Subject #24
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This is Subject #24, VHHHHHHBHHMi, Age 15,
2, Track A.
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The date is"1 August 5, 1964.
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We are recording on Tape #10,
The time is - ah - 3:00C
2sOOf'
2:00^ Earnest has had two years of Algebra and a year of Plane Geometry and has
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just graduated from the 10th grade; Instructions: We are interested in how
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people solve problems. This experiment is not designed to test your problefoi
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solving ability, simply to discover what methods you would use to attack an Algebra
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word problem. There are no time limits on any of these problems. t% much more
interested in kiteuiLLug how you*go about solving them than how well you do/
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solutions. Please try to get the right answer, but concentrate on'5 explaining
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how you do it as you go along. We ask that you talk your way through the problems
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explaining each step as you go . This should not handicap your problem solving
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in any way. In fact, it might actually help it. Please try to keep talking at
all times.1 If you're puzzled, explain what is puzzling you. i If you're re-reading
the problem, re-read thefjprobSem $ e that you're K^reading alofid.
All the
problems^involve one Algebraic equasion with one unknown. Now, some people
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usinc
fyattempt to solve these problems mdclPtwo unknowns . They can be done that way,
but, ah, just for the sake of having everyone do it the same way, we ask you to
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use one unknown. Every problem can be reduced to a single'equasion with one
unknown.
Ah~*-
You may be used to using a scratch pad and paper to do Algebra
problems and you mostly
have them provided. To take the place of that, we
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have a blackboard in the front of the room.f I'll write down bits of information
>qpBX want to rembember, parts of the equasion - anything that you need to keep track cf
I'll write it down on the board.
tell me when to write.
Please explain what you JGBBJ& want written and
The problems will be presented to you on 5 by 9 cards.
Read the problem aloud to begin.
After you've read the problem aloud and started
solving it, I won't ask any more, answer any more questions.
the rulesof the game and please don't ask xany.
This is against
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I may make an occasional comment from time to time, but, ah, I will not answer
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questions, nor will I give encouragement, jmrxirinht or provide additional infor-
mation.
When you are through, please say so and ask for the next problem.
wait, I won't say anything.
I'll
I don't want to interrupt^your solution before you 1 re
dene.. Please try to keep the following four instructions particularly in mind:^
Keep talking, read each problem aloud/ do not ask questions of the experimenter,
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specify 1 clearly what you want written on the board and signify when you ?3 are through.
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The completion of each task is to write cut an equasion, which if you did the
actual arithmetic, would give you the correct answer.
the arithmetic yyourself.
We don f t ask that you do
In fact, we prefer that you didn't.
up with a numerical answer.
Don't try to come'3
This is not what we're looking for.
All we want
is an equasion
A's^anoequals and plus signs ^that^Hs all-. The right equasion
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but, ah, you're not supposed to figure out the answer yourself. Are the instructions
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clear?' Any questions? W^' No? CK. I'll give you three practice problems for
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you to warm up on. This is practice problem #1;
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If a certain number is multiplied by 6 and the product is increased by 44 > the
result is 68.
equals 68.
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Find the number.
Let X equal the number.
Ah,
C& plus 44'
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Is that what you want? J
—That
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This is practice ppoblem #2.
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If 3 more^than a certain niimber/aivided by 5* the result is the same as twice^
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the number diminished by 12. What is7the number?
Let X be the certain
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niimber. 5
X divided by 5 plus 3 equals 2X minus 12.
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Will you dictate that to m»? *
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X divided by 5 plus 3 equals5 * 2X minus 12.
S:
Now, will you please explain now you did that?
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Well, the problem states that 3 more than a certain number divided by 5* so X is
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the certain number divided by 5> so you add 3 to it - and the result is the same
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as twice the same number minus 12.
E:
Allright.
S:
E:
The first part?
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Uhuhm. Would you read the first part aloud?
S:
If 3 more than a certain number is divided by 5
Could you give me another interpretation of the first part of that?
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Cf the first part of the ,,itM $-trtr.^f
of the problem.
*/
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It could be X^'plus 3 divided by 5.
