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Math Help - Do you like Fractions?

  1. #1
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    Do you like Fractions?

    A recipe calls for 3/5 lb of pasta for a salad. How much pasta should be used for 1/2 of the recipe?
    I used 3/5 divided to 2 and I got the outcome of 3/10. Is this correct?
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  2. #2
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    Why do you doubt?

     <br />
\frac{3}{5} \cdot \frac{1}{2}\;=\;\frac{3}{10}<br />
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    Quote Originally Posted by TKHunny View Post
    Why do you doubt?

     <br />
\frac{3}{5} \cdot \frac{1}{2}\;=\;\frac{3}{10}<br />
    Because the way the person wrote the question is so confusing. While I sort of knew it has to be dividing, I didn't know what to divide with... If I were to divide 1/2 I would have to flip the equation around. BUT if I divide it to number 2 then I would have to flip 2 to 1/2.
    I still don't know why the hell did he put that 1/2 there.
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    Quote Originally Posted by BrilliantLegacy View Post
    it has to be dividing
    That's just wrong. Please practice more exercises and get more familiar with algebraic manipulation.
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    Quote Originally Posted by TKHunny View Post
    That's just wrong. Please practice more exercises and get more familiar with algebraic manipulation.
    Alright. Was the answer right? And can you explain why did I divide them and such......
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  6. #6
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    Quote Originally Posted by BrilliantLegacy View Post
    Alright. Was the answer right? And can you explain why did I divide them and such......
    Yes, the answer is correct. We can see this intuitively because \frac {3}{5} =  \frac {6}{10} and half of \frac {6}{10} is simply  \frac {3}{10}.

    But I understand that you're unsure with the "mechanical" manipulations of fractions. In the former post (I forget who posted it), we saw that the problem was solved using multiplication, even though by intuition if you want one half of something, you divide. Well, here's the thing. Division is multiplication. Anything that you can write as a division, you can write as a multiplication as well.

    Revisiting the problem, we see that \frac {3}{5} \cdot \frac {1}{2} = \frac {3}{10}. If we multiply something by \frac {1}{2}, we will get an output of half that thing because we are multiplying by a number less than one. You know that if you multiply by 1, you get the same number. If we multiply by a number larger than 1 (as long as the number we're multiplying is also greater than 1), we get a larger number. But if we multiply it by less than 1, we will obviously get something smaller (for now, we are not considering negative numbers--0 is our endpoint). So if we multiply something by 