SAT question of the day
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To make an orange dye, 3 parts of red dye are mixed with 2 parts of yellow dye. To make a green dye, 2 parts of blue dye are mixed with 1 part of yellow dye. If equal amounts of green and orange are mixed, what is the proportion of yellow dye in the new mixture? a. 3/16 b. 1/4 c. 11/30 d. 3/8 d. 7/12 -- From the SAT question of the day email I get as Ian signed me up as well to get these questions. Now, he figured this out (good for him) but it stumped me because I view the concept of "parts" to be abstract, making it impossible to equate "equal amounts of green and orange". I guess that's what I get for dealing with object oriented programming languages and always thinking too hard about math word problems. I guess if you consider "part" as a variable, like in: 5po=3pr + 2py 3pg=2pb + 1py then the "p" gets completely factored out. But in my thinking, the "parts" for making orange can be very different than the "parts" for making green. Which is another thing that I always had a problem with in word problems. If something can be completely factored out in the math, then why is it even used as a word in the problem? I've always attached meaning to the words in a math problem, when in reality, a lot of those words simple disappear in the math expressions. Wierd. Oh well, back to my abstractions and other imaginary worlds that I live in. Marc
Marc Clifton wrote:
But in my thinking, the "parts" for making orange can be very different than the "parts" for making green.
you are correct... the trick in word problems is writing them down correctly.... parts is a volume, so you can't just remove it per se. the first goal is to get equal volumes of liquid. In Orange you have 5parts, and in green you have 3 parts, assuming parts are pints or litres is irrelevant, you have unequal proportions and the word problem ended with... "If equal amounts of green and orange are mixed" so parts being a unit of measure, not a variable, rethink your problem. if you had 15 parts of orange and 15 parts of yellow, you would have 3 mixes of parts to get orange, and 5 mixes of part to get green and total volume would be 30 parts. you can't treat parts as a variable, but you do have to remember it is a unit of volume. when you store a unit of volume it usually goes in the variable name since there is no way to tell an integer or a double that it represents meters or liters.
_________________________ Asu no koto o ieba, tenjo de nezumi ga warau. Talk about things of tomorrow and the mice in the ceiling laugh. (Japanese Proverb)
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To make an orange dye, 3 parts of red dye are mixed with 2 parts of yellow dye. To make a green dye, 2 parts of blue dye are mixed with 1 part of yellow dye. If equal amounts of green and orange are mixed, what is the proportion of yellow dye in the new mixture? a. 3/16 b. 1/4 c. 11/30 d. 3/8 d. 7/12 -- From the SAT question of the day email I get as Ian signed me up as well to get these questions. Now, he figured this out (good for him) but it stumped me because I view the concept of "parts" to be abstract, making it impossible to equate "equal amounts of green and orange". I guess that's what I get for dealing with object oriented programming languages and always thinking too hard about math word problems. I guess if you consider "part" as a variable, like in: 5po=3pr + 2py 3pg=2pb + 1py then the "p" gets completely factored out. But in my thinking, the "parts" for making orange can be very different than the "parts" for making green. Which is another thing that I always had a problem with in word problems. If something can be completely factored out in the math, then why is it even used as a word in the problem? I've always attached meaning to the words in a math problem, when in reality, a lot of those words simple disappear in the math expressions. Wierd. Oh well, back to my abstractions and other imaginary worlds that I live in. Marc
Part in this case means same sized container of indeterminate size, essentially unit of measure. The important thing in this setup is that the container used is the same size throughout, like a scoop or a coffee cup. That terminology is used in re-sizing a recipe for example. 5 parts flour to 1 part of sugar. Part can be 1 cup or 10 cups depending how much you are making. I'll pre-mix pancakes and use those types of ratios when I'm mixing it up ahead of time. That way I don't have to grab a specific measuring cup, just a large enough bowl. Math equations exist independently of the models, so the equations you wrote down can be understood as lines on a plane, or ratios of mixed components. Another one would be two lines through the origin intersecting with a third line. Math is the study of the abstract systems without considering the models. What gets really strange is when the same relationships can be re-used within the same model. In projective geometry, the geometry used to generate projections of 3d objects on a 2d surface. Statements like: Between any 2 points there is one line have a corresponding dual interpretation: every 2 lines intersect in one point. Every theorem about lines has a corresponding dual theorem about points. It's the same theorem you are just plugging in different "parts" that you are operating on. It's all very generic and functional programming style.
This blanket smells like ham
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Marc Clifton wrote:
Wierd.