E:
OK.' L
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T£e difference between two numbers is 12 and 7 times the 3 smaller number exceeds
This is practice problem
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the greater by 30. Find the number;ift/fite^)
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7 times the smallest number exceeds
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the greater by 30
So let X*equal the scalier number* 7X ? minus 30 v?is your
larger number E:
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so'X minus 7X^rninus 30'equals 12.
CK. y Could you explain that to me"
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Well, the problem^states that 7 times the smaller number exceeds the greater by
30.b So X is the smaller number Land 7 times the smaller number minus 30 ?would be
y
the larger number.
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And the difference between these two numbers is 12/ *
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Ah, Could you go through that again. I - I I haven't quite, I don f t know the
problem myself.
_ _
That might be right.
S:
It says 5 7 times the smaller number exceeds the greater by 30. y
E:
Ohuh.
S:
So, if the smaller number is X, then the larger number would be 7 times X minus 30.
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The difference
between these two numbers is 12/]??
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Now, let f s go on to the actual problems. This is problem $1.
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A man has 7 times as many quarters than he has dimes/y The value of the dimes
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exceeds the vaiue^of the quarters by two fifty. How 4many has he of each coin?
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Let 7 be the number cf dimes/1 " /]Let X be the number of dimes. The 7X would be
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the number of quarters.
down, Just
If there's anything you want written, tell me.
S:
What?
E:
Allright.
E:
If you want any of that written on the board.*
S:
Yeah, you can write it.
E:
What's that?
S:
You can write it on the board.
E:
What?
S:
X equals the dimes.
?x equals the quarters.
exceeds the value cf the quarters by two fifty.
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The value of the dimes
It would be X
minus
?X
equals
E:
Would you please explain that solution?
St
Well, it says that the value of the dimes exceeds the value of the quarters by
two fifty so if you subtract
the value of the quarter^ from the value of the
dimes, you get two fifty.
E:
And what is X r'
S:
X is the number of dimes.
E:
What about the dimes?
S:
Their value is
E:
Yes, but wkfet is X again?
S:
The number of dimes.
E:
The number cf dimes.
S:
Naw.
E:
Which last'part.
S:
How many has he of each coin.
E:
Well, you get it by solving the equasicn, perhaps by^ke X,
e.
w
It's
two fifty i^cre than the value of the quarters.
Allright.
How many has he of each coin?
Is that your solution?
I don f t think I know how to get that last part.
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But, you're net asked to do that, you./ We don't, we're not interested in numerical
answers.
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Ch.
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This doesn't' - ah -
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Ah - I WQO ~ I ;*as trying to figure out the equasion.
Was that the question?
To tell how many coins he had.
£:
Well, that f s the one*
Wh Why did you write that one down?
S:
That's to the second statement.
A man has 7 times as many quarters as he has dimes
and the value of the dimes exceeds the value of the quarters by two fifty.
I don't understand this problem.
E:
What don't you understand about it?
3:
Well, it days he has 7 times as many quarters as dimes.
Then it says the value
of the dimes exceeds the value of the quarters by two fifty.
E:
Yes.
3o.
S:
So, if he has 7 times as many quarters as dimes, how can the dimes exceed the
quarters by two fifty?
That's an interesting question.
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Uhumm.
Is there an equasion for it anyway?
S:
N0.
E:
OK.
S:
Mr. Stewart decided to invest four thousand dollars, some at 3% and the rest at
.I.et's try problem $2. "'
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k%.
How much should he invest at each rate to produce equal income?
Let'X be the number he - ah - the amount he invested at 3%*
be the number he invested at 4$.
tiroes 3 hundredths
So it would be X
400.
And
$4,000 minus X
So, ah,
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equals
four thousand minus X
times
That's the equasicn.
E:
Okay, would you explain that to mw?
S:
Well, that's ah/ he wants to invest four thousand, some at 3% and the rest at
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k% and it says hew much should he invest at each rate to produce equal income. So
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you g^4 X equalf what - what he invests at 3% and 4,000 minus X would be what f s
left, and he would invest that at 4^*
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CK.
Problem #3.'
