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:
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
If the proportion of yellow dye in the orange mixture is 2/5, and the proportion of yellow dye in the green mixture is 1/3, wouldn't the proportion of yellow dye in the new mixture be 3/8?
"Normal is getting dressed in clothes that you buy for work and driving through traffic in a car that you are still paying for, in order to get to the job you need to pay for the clothes and the car and the house you leave vacant all day so you can afford to live in it." - Ellen Goodman
"To have a respect for ourselves guides our morals; to have deference for others governs our manners." - Laurence Sterne
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PIEBALDconsult wrote:
I before E except after C and W or before GH.
Albeit you might be absolutely correct, I'm not agreeing with you! We were anteing up for a bet, and I think you are actually wrong. I think any foreigners on CP might agree with me, unless you counterfeit the results. I had a seizure just thinking about it. The atheists were particularly upset. So, I put on my beige shorts, and proclaimed "anchors aweigh" as I ate my braunschweiger (for protein) and marbleized crumpets with my cup of caffeine. "We're going canoeing!" I said, clueing others in on our daily adventure. Just don't get into the counterfeit canoe. The water's fine, because it's deionized - thank the blueish deity (no, it's not from dyeing his skin, but any einstein would know this). And it's not a blue holstein, either. Weird indeed. I had to forfeit because my heirs had heisted a canoe earlier. So I was looking through the kaleidoscope as I fell on my keister. After kneeing the monseigneur, I changed into other leisureware (I was caught peeing in my shorts). But that's neither their here nor their there. The reigning captain reignited the debate on whether I should be reimbursed, as I was reinfecting the earlier wound. I told him I would reinitiate a suit, as I had reimaged the earlier wound (had to reink the printer first). That's when he unleashed the rottweilers! I was seeing more clearly now. I thought I was having a seizure, but then I noticed the seismometer was seizing. The sheilas sat on it as they were shoeing their feet while surveilling sightseeing surroundings. Thereinafter, I decided I was tieing (M-W) my own shoes, or would resort to tippytoeing. Wherein I decided, with veins bulging, it's a good thing we have simple rules to live by, especially for weirdos with weiners, eh? :laugh:
Gary
:-\ We're all just skiers on a skein. English is the C++ of human languages. Tons of strange, ugly syntax and irregular forms, hampering conciseness and forcing the use of many painful idioms rather than simple rules (even the simple heuristics are full of exceptions). But everybody bloody uses it. :confused:
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Andy Brummer wrote:
lol, I'm kinda partial to gray myself.
Um, five parts black, five parts white? :confused:
Gary
If they are of equal quantities, you reduce, so "5 parts to 5 parts" would properly be represented as "1 part to 1 part", or more accurately "equal parts". Think of it as being fractional. In your example, 5+5 is 10, so each part is 5/10 (ot 1/2) of the whole. If it was 3 parts to 9 parts, then the first would be 3/12 (1/4) and the second 9/12 (3/4) making it actually be 1 part to 3 parts. Parts is always a unit of equals measure in the comparison, so substitue in your mind your favorite unit of measure (grams, ounces, whatever) in place of the word parts. Thats what makes the problem so easily solved as 11/30ths. Half of the mixture is 1/3 yellow (1 yellow to 2 blue) and the other is 2/5 yellow (2 yellow to 3 red). So, 1/3 of 1/2 is 1/6 and 2/5 of 1/2 is 1/5. Meaning the total yellow is 1/6 + 1/5 or 5/30 + 6/30 = 11/30.
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Marc Clifton wrote:
Part in this case means same sized container of indeterminate size, essentially unit of measure.
From wikipedia - The concept of "part" is not easy to define though it is intimately connected to the concept "whole". From dictionary.com - 1. a portion or division of a whole that is separate or distinct; piece, fragment, fraction, or section; constituent: the rear part of the house; to glue the two parts together. See, you should have known to use dictionary.com reference 5 - NOT reference 1. It was so clear. To answer the test question, yes, one can _imply_that the parts are equally sized, especially if the question was word comprehension. However, it was math, and the question should have qualified the meaning by stating "equal parts". Its a bad question if there can be multiple correct answers. Step 1: read question. Step 2: Decide what assumptions are made in the question. Step 3: Answer accordingly. Example: I have a gallon of Green dye, and a quart of Orange dye. Combine together - how many parts of yellow dye are in the whole? :confused:
Gary
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If they are of equal quantities, you reduce, so "5 parts to 5 parts" would properly be represented as "1 part to 1 part", or more accurately "equal parts". Think of it as being fractional. In your example, 5+5 is 10, so each part is 5/10 (ot 1/2) of the whole. If it was 3 parts to 9 parts, then the first would be 3/12 (1/4) and the second 9/12 (3/4) making it actually be 1 part to 3 parts. Parts is always a unit of equals measure in the comparison, so substitue in your mind your favorite unit of measure (grams, ounces, whatever) in place of the word parts. Thats what makes the problem so easily solved as 11/30ths. Half of the mixture is 1/3 yellow (1 yellow to 2 blue) and the other is 2/5 yellow (2 yellow to 3 red). So, 1/3 of 1/2 is 1/6 and 2/5 of 1/2 is 1/5. Meaning the total yellow is 1/6 + 1/5 or 5/30 + 6/30 = 11/30.
