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#1
Old 12-20-2004, 11:29 AM
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Figuring out this riddle using algebra

I've known this riddle for years and I can figure out similar riddles in my head using trial and error, but how do I figure this out using an algebreic formula?

You bought two pencils in a stationery store, an expensve pencil and a cheap pencil. The expensive pencil costs a dollar more than the cheap pencil. All together, you spent $1.10. How much was the cheap pencil?
#2
Old 12-20-2004, 11:38 AM
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Let x be the price of the cheaper pencil, and y the price of the more expensive pencil.

Then the more expensive pencil is a dollar more expensive than the cheap one, so y = x+ 1. Also, the total price is $1.10, so x+y = 1.1.

Then substitute the first equation into the second to get:
x+y = 1.1
x+ (x + 1) = 1.1
2x = .1
x = .05

So the cheaper pencil costs $0.05 or 5 cents, and the more expensive pencil is $1.05.
#3
Old 12-20-2004, 11:38 AM
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x + y = 1.10 [All together, you spent $1.10.]
x = y + 1.00 [The expensive pencil costs a dollar more than the cheap pencil.]

substitute the second equation into the first:

y + 1.00 + y = 1.10

2y = .10

y = .05

x = 1.05 (from the first equation)
#4
Old 12-20-2004, 11:39 AM
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Quote:
Originally Posted by x-ray vision
I've known this riddle for years and I can figure out similar riddles in my head using trial and error, but how do I figure this out using an algebreic formula?

You bought two pencils in a stationery store, an expensve pencil and a cheap pencil. The expensive pencil costs a dollar more than the cheap pencil. All together, you spent $1.10. How much was the cheap pencil?
X + Y = 1.10
X - Y = 1.00

Since both equations are true, you can "add" them (Mathochist will be along with a proof I'm sure ) and get

2X + 0Y = 2.10

And therefore X = $1.05 (clearly X is the expensive pencil). Plug that result into either of the first two equations to solve for Y. You can also subtract the equations from each other to solve for Y first.
#5
Old 12-20-2004, 12:03 PM
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Quote:
Originally Posted by Jurph
Mathochist will be along with a proof I'm sure
I try not to respond to posts which smell too strongly of homework, creative backstories or not.
#6
Old 12-20-2004, 12:22 PM
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Whatever. Maybe you shouldn't bother posting about any math questions then. What kind of math question can't smell of homework?
#7
Old 12-20-2004, 12:25 PM
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Interesting. I saw the problem a little differently than everyone else. When I worked the problem in my head, since we are dealing with two like items (pencils), I treated them both as x:

x + (x+$1) = $1.10
2x = $1.10 - $1.00
x = $.05

It never ocurred to me to treat one pencil as x and another as y as they are both the same items and can be directly related, thus eliminating the need for simultaneous equations as in Jurph's solution.

/hijack

what is the proper naming convention for variable assignment?

hijack/
#8
Old 12-20-2004, 12:46 PM
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There is no proper naming convention for variables any more. One of the early French algebraists had a system of always starting from the back of the alphabet for variables, and from the front of the alphabet for constants. In this crass and shoddy modern world, however, people have stopped holding strictly to those rules, and mathematicians instead simply pick variables that should be easy for the audience to remember.
#9
Old 12-20-2004, 12:47 PM
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Quote:
Originally Posted by Tanach
Interesting. I saw the problem a little differently than everyone else. When I worked the problem in my head, since we are dealing with two like items (pencils), I treated them both as x:

x + (x+$1) = $1.10
2x = $1.10 - $1.00
x = $.05

It never ocurred to me to treat one pencil as x and another as y as they are both the same items and can be directly related, thus eliminating the need for simultaneous equations as in Jurph's solution.

/hijack

what is the proper naming convention for variable assignment?

hijack/
Actually, you're not treating them both as x, you can't do that because they aren't the same price. You're just never formally declaring y but 'inlining' the expansion of the more expensive pencil as x + $1 In a sense, you're doing the same work as solving the simultaneous equations, you just don't notice it because the second equation (y = x + $1) is so simple you can substitute it into the other equation directly (and because the second equation is pretty simple too.)

Not sure there are any formal naming conventions for algebra word problems. Standard practice is to assign all important unknown quantities on the series x, y, z, w, v, u as they occur to you. If you realize you need more than six unknowns, it's generally common to group them and use numbers to differentiate them... x1, x2, x3 and so on.

Did that make any sense??
#10
Old 12-20-2004, 12:49 PM
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Quote:
Originally Posted by Tanach
Interesting. I saw the problem a little differently than everyone else. When I worked the problem in my head, since we are dealing with two like items (pencils), I treated them both as x:

x + (x+$1) = $1.10
But you just did the first step, substitution, in your head - it was the same exact technique.
#11
Old 12-20-2004, 01:24 PM
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Quote:
Originally Posted by x-ray vision
You bought two pencils in a stationery store, an expensve pencil and a cheap pencil. The expensive pencil costs a dollar more than the cheap pencil. All together, you spent $1.10. How much was the cheap pencil?
You are really getting reamed by your office supply store. Ticonderoga #2 pencils go for $11.49 for 72 pencils!

Cite: http://staples.com/Catalog/Brows...e=1&Sku=138180

Quote:
World class wood quality!