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A stream flows at the rate cf 2 miles per hour. A launch can go at the rate of
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8 miles per hour in still water. How far down the stream does a launch go and
return if the upstream trip takes one half as much time as the downstream trip?
Well, it says that the water flows at 2 miles per hour and the
launch can go at 8 niles per hour in still water-
So "> says that - so - the
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speed going downstream would be
8 miles plus the 2 miles per hour of the stream.
So, it f s - if the half- if the upstream takes half as much time as the downstream
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trip,
X would be th- the time.^
That would be one half taaase X times
8 minus 2 equals X times 8 plus 2.
E:
Would you explain that to me?
S:
Well, his rate upstream would be the 8 irdles per hour in still water minus the
2 irdles per hour'of the
of the current of the stream.
So, since the time is X
and it takes him c.na half'the time upstream, the one half X would be the time,
times the speed which wottitPbe^ 8 minus 2. ifccSays the downstream came - trip
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takes X then his speed would be 8 plus X, I mean 8 plus 2.
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Let's see, ' I dcn't know that^ l -
Oh.
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(unintelligible)
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How far down the
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stream does the launch go and return if the upstream trip takes half as much time
as the downstream trip.
E:
Why did you multiply the time^'by the rate?
S:
The time$ timesthe rate.' ^_,^ ;V
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distance.
E:
4k, you're right.
S:
Rate timet time "equals distance.
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Rate xx&
time equals
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Rate times time equals distance/
The distance would equal
The distance would
equal the correct amount of time.
E:
OK.
S:
Well, he would have to go f as far downstream as he did to c me back upstream at the
Mxii, what I want to know is why are those two quanitities equal?
same point, so it should be equal.
E:
OK. 4 Problem ^4^
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A board is sawed into two pieces.
One piece is 2/3 as long as the whole board.
and is ex, and was exceeded in length by the second piece by 4 feet.
was the board before it was cut?
How long
It says that one piece was 2/3 &8 long" as the
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whole board -
so the whole board - let the whole board equal X and 2/3Xis equal
length of the piece,
so*2/3X plus 4
and then it says the second piece exceeds that by 4 feet,
would be the second piece.
So
2 times 2/3 X plus 4 would
equal the whole board......
E:
What's your equasion?
Si
Minus.
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And 2/3X
Write down ah - 2/3 X equals length of cne piece.....
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plus 4 equals the length of the second piece......
So, that would be'2 - 2 times 2/3X plus 4 equals X.
E:
You mean 2 times 2/3 X
S:
2/3 X.
E:
Huh?
what do you want me to multiply by 2?
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Yeah, plus 4
E:
OK.
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Well, it says that the one piece is 2/3 as long as the whole board, so if the
equals' thl^linK^fbftiJhe whole board which is'X.
Would you explain that to me?
whole boara is X, thai that piece would be 2/3X.
by the second piece !by 4 feet.
Then it says,it was, ±fc exceeded
So it would be 2/3 X plus 4.
So both of those
added together would equal the whole board, which is X.
E:
GfU
That was #/*.
Last problem.
S:
A car radia - radiator contains exactly one liter of a 90^ alcohol-water mixture.
What quantity of water will change the liter to 80;&£ Alcohol-water mixture?
E:
DC you know what a liter is?
S:
Yes.
E:
In volume.
S:
It f s a measurement of water.
In volume.
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- muiiible-Let f s see.
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Let X be the quartity of water.
The car radiator contains exactly one liter of a 90$ alcohol-water mixture.
That would be 90^'of the alcohol-water mixture.
It will be 90$ of *fee alcohol-
water mixture plus X water would equal ah - 80$ water-alcohol mixture.
E:
What's your equasion?
S:
90/S
E:
Would you explain that.
S:
Well, it says that if a radiator contains one liter - ih - 903> water-alcohol
asalwajBr mixture plus X would equal 80$ water-alcohol mixture.
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ir.ix.ture.
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We±±, it says what quartl ty of water will change it to ah - liter cf -
the liter to 80$ alcohol-water mixture!