Apparently you're not too good at spelling, either. Bummer. ;P
"A Journey of a Thousand Rest Stops Begins with a Single Movement"
Amusingly enough, if all of us college graduates had to retake the SAT's now, we'd probably all fail and never get admitted a second time round :)
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Marc Clifton wrote:
Wierd.
Apparently you're not too good at spelling, either. Bummer. ;P
"A Journey of a Thousand Rest Stops Begins with a Single Movement"
Roger Wright wrote:
Apparently you're not too good at spelling, either. Bummer.
The way weird is spelled is always wierd to me. Marc
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Part in this case means same sized container of indeterminate size, essentially unit of measure. The important thing in this setup is that the container used is the same size throughout, like a scoop or a coffee cup. That terminology is used in re-sizing a recipe for example. 5 parts flour to 1 part of sugar. Part can be 1 cup or 10 cups depending how much you are making. I'll pre-mix pancakes and use those types of ratios when I'm mixing it up ahead of time. That way I don't have to grab a specific measuring cup, just a large enough bowl. Math equations exist independently of the models, so the equations you wrote down can be understood as lines on a plane, or ratios of mixed components. Another one would be two lines through the origin intersecting with a third line. Math is the study of the abstract systems without considering the models. What gets really strange is when the same relationships can be re-used within the same model. In projective geometry, the geometry used to generate projections of 3d objects on a 2d surface. Statements like: Between any 2 points there is one line have a corresponding dual interpretation: every 2 lines intersect in one point. Every theorem about lines has a corresponding dual theorem about points. It's the same theorem you are just plugging in different "parts" that you are operating on. It's all very generic and functional programming style.
This blanket smells like ham
Andy Brummer wrote:
Part in this case means same sized container of indeterminate size, essentially unit of measure.
Exactly.
Andy Brummer wrote:
The important thing in this setup is that the container used is the same size throughout
That's where my brain goes "clunk". It doesn't say that. It's an assumption! Marc
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Part in this case means same sized container of indeterminate size, essentially unit of measure. The important thing in this setup is that the container used is the same size throughout, like a scoop or a coffee cup. That terminology is used in re-sizing a recipe for example. 5 parts flour to 1 part of sugar. Part can be 1 cup or 10 cups depending how much you are making. I'll pre-mix pancakes and use those types of ratios when I'm mixing it up ahead of time. That way I don't have to grab a specific measuring cup, just a large enough bowl. Math equations exist independently of the models, so the equations you wrote down can be understood as lines on a plane, or ratios of mixed components. Another one would be two lines through the origin intersecting with a third line. Math is the study of the abstract systems without considering the models. What gets really strange is when the same relationships can be re-used within the same model. In projective geometry, the geometry used to generate projections of 3d objects on a 2d surface. Statements like: Between any 2 points there is one line have a corresponding dual interpretation: every 2 lines intersect in one point. Every theorem about lines has a corresponding dual theorem about points. It's the same theorem you are just plugging in different "parts" that you are operating on. It's all very generic and functional programming style.
This blanket smells like ham
Andy Brummer wrote:
The important thing in this setup is that the container used is the same size throughout
I don't think that's entirely the case; different units could be used for each step. Gallons could be used for mixing orange, liters could be used for mixing green, and barrels could be used for mixing orange and green with no trouble. But you mustn't mix units within each step. A problem would occur if one tried to mix one gallon of orange to one liter of green.
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Andy Brummer wrote:
Part in this case means same sized container of indeterminate size, essentially unit of measure.
Exactly.
Andy Brummer wrote:
The important thing in this setup is that the container used is the same size throughout
That's where my brain goes "clunk". It doesn't say that. It's an assumption! Marc
From dictionary.com: 5. any of a number of more or less equal quantities that compose a whole or into which a whole is divided: Use two parts sugar to one part cocoa. It's used whenever you are talking about ratios of quantity, like in chemistry or baking. It doesn't have to say that explicitly because that's what the word means. Also when I said same size throughout, that doesn't actually hold true. It could be gallons to mix the first batch, pints to mix the second and ounces to mix the third. The important part is the ratios when quantities are mixed.
This blanket smells like ham
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Andy Brummer wrote:
The important thing in this setup is that the container used is the same size throughout
I don't think that's entirely the case; different units could be used for each step. Gallons could be used for mixing orange, liters could be used for mixing green, and barrels could be used for mixing orange and green with no trouble. But you mustn't mix units within each step. A problem would occur if one tried to mix one gallon of orange to one liter of green.