* Real wood means a sturdy pencil and easy sharpening
* Smooth writing lead lubricates surfaces as you write
* Yellow, hexagon-shaped barrel with top-quality pink eraser
* Certified non-toxic
#12
Old 12-20-2004, 01:30 PM
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Quote:
Originally Posted by vetbridge
You are really getting reamed by your office supply store. Ticonderoga #2 pencils go for $11.49 for 72 pencils![/url]
Only if he buys the expensive pencils. That's still more than three times the price of the cheap ones.
#13
Old 12-20-2004, 02:19 PM
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Quote:
Originally Posted by x-ray vision
Whatever. Maybe you shouldn't bother posting about any math questions then. What kind of math question can't smell of homework?
Sorry, I forgot that to most people math is only good as a source of pointless exercises and "riddles" poseable as homework problems.

This is such a standard example of algebra that I have great difficulty believing that you can figure out "similar examples" in your head and have trouble with this one.
#14
Old 12-20-2004, 02:25 PM
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I did it similarly to Tanach. Basically, I thought, "well, the price of this pencil, plus (the price of this pencil plus a dollar), equals a buck ten." I know that's functionally equivalent to what the other solutions said, but it stayed a word problem in my head a little longer than for other folks .

Daniel
#15
Old 12-20-2004, 02:40 PM
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Quote:
Originally Posted by Q.E.D.
Only if he buys the expensive pencils. That's still more than three times the price of the cheap ones.

Touche!

That's what I get for failing to show my work

Happy Holidays
#16
Old 12-20-2004, 04:15 PM
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Quote:
Originally posted by Mathochist
This is such a standard example of algebra that I have great difficulty believing that you can figure out "similar examples" in your head and have trouble with this one.[/quote]

Like I said, I use trial and error. I did it something like this:

Okay, the obvious answer is 10 cents. A dollar more than 10 cents is $1.10, $1.10 plus 10 cents is $1.20 - nope, that can't be right, 10 cents is too much. Lets cut it in half and try 5 cents. 5 cents plus a dollar more ($1.05) is $1.10, yup, I got it.

This is the way I've always solved these sort of problems and I'm embarresed to say, it still seems easier to me than the above formulas. I know if the problem is more difficult, I would definitely need a formula. That's why I asked. FWIW, I've been a member in good standing here for over two years and I wouldn't use these boards to cheat on homework (and I haven't been in school for years).

I don't think I'll be able to do other problems like this the way silverfish and John Mace solved it, but the way Jurph and Tenach explained it makes sense to me. Thanks everyone.
#17
Old 12-20-2004, 05:28 PM
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Quote:
Originally Posted by vetbridge
You are really getting reamed by your office supply store. Ticonderoga #2 pencils go for $11.49 for 72 pencils!

Cite: http://staples.com/Catalog/Brows...e=1&Sku=138180
But it's not really out of line for high-end drafting/drawing pencils at $10.77 a dozen.

http://staples.com/catalog/searc...IpClassCount=7
#18
Old 12-20-2004, 07:29 PM
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Quote:
Originally Posted by x-ray vision
I've known this riddle for years and I can figure out similar riddles in my head using trial and error, but how do I figure this out using an algebreic formula?
Your roentgen tube has gone bad.

It is not a riddle. It is a word problem in mathematics!
#19
Old 12-20-2004, 08:17 PM
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Quote:
Originally Posted by x-ray vision
I don't think I'll be able to do other problems like this the way silverfish and John Mace solved it, but the way Jurph and Tenach explained it makes sense to me. Thanks everyone.
As some have already noted, these are really all the same (or very nearly so) way of solving the problem. The standard technique to solve these problems is to simply assign variables (x, y, etc.) to the unknowns and then translate the wording into equations (I tried to highlight that by putting the exact wording next to the equations). Then you solve for one of the variables (in terms of the other) in one equation and substitute it into the other equation. At that point you only have one variable-- you solve for it, then substute the answer into the other equation and solve for the other variable.

Once you do a few that way it's a piece of cake.
#20
Old 12-21-2004, 09:42 AM
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Quote:
Originally Posted by John Mace
As some have already noted, these are really all the same (or very nearly so) way of solving the problem. The standard technique to solve these problems is to simply assign variables (x, y, etc.) to the unknowns and then translate the wording into equations.
True. For myself, I find single-variable algebra to be far, far easier than multiple-variable algebra, and so I'll always try to get to single-variable before putting it in equation form. But that's just an idiosyncrasy of my head, I think.

Daniel
#21
Old 12-21-2004, 11:49 AM
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Quote:
Originally Posted by vd
But it's not really out of line for high-end drafting/drawing pencils at $10.77 a dozen.

http://staples.com/catalog/searc...IpClassCount=7

My mistake. I assumed that someone having trouble figuring out the cost of a couple pencils would be looking more for low-end number twos.

J/K of course
#22
Old 12-21-2004, 12:15 PM
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This question (and questions very similar to it) are pretty common basic algebra word problems, because they're straightforward if you take the algebraic approach, but if you try to answer them just off the top of your head, it's easy to get the wrong answer—to jump to the conclusion that x-ray vision started with in Post #16? (something like that—@#!%* Preview won't show post numbers) but then not to notice that it wouldn't work.

Quote:
Originally Posted by Jurph
X + Y = 1.10
X - Y = 1.00

Since both equations are true, you can "add" them (Mathochist will be along with a proof I'm sure )
I'm not Mathochist, but in case anybody's still wondering why you can "add" two equations together:

It's legal to add the same number to both sides of an equation, right? So I could take an equation like X + Y = 1.10 and add the same number to both sides, and it would be equivalent to the equation I started with.

Now the second equation is telling me that X - Y and 1.00 are equal—that they represent the same number. When I add the two equations together, adding X-Y to X+Y and 1.00 to 1.10, I'm just adding the same number to both sides of X + Y = 1.10. So the resulting equation is true whenever the original two were.
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