So, if you add X 'amount of water to 90$-
to a 9C$ water-alcohol mixture, then you'd get a 80/1 alcohal-water mixture.
E:
OK.
Now I f d like for you to go to the board if you would.
rase off.
I'm going to read you three of the problems you've just solved and what I f d like
you to do is to draw a diagram cr picture that represents all the information
in the problem.
That is, draw a diagram that you could use f to explain this
problem to someone who hasn't seen the written problem.
sawed into two pieces.
CK?
A board is
One piece was 2/3 as long as the whole board and was
exceeded in length by the second piece by 4 feet.
How long was the board before
it was cut?
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(softly)- the whole board equals X.
You cut the board'into 2/3 the length of the
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whole board.
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And the second piece was ~ exceeded by the first piece by
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CK.
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The second piece.
E:
The secona piece.
Now -ah'- which piece is longer?
Could you draw another board just below that, please?
how, could you draw a cut that looks like it divides "the board in/nan 2/3 and
one - 2/3, please.
/
the 2/3 point. 8K.
beard.
Draw,
draw a cut in that board that divides it at about
Now, which section of the board is 2/3 the length of the
You want to write that down there.
Which piece is longer?
S:
This piece.4
E:
You Just told me the other piece was longer.
S:
That's what the problem says, the second piece exceeds the first piece by 4 feet
E:
tlhuh.*
S;
Now if it exceeds the first piece by 4 feet, then it's the longest piece.
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-9Well, that's true.
But, ah, could you show me that on the, on the second piece
that you've drawn?
S:
I can't do that.
E:
Why?*
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S:
This is 2/3 of the board.
f
uhum.
S:
Then it couldn't be greater than 2/3 of the board.
E:
Puzzling, isn't it?
(unintelligible)
The next piece——
Are you sure you heard the problem correctly?
Cne piece was 2/3 as long as the whole board and was exceeded in length by the
Which of the two pieces was longer? ""
second piece by 4 feet^
S:
Well, the problem says the second piece is longer.
.„_,,_
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longer.
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————..,_.._„
E:
Let's try problem #3.
A stream flows "at the rate of
A launch can go at the rate of 8 miles per hour in still water,
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How far down the stream does the launch go and return if the upstream trip takes
2 miles per hour.
one half as much time as the downstream trip?
S:
This is the starting point.
E:
Uhuh.
S:
The stream is flawing 2 miles
per hour.
per hour.
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Thi^—ie-a launch;traveling 8 miles
I&ts going downstream, it would be 8 miles per hour plus the 2 miles
per hour of the stream.
E:
Uhuh.
S:
Coming back it will be traveling 8 miles per hour"minus the 2 miles per hour
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of the streanu The problem said that it would take half as much time for the
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upstream trip than the downstream.
E:
That's rigit**
S:
So, it would be one half X minus 2,
8 minus two,
one half X
eauals V
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Show me what - ah
What is X?
uan you show X in that drawing?
X is the time.
S:
X is the distance./
E:
OK.
S:
I put the distance ——
E:
Uhum."
S:
It would be ah -
E:
Tell me.
Would you write me ah -
What is that distance?
Well, can you express it just in the terms of time.
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minus 8 minus 2. orlt would be X times 8 plus
X
Do you travel - which way do you travelfiwfeeR-ye^pa going
thi« way?
Downstream or upstream?
S:
Downstream
E:
And, so therefore, which trip will take* more time?
S:
Upstream.
E:
What's the problem say?
S:
Let's see,
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he went faster when he went downstream than when he went upstream, the
Do you remember?
downstream trip wouldn't take as much time.
The problem doesn't say that does it?
E:
I see/
S:
It says the upstream trip does.
E:
Which do you think is right?
S:
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The downstream trip isxadt^Wt.took less time.
E:
Well, do you think the problem is wrong.
S:
Tos.
E:
Stating the trip/
1> gftieftHxtfc
stating the trip.
OK.
Let's try #5.'
A car radiator' contains exactly one liter of 90$ water-alcohol rrdxture.
What
quantity of water will change the liter to/80$, alcohol-water mixture?
Draw me
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a picture of that.