Right, I just thought it might be more confusing to bring up 2 concepts at the same time.
This blanket smells like ham
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Right, I just thought it might be more confusing to bring up 2 concepts at the same time.
This blanket smells like ham
Ah, you know your audience. :-D
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From dictionary.com: 5. any of a number of more or less equal quantities that compose a whole or into which a whole is divided: Use two parts sugar to one part cocoa. It's used whenever you are talking about ratios of quantity, like in chemistry or baking. It doesn't have to say that explicitly because that's what the word means. Also when I said same size throughout, that doesn't actually hold true. It could be gallons to mix the first batch, pints to mix the second and ounces to mix the third. The important part is the ratios when quantities are mixed.
This blanket smells like ham
Andy Brummer wrote:
It's used whenever you are talking about ratios of quantity, like in chemistry or baking. It doesn't have to say that explicitly because that's what the word means.
Interesting. Well. Yet again, I discover how warped my view of the world is. :) Thanks! Nice to know I can learn something 30 years after it was important to learn it. :-D Marc
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To make an orange dye, 3 parts of red dye are mixed with 2 parts of yellow dye. To make a green dye, 2 parts of blue dye are mixed with 1 part of yellow dye. If equal amounts of green and orange are mixed, what is the proportion of yellow dye in the new mixture? a. 3/16 b. 1/4 c. 11/30 d. 3/8 d. 7/12 -- From the SAT question of the day email I get as Ian signed me up as well to get these questions. Now, he figured this out (good for him) but it stumped me because I view the concept of "parts" to be abstract, making it impossible to equate "equal amounts of green and orange". I guess that's what I get for dealing with object oriented programming languages and always thinking too hard about math word problems. I guess if you consider "part" as a variable, like in: 5po=3pr + 2py 3pg=2pb + 1py then the "p" gets completely factored out. But in my thinking, the "parts" for making orange can be very different than the "parts" for making green. Which is another thing that I always had a problem with in word problems. If something can be completely factored out in the math, then why is it even used as a word in the problem? I've always attached meaning to the words in a math problem, when in reality, a lot of those words simple disappear in the math expressions. Wierd. Oh well, back to my abstractions and other imaginary worlds that I live in. Marc
How I solved the problem: Yellow = 2/5 of Orange dye. Yellow = 1/3 of Green dye. 1/2 Orange + 1/2 Green = new mixture. New mixture contains (1/2 * 2/5) + (1/2 * 1/3) yellow. New mixture contains 11/30 yellow. I've always been good at visualizing and solving math problems in my head, but I haven't really had to do that much of it recently, with me mostly doing WPF user interface design as of late.
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Roger Wright wrote:
Apparently you're not too good at spelling, either. Bummer.
The way weird is spelled is always wierd to me. Marc
I before E except after C and W or before GH.
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I before E except after C and W or before GH.
PIEBALDconsult wrote:
I before E except after C and W or before GH.
There's a joke about GWB in there somewhere! Marc
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Part in this case means same sized container of indeterminate size, essentially unit of measure. The important thing in this setup is that the container used is the same size throughout, like a scoop or a coffee cup. That terminology is used in re-sizing a recipe for example. 5 parts flour to 1 part of sugar. Part can be 1 cup or 10 cups depending how much you are making. I'll pre-mix pancakes and use those types of ratios when I'm mixing it up ahead of time. That way I don't have to grab a specific measuring cup, just a large enough bowl. Math equations exist independently of the models, so the equations you wrote down can be understood as lines on a plane, or ratios of mixed components. Another one would be two lines through the origin intersecting with a third line. Math is the study of the abstract systems without considering the models. What gets really strange is when the same relationships can be re-used within the same model. In projective geometry, the geometry used to generate projections of 3d objects on a 2d surface. Statements like: Between any 2 points there is one line have a corresponding dual interpretation: every 2 lines intersect in one point. Every theorem about lines has a corresponding dual theorem about points. It's the same theorem you are just plugging in different "parts" that you are operating on. It's all very generic and functional programming style.
This blanket smells like ham
Andy Brummer wrote:
Part in this case means same sized container of indeterminate size, essentially unit of measure. The important thing in this setup is that the container used is the same size throughout, like a scoop or a coffee cup. That terminology is used in re-sizing a recipe for example. 5 parts flour to 1 part of sugar. Part can be 1 cup or 10 cups depending how much you are making. I'll pre-mix pancakes and use those types of ratios when I'm mixing it up ahead of time. That way I don't have to grab a specific measuring cup, just a large enough bowl.