Draugnar wrote:
Thats what makes the problem so easily solved
Something wrong. I've never put either white nor black in my concrete. 3 buckets brown sand, 5 buckets brown rocks, 1 bucket grey mortar mix, couple buckets of clear water. 8/11 of Brown + 1/11 of gray + 2/11 of clear = gray No white, no black... WTF Equation is broke - concrete not broke. :omg:
Draugnar wrote:
Meaning the total yellow is 1/6
Hmmm, no yellow (or red or blue) in concrete.
Gary
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Is it not?
We are a big screwed up dysfunctional psychotic happy family - some more screwed up, others more happy, but everybody's psychotic joint venture definition of CP
blog: TDD - the Aha! | Linkify!| FoldWithUs! | sighistOnly within each step. You can use quarts when making the orange, liters when making the green, and cupps when making the final 1:1 equal mixture of (brown?) paint. It doesn't mater outside of each step as the whole question is about conceptualizing ratios in the "real" world. One assumption everyone is making is that the SAT math is only math. It is actually an analytical test and part of analytics is interpretation.
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I before E except after C and W or before GH.
Back when I learned it, it was a little jingle..."i before e, except after c, or when sounding like a, as in neighbor or weigh.
Tech, life, family, faith: Give me a visit. I'm currently blogging about: Upon this disciple I'll build my new religion? The apostle Paul, modernly speaking: Epistles of Paul Judah Himango
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Marc Clifton wrote:
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
If the proportion of yellow dye in the orange mixture is 2/5, and the proportion of yellow dye in the green mixture is 1/3, wouldn't the proportion of yellow dye in the new mixture be 3/8?
"Normal is getting dressed in clothes that you buy for work and driving through traffic in a car that you are still paying for, in order to get to the job you need to pay for the clothes and the car and the house you leave vacant all day so you can afford to live in it." - Ellen Goodman
"To have a respect for ourselves guides our morals; to have deference for others governs our manners." - Laurence Sterne
No, you multiply, not add, and remember that each is only HALF of the total mixture. 2/5 * 1/2 = 2/10 = 1/5 (reduction). 1/3 * 1/2 = 1/6. 1/6 + 1/5 = 5/30 + 6/30 (to add fractions, you must have equal denominators, which requires multiplying the numerators by their respective factors that makes each fractions denominator equal, in this case 5 and 6 respectively). 5/30 + 6/30 = 11/30. Please tell me you don't develop software for an engineering firm, or if you do, tell me what they have built/designed so I know never to get near their products. Oh, if you work for a bank or insurance comapny, also let me know so I know to avoid their services and to short their stock. :laugh:
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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
Yes, p gets factored out. 5o = 3r + 2y 3g = 2b + 1y So any amount of orange is (proportion) 3/5 r and 2/5 y. Hence: o: 3r/5 + 2y/5 g: 2b/3 + 1y/3 Equal amounts o and g = o/2 + g/2 = 9r/30 + 6y/30 + 10b/30 + 5y/30 = 9r/30 + 10b/30 + 11y/30 so the proportion of yellow in any mix of equal parts orange and green is 11/30th. It's not rocket science; it's simple high school algebra. Jim
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No, you multiply, not add, and remember that each is only HALF of the total mixture. 2/5 * 1/2 = 2/10 = 1/5 (reduction). 1/3 * 1/2 = 1/6. 1/6 + 1/5 = 5/30 + 6/30 (to add fractions, you must have equal denominators, which requires multiplying the numerators by their respective factors that makes each fractions denominator equal, in this case 5 and 6 respectively). 5/30 + 6/30 = 11/30. Please tell me you don't develop software for an engineering firm, or if you do, tell me what they have built/designed so I know never to get near their products. Oh, if you work for a bank or insurance comapny, also let me know so I know to avoid their services and to short their stock. :laugh:
Doing the math is easy. Formulating the problem itself is an entirely different matter, especially if assumptions and opinions are taken into consideration. If one unit of orange is 3 parts red plus 2 parts yellow, it contains 5 parts. Similarly, if one unit of green is 2 parts blue plus 1 part yellow, it contains 3 parts. Therefore, wouldn't one unit of the new mixture contain one unit of orange plus one unit of green, for a total of 8 parts, with 3 of those parts being yellow? O = R R R Y Y G = B B Y OG = R R R B B Y Y Y Where does the HALF come into play?
