to do your home work. Besides which, you have the worth of the beads now, so it is just simple maths to work out the 5 items bead values.

[dimava]

1. CD------------$16
2. Video Game----$50
3. Book----------$20
4. Pack of Pens--$3
5. Head Phones---$10


so is this correct:

1. P = $0.16
W = $0.08

2. now I forogt, do I do 0.24 * 50 or do I do 50 / 0.24?

[HarryW]

Aha I know why they're different: Iain's figures are taking white beads as worth 2 purple beads, an mine are the opposite - purple beads worth double what white beads are. So my figures are correct, and Iain's figures are just another solution to the equation 300p + 60w = 50

Dimava: what do we have to do, write it out, fly over there and hand it in for you? Do it yourself!
We did the hard bit!

Cheeky bugger

Incidentally since some advanced programming like 3D stuff is heavy on the maths side, you really ought to learn this.

[Sam Finch]

the reason you get lots of solutions is that you've only got one equation in 2 unknowns, the other is just a guideline.

so

300p + 60w = 50

w = (50/60) - (300/60)p 
  = (5/6) - 5p

we can choose any value of p and get the corresponding value of w.


obviously we need to choose values where w >= 0 and p >=0

so If we find the value for p where w = 0 we get

0 = 5/6 - 5p

5p = 5/6
p = 1/6


so p can be anywhere in the range 0 <= p <= 1/6



this allows you to work out all possible solutions to the problem.

If we impose the limit of p ~ 2w then just pick a value of p close to $1.56 (Ian's solution) and work out the value of w.


to find the solution to the second problem (how many whit/purple beads for something worth $10) it easy.

we already know the solution to something costing $50

so for something costing $10 we need a fith as many beads of each colour.

eg 60 purple and 12 white.

that solution is the only solution garanteed to be correct because we don't know the exact ratio of the values of the beads.

[Iain17]

Opps Harry, that will teach me to read the question.

Still, he he deserves a wrong answer if he is aksing people to do his homework for him.

[dimava]

Thanks to every one!!! and I'm not asking you to do this homework for me, I'm just asking for help becuase I dont understand how to do it. (BTW I'm in 8th grade not 10th)

[HarryW]

Hey I just noticed this thread is 'The Ultimate Brian Teaser'. Well where's Brian? And how dare you tease him, what did he ever do to you?

Secondly, I really do object to the overuse of the word 'ultimate' nowadays. Ultimate means last! Is this the last one? Is it? It says 'the ultimate games console' on my old megadrive box but was it? I should go to trading standards!

[dimava]

to me ultimate means "The Best/Hardest" for that time being

[HarryW]

That's just it, the word 'ultimate' doesn't allow the prospect of anything ever being greater, better, you get the idea. It's final! Think of the words that are similar - 'ultimatum' meaning the final set of terms that will be presented, 'ultima' meaning tha last syllable of a word.

It's a pet hate of mine, the way people use the word 'ultimate'. That and television licenses, don't get me started on those!

I am using ultimate in my new Biz name....
I am not anouncing the name here, because i haven't registered the domain yet.

Harry, do you think that most people would know the correct definition of ultimate??
not really, they would just think that its the best, or something.

And anyway... it sounds cool:
Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate Ultimate

oh crap, I've said it too many times, now it doesn't even seem like a real word