Yeah and similar to Marc's issue with it, it's a concept that entirely breaks down when you follow it to make bread or many other pastry chef type recipes because a cup of flour isn't a meaningfully accurate amount in a bread formula. Serious bread formulas are always in "bakers percentage" by mass. Which makes me wonder how bread formulas will be adapted in the future for bakers on mars and in microgravity etc. Hmmm...maybe a get rich patent idea in there somewhere. An inexpensive, lightweight device for the home cook to measure mass accurately in varying amounts of gravity :)
When everyone is a hero no one is a hero.
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Andy Brummer wrote:
Part in this case means same sized container of indeterminate size, essentially unit of measure. The important thing in this setup is that the container used is the same size throughout, like a scoop or a coffee cup. That terminology is used in re-sizing a recipe for example. 5 parts flour to 1 part of sugar. Part can be 1 cup or 10 cups depending how much you are making. I'll pre-mix pancakes and use those types of ratios when I'm mixing it up ahead of time. That way I don't have to grab a specific measuring cup, just a large enough bowl.
Yeah and similar to Marc's issue with it, it's a concept that entirely breaks down when you follow it to make bread or many other pastry chef type recipes because a cup of flour isn't a meaningfully accurate amount in a bread formula. Serious bread formulas are always in "bakers percentage" by mass. Which makes me wonder how bread formulas will be adapted in the future for bakers on mars and in microgravity etc. Hmmm...maybe a get rich patent idea in there somewhere. An inexpensive, lightweight device for the home cook to measure mass accurately in varying amounts of gravity :)
When everyone is a hero no one is a hero.
John C wrote:
Yeah and similar to Marc's issue with it, it's a concept that entirely breaks down when you follow it to make bread or many other pastry chef type recipes because a cup of flour isn't a meaningfully accurate amount in a bread formula. Serious bread formulas are always in "bakers percentage" by mass. Which makes me wonder how bread formulas will be adapted in the future for bakers on mars and in microgravity etc.
Yeah, but they are in percentage by mass, part isn't restricted to volume it can be mass as well. It's equivalent to unit of measure in this definition. Anyway, unless you are measuring flour on Earth, salt on Venus and water on Mars, it's going to workout just fine.
John C wrote:
Hmmm...maybe a get rich patent idea in there somewhere. An inexpensive, lightweight device for the home cook to measure mass accurately in varying amounts of gravity
Yeah, I think I might be able to come up with something like that. :laugh:
This blanket smells like ham
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John C wrote:
Yeah and similar to Marc's issue with it, it's a concept that entirely breaks down when you follow it to make bread or many other pastry chef type recipes because a cup of flour isn't a meaningfully accurate amount in a bread formula. Serious bread formulas are always in "bakers percentage" by mass. Which makes me wonder how bread formulas will be adapted in the future for bakers on mars and in microgravity etc.
Yeah, but they are in percentage by mass, part isn't restricted to volume it can be mass as well. It's equivalent to unit of measure in this definition. Anyway, unless you are measuring flour on Earth, salt on Venus and water on Mars, it's going to workout just fine.
John C wrote:
Hmmm...maybe a get rich patent idea in there somewhere. An inexpensive, lightweight device for the home cook to measure mass accurately in varying amounts of gravity
Yeah, I think I might be able to come up with something like that. :laugh:
This blanket smells like ham
Andy Brummer wrote:
Anyway, unless you are measuring flour on Earth, salt on Venus and water on Mars, it's going to workout just fine.
True but the end result is often a desired volume of bread "loaves" so I guess you'd have to experiment a bit to find out how to get 10 loaves out of an earth bread formula when you're on Mars. Hmm...now that I think about it the bread would probably rise at a different rate in different gravity and air pressure environments as well. I'd really like to send up a starter ball of dough with the space shuttle and have them put it somewhere out of the way with no air currents so it just floats there as it expands. Then put it in the airlock and pump out the air, it would probably expand to the entire inside of the lock. :)
When everyone is a hero no one is a hero.