"Normal is getting dressed in clothes that you buy for work and driving through traffic in a car that you are still paying for, in order to get to the job you need to pay for the clothes and the car and the house you leave vacant all day so you can afford to live in it." - Ellen Goodman
"To have a respect for ourselves guides our morals; to have deference for others governs our manners." - Laurence Sterne
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Doing the math is easy. Formulating the problem itself is an entirely different matter, especially if assumptions and opinions are taken into consideration. If one unit of orange is 3 parts red plus 2 parts yellow, it contains 5 parts. Similarly, if one unit of green is 2 parts blue plus 1 part yellow, it contains 3 parts. Therefore, wouldn't one unit of the new mixture contain one unit of orange plus one unit of green, for a total of 8 parts, with 3 of those parts being yellow? O = R R R Y Y G = B B Y OG = R R R B B Y Y Y Where does the HALF come into play?
"Normal is getting dressed in clothes that you buy for work and driving through traffic in a car that you are still paying for, in order to get to the job you need to pay for the clothes and the car and the house you leave vacant all day so you can afford to live in it." - Ellen Goodman
"To have a respect for ourselves guides our morals; to have deference for others governs our manners." - Laurence Sterne
But the last step said EQUAL amounts of Orange and Green (in other words, half of the final mix is orange and half blue) were mixed together, not the TOTAL amount of both mixtures. Your equation would only work if 5 parts of orange were mixed with 3 parts of green and 5 and 3 aren't equal. You have to read the ENTIRE question. Assuming you use all of the Green (and you used the same unit of measure when mixing the components), you will only use 3/5ths of the Orange. Change parts to ounces and work it through. 2 ounces of yellow and 3 ounces of red make 5 ounces of orange. 1 ounce of yellow and 2 ounces of blue wake 3 ounces of green. Because we are using equal amounts of orange and green (read the question closely, it states equal amounts) the most we can make is six ounces, 3 green and 3 orange. So, we have 1 ounce of yellow in the green we use and 1 1/5 ounce of yellow in the red (2 ounces * 3/5 as we are using only 3 of the 5 ounces of orange). So, 2 1/5 ounces of yellow out of 6 ounces of the final color. 2 1/5 = 11/5. 11/5 of 6 is 11/5*1/6 = 11/30. If you don't get it now, I don't know if I can help further. You need to go take remedial mathematics and learn fractions.
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Yes, p gets factored out. 5o = 3r + 2y 3g = 2b + 1y So any amount of orange is (proportion) 3/5 r and 2/5 y. Hence: o: 3r/5 + 2y/5 g: 2b/3 + 1y/3 Equal amounts o and g = o/2 + g/2 = 9r/30 + 6y/30 + 10b/30 + 5y/30 = 9r/30 + 10b/30 + 11y/30 so the proportion of yellow in any mix of equal parts orange and green is 11/30th. It's not rocket science; it's simple high school algebra. Jim
I learned that LONG before high school. Try 6th or 7th grade. Honestly, who here didn't take pre-algebra in 7th grade and algebra before they started high school? I've always thought decent software developers were amongst the smartest of the populace and that we all had completed Calc 101 (college level) our senior year of high school. Most of the AP students at my high school did so we could go to Calc 201 first quarter of our freshman year of college, or just take a break from advanced mathematics for a year (I took the AB and BC exam and skipped 101 and 201).
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But the last step said EQUAL amounts of Orange and Green (in other words, half of the final mix is orange and half blue) were mixed together, not the TOTAL amount of both mixtures. Your equation would only work if 5 parts of orange were mixed with 3 parts of green and 5 and 3 aren't equal. You have to read the ENTIRE question. Assuming you use all of the Green (and you used the same unit of measure when mixing the components), you will only use 3/5ths of the Orange. Change parts to ounces and work it through. 2 ounces of yellow and 3 ounces of red make 5 ounces of orange. 1 ounce of yellow and 2 ounces of blue wake 3 ounces of green. Because we are using equal amounts of orange and green (read the question closely, it states equal amounts) the most we can make is six ounces, 3 green and 3 orange. So, we have 1 ounce of yellow in the green we use and 1 1/5 ounce of yellow in the red (2 ounces * 3/5 as we are using only 3 of the 5 ounces of orange). So, 2 1/5 ounces of yellow out of 6 ounces of the final color. 2 1/5 = 11/5. 11/5 of 6 is 11/5*1/6 = 11/30. If you don't get it now, I don't know if I can help further. You need to go take remedial mathematics and learn fractions.