Should I draw a radiator or can I draw a bottle?
E:
A bottle is fine/
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This is 90^." This long line.'
90^
X
is the water that
you 1 re going to add to it.
E:
Allright.
Now, could you tell me - ah - what is the volume of the solution in
the second - in the bottle on your right? The one you have labeled 80$.
S:
This?
E:
Yeah.
S:
Well, it would take'
E:
One liter plus what?
S:
Plus X?
E:
Plus X.
S:
9 times as much water." '
E:
What was that again?
-S:
OK.
one
--
liter
plus the volume
of the water
Could you tell me how much alcohol there is in the first bottle?
9 times as much water.
E:
And how much - ah how much is that?
S:
Ah,
Kow many liters would that be?
E:
9/10.
,/
9/10 of.
S:*
8/10 t>f the whole solution.' * It would be 8/10'of one liter plus X.
.E:
*
S:
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What was the volume?
Te^l me.
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How much alcohol ~ ah - in the bottle on the right?
Did you change "the amount of alcohol between the first and second bottle?
No.
S:
Kow imch'alcohol is there in the last bottle?
S:
^>ame thing.
E:
Which was what.
S:
96$
E:
•9Q^/ X 1$ you know what the definition cf percent is?
No - it would be —— the same as the first.
9/10 liter/
Qould you write'one for me?
S:
Difinition of percent.
E:
Well, explain percent to me." Explain it to me in terms of this particular problem,
how
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is.
I don't know what you mean/
What does 90^ alcohol mean?
S:
It means that 9/10 of this solution is alcohol.
E;
How would you find the percent if you were given the number of liters of alcohol
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and - ah - the number of liters of water in the first part.
find the percent?
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E:
Hmmm - the percent would be.
9
Over 'what?
3:
A hundred.
E:
S:
Over a hundred?
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Perc ent of- ——
E:
Now I want you to try an actual case.
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How would you
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the amount of alcohol over a hundred.
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one liter of alcohol.
Suppose you had a - two liters of water and
Get the percent of alcohol/
S:
Hew many liters of alcohol?
E:
One.
S:
Well, the whole solution would be ~ it would be ~ the alcohol would be i^ifi 1/3
;';••;*? * ' • • '"'
the alcohol content would be 1/3.
E:
That f s 33 and a third.
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E:
Yes, it f s a fraction.' A fraction consisting of what?
S:
Percent is a - 33 and a third.
.£:
No.
S:
Well, percent means hundreds and 1/3 of a hundred is 33 and a third.
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Alarm*
Well, where f d you get the 1/3 is the question.
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1/3 is the arsount of alcohol cause it f s 2 of water and 1 of alcohol - and altogether
S:
What does it mean?
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Now does that help you to see what percentage means?
It f s a fraction. $
E:
CK.
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But, how did you get it, that's what I ! IL interested in.
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it makes 3.
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E:
So wfcer^tr tixe 1/3.
S:
I said not, ttCSBHR- -~—— —
E:
It is.
S:
V/ell, the amount of alcohol
E:
Well, that's still one.
.
liter.
Did
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That f s one.
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The amount of alcohol in the whole mixture is one
Well, I made it a point
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I got interested in it.
For your own
reference - ah - you'll notice that a percentage always involves a ratio and in
this case it's the ratio of the amount of alcohol to the total volume of the
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solution. You see, it's a comparison - it's 1 to -three or in that case, 9 to 1 -
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or 9/10 to one.
That f s a different thing for measuring volume sc your solutipn here,
you're adding, you're adding a certain volume'that you don f t know.
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llal-ti^^Si^ aga *
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This/^percentageerlally means that you Subtract
Do you think you could solve the problem doing that?
Solve their, hew.
Doin£ what I was doing.
Solve them <--
Find out how much you have to add *———•*---
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(unintelligible)
9^*8 9v^ percent of what?
You see, when you're taking'a percentage of
something, you always have to ask what the question, what percentage of what, part
of what whole.
Percentage always expresses part similar to whole.
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*x
there's no clear meaning of what part - 10^ of sc mething is.
In that case,
/*
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Well,'
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