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To make an orange dye, 3 parts of red dye are mixed with 2 parts of yellow dye. To make a green dye, 2 parts of blue dye are mixed with 1 part of yellow dye. If equal amounts of green and orange are mixed, what is the proportion of yellow dye in the new mixture? a. 3/16 b. 1/4 c. 11/30 d. 3/8 d. 7/12 -- From the SAT question of the day email I get as Ian signed me up as well to get these questions. Now, he figured this out (good for him) but it stumped me because I view the concept of "parts" to be abstract, making it impossible to equate "equal amounts of green and orange". I guess that's what I get for dealing with object oriented programming languages and always thinking too hard about math word problems. I guess if you consider "part" as a variable, like in: 5po=3pr + 2py 3pg=2pb + 1py then the "p" gets completely factored out. But in my thinking, the "parts" for making orange can be very different than the "parts" for making green. Which is another thing that I always had a problem with in word problems. If something can be completely factored out in the math, then why is it even used as a word in the problem? I've always attached meaning to the words in a math problem, when in reality, a lot of those words simple disappear in the math expressions. Wierd. Oh well, back to my abstractions and other imaginary worlds that I live in. Marc
The intrinsic assumptions here are that: (A) The same measure can and is applied to all colors. (B) The measure is additive, i.e. mixin A parts of X and B parts of Y gives (A+B) parts of whatever. Given that, the measure can be anything, a weight, a volume, the amount of paint Shlemiel the painter[^] uses on day 15. ('Part' can not be, however, the number of days Shlemiel needs to use the amount, since it's not linear, so mixing would not be additive) So, 5 parts of orange are contain 2 parts of yellow, so one part of orange dye contains 2/5 parts of yellow. Similary, 1 parts of green contain 1/3 part of yellow. 1 parts of yucky contain 1/2*(2/5+1/3) = 11/30 yellow. (c)
Math text questions always contain such assumptions, and recognizig them is an intrinsic part of solving such a question. A common problem of quite some bright minds is not recognizing the intrinsic assumptions, because their filter is crystal clear on the algebra/calculus stuff, but pithc black on the "common sense" part. In a bad education system, they simply need to know which question patterns are subject of this test, and which pattern is this? Simple and boring, because they see the pattern long before they understood the "real world problem" the question is trying to pose . In a good education system, they are immensely more challenged, and need some kind of reverse reasoning: (A) What kind of result is expected? (a value? a formula? a proof a solvability verdict?) (B) Consider all hidden assumptions necessary to arrive at such a result (C) Pick the assumptions that are most simple while requiring all - or the majority - of the information given. This contains two other hidden assumptions: the question is solvable with the information given, and there is no - or not much - redundant information. This is (often much) more complicated than applying common sense - because there's an infinite pool of possible assumptions. A simple fractional arithmetics question, solved in 30 seconds by some guy who needs five minutes to recognize the pattern, suddenly becomes open ended:
If we drop the "additive measure" requirement: Be O=O(R,Y) the number of parts of orange dye acquired from mixing R parts of red and Y parts of yellow. Similary, define G=G(B,Y) and U=U(O,G). this gives U = U
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How I solved the problem: Yellow = 2/5 of Orange dye. Yellow = 1/3 of Green dye. 1/2 Orange + 1/2 Green = new mixture. New mixture contains (1/2 * 2/5) + (1/2 * 1/3) yellow. New mixture contains 11/30 yellow. I've always been good at visualizing and solving math problems in my head, but I haven't really had to do that much of it recently, with me mostly doing WPF user interface design as of late.
Show off... I did it as (2/5 + 3/5) + (1/3 + 2/3) == (6/15 + 9/15) + (5/15 + 10/15) looking at the numerators only, I see 30 parts, 11 of which are yellow.
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Andy Brummer wrote:
Anyway, unless you are measuring flour on Earth, salt on Venus and water on Mars, it's going to workout just fine.
True but the end result is often a desired volume of bread "loaves" so I guess you'd have to experiment a bit to find out how to get 10 loaves out of an earth bread formula when you're on Mars. Hmm...now that I think about it the bread would probably rise at a different rate in different gravity and air pressure environments as well. I'd really like to send up a starter ball of dough with the space shuttle and have them put it somewhere out of the way with no air currents so it just floats there as it expands. Then put it in the airlock and pump out the air, it would probably expand to the entire inside of the lock. :)
When everyone is a hero no one is a hero.
Why bake bread in space anyway, you have freeze dried ice cream and tang. Mmmmmmmmmmmmmm. Tang.
This blanket smells like ham
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Andy Brummer wrote:
It's used whenever you are talking about ratios of quantity, like in chemistry or baking. It doesn't have to say that explicitly because that's what the word means.
Interesting. Well. Yet again, I discover how warped my view of the world is. :) Thanks! Nice to know I can learn something 30 years after it was important to learn it. :-D Marc
You never know when you'll have to make concrete with cement, sand, gravel and water. You already have solar power, you're halfway to having your own compound already.
This blanket smells like ham