Draugnar wrote:
But the last step said EQUAL amounts of Orange and Green (in other words, half of the final mix is orange and half blue) were mixed together, not the TOTAL amount of both mixtures.
Therein lies the confusion. Equal could mean anything, not just half. While half a unit of orange plus half a unit of green makes a whole unit of orange/green, you could also say that 1 unit of orange plus 1 unit of green makes a whole unit of orange/green. Regardless of what you add together, the result would still be a whole unit of orange/green. What you describe makes sense, depending on how you interpret the question, but you could still argue that one unit of orange is equal to one unit of green in terms of weight and volume.
"Normal is getting dressed in clothes that you buy for work and driving through traffic in a car that you are still paying for, in order to get to the job you need to pay for the clothes and the car and the house you leave vacant all day so you can afford to live in it." - Ellen Goodman
"To have a respect for ourselves guides our morals; to have deference for others governs our manners." - Laurence Sterne
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Draugnar wrote:
But the last step said EQUAL amounts of Orange and Green (in other words, half of the final mix is orange and half blue) were mixed together, not the TOTAL amount of both mixtures.
Therein lies the confusion. Equal could mean anything, not just half. While half a unit of orange plus half a unit of green makes a whole unit of orange/green, you could also say that 1 unit of orange plus 1 unit of green makes a whole unit of orange/green. Regardless of what you add together, the result would still be a whole unit of orange/green. What you describe makes sense, depending on how you interpret the question, but you could still argue that one unit of orange is equal to one unit of green in terms of weight and volume.
"Normal is getting dressed in clothes that you buy for work and driving through traffic in a car that you are still paying for, in order to get to the job you need to pay for the clothes and the car and the house you leave vacant all day so you can afford to live in it." - Ellen Goodman
"To have a respect for ourselves guides our morals; to have deference for others governs our manners." - Laurence Sterne
And because they are equal, your equation doesn't work. It isn't interpretation at all. The two parts of orange and green are equal. Equal means equal. There is no other interpretation. your original 3/8s means the orange and the green would not be equal amoutns because you are assuming the initial parts are equal, making one 2/3 greater than the other. If the inital parts aren't equal so that the final volums are (say you use 3/5 of an ounce for each part in the orange and one ounce for each part in the green) then it still wouldn't be 3/8 of the solution, because 2 of those 3 parts were only a sum total of 6/5 of an ounce where as the other 1 of those three parts was a whole ounce. You seriously need to retake pre-algebra. This isn't that difficult and it makes me afraid to even think what you code might do if involved in monetary or engineering equations.
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PIEBALDconsult wrote:
I before E except after C and W or before GH.
Albeit you might be absolutely correct, I'm not agreeing with you! We were anteing up for a bet, and I think you are actually wrong. I think any foreigners on CP might agree with me, unless you counterfeit the results. I had a seizure just thinking about it. The atheists were particularly upset. So, I put on my beige shorts, and proclaimed "anchors aweigh" as I ate my braunschweiger (for protein) and marbleized crumpets with my cup of caffeine. "We're going canoeing!" I said, clueing others in on our daily adventure. Just don't get into the counterfeit canoe. The water's fine, because it's deionized - thank the blueish deity (no, it's not from dyeing his skin, but any einstein would know this). And it's not a blue holstein, either. Weird indeed. I had to forfeit because my heirs had heisted a canoe earlier. So I was looking through the kaleidoscope as I fell on my keister. After kneeing the monseigneur, I changed into other leisureware (I was caught peeing in my shorts). But that's neither their here nor their there. The reigning captain reignited the debate on whether I should be reimbursed, as I was reinfecting the earlier wound. I told him I would reinitiate a suit, as I had reimaged the earlier wound (had to reink the printer first). That's when he unleashed the rottweilers! I was seeing more clearly now. I thought I was having a seizure, but then I noticed the seismometer was seizing. The sheilas sat on it as they were shoeing their feet while surveilling sightseeing surroundings. Thereinafter, I decided I was tieing (M-W) my own shoes, or would resort to tippytoeing. Wherein I decided, with veins bulging, it's a good thing we have simple rules to live by, especially for weirdos with weiners, eh? :laugh:
Gary
While I agree they are words most of your exceptions are due to puting a prefix or a postfix on the root word. Naturally any verb that begins in an i can form a "ei" combo by adding a re- in front, or one that ends in a e with an -ing. The others though a amusing none the less. Also alot/most you'll note were borrowed from french or german, I challenge you to repeat said joke only using words derived from old english. ;P P.S. why did the US decide on math and vocab as the standard for college admissions anyways? Say your planning on doing pure math. Sure your math scores should be through the roof, but your need for vocab is limited as you need a small subset of vocab to be able to do your job.
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While I agree they are words most of your exceptions are due to puting a prefix or a postfix on the root word. Naturally any verb that begins in an i can form a "ei" combo by adding a re- in front, or one that ends in a e with an -ing. The others though a amusing none the less. Also alot/most you'll note were borrowed from french or german, I challenge you to repeat said joke only using words derived from old english. ;P P.S. why did the US decide on math and vocab as the standard for college admissions anyways? Say your planning on doing pure math. Sure your math scores should be through the roof, but your need for vocab is limited as you need a small subset of vocab to be able to do your job.
deltalmg wrote:
most of your exceptions are due to puting [putting] a prefix or a postfix on the root word.
Nope. Count em. I left most of the re-i and e-ing words out. I needed some for context, however, to make the story flow. College is for all disciplines, not just math and engineering. Why do assume ALL programmers need math? No math is needed to build a web site (for-loops excepted). Similar question, why does a language major need to know math??
deltalmg wrote:
P.S. why did the US decide on math and vocab as the standard for college admissions anyways?
I dunno. That was before my time. I'm not a history buff. :)
deltalmg wrote:
Say your [you're] planning on doing pure math...but your need for vocab is limited
Vocab is important, none the less, so students can hear, read and understand the homework assignments. X|
Gary
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Back when I learned it, it was a little jingle..."i before e, except after c, or when sounding like a, as in neighbor or weigh.
Tech, life, family, faith: Give me a visit. I'm currently blogging about: Upon this disciple I'll build my new religion? The apostle Paul, modernly speaking: Epistles of Paul Judah Himango
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26 (equal) parts of 75 total (equals) parts of 26/75 of the formula is yellow or 34 2/3% is yellow or, for a really complex breakdown... 26 (equal) parts of yellow 40 (equal) parts of blue 9 (equal) parts of red
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Draugnar wrote:
26 (equal) parts of yellow
That is one of about a gazillion correct answers. :) There are also 39 (equal) parts of yellow.
Gary
What ARE you smoking? There is only one correct answer: 1 gallon = 4 quarts. Therefore, the Green comprise 4 of the 5 parts of this equation, while the Orange comprises 1 of the 5 parts. 1/3(Yellow in Green) * 4/5(Green in final solution) = 4/15(Yellow in final solution from Green) 2/5(Yellow in Orange) * 1/5(Orange in final solution) = 2/25(Yellow in final solution from Orange) 4/15(Yellow) + 2/25(Yellow) = 20/75(Yellow) + 6/75(Yellow) = 26/75(Yellow) The only way you could get 39 parts yellow is if you deliberately wanted to confuse your readers by not keeping the ratios as reduced fractions. This would require both the numerator and the denominator to be multipled by 1.5, which would give a ratio of 39/112.5, which is not only NOT REDUCED, but it has a decimal as part of a fraction, which is very bad form, kind of like making the claim thatg mathematics is not determinable. Come on, I thought people were supposed to know at least a little bit about elementary (as in the level of schooling, not as in a synonym for "basic") mathematics around here.
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But that is o so wrong. how can
0 == 0prove or solve anything? :wtf: you should have used real numbers here.Luc Pattyn [Forum Guidelines] [My Articles]
This month's tips: - before you ask a question here, search CodeProject, then Google; - the quality and detail of your question reflects on the effectiveness of the help you are likely to get; - use PRE tags to preserve formatting when showing multi-line code snippets.
Well, let's see..... if you have an equation that reduces to 0==0, then you know (depending on the equation being solved) either the input was correct, or the variables don't affect the outcome. If you assigned values to all the variables and found that 0==0, then there aren't any contradictions in the equation. If you left values out of the variables, and simplifying the equation made the variables go away, then the variables cancelled each other out, which means they didn't mtter. Not that that has anything to do with the post you were replying to. Why did you even post a statement about 0==0? The previous post reduced to 1==1, which should always be the case. And I believe all those fractions that were used were real numbers. "Of course, I could be wrong..... but I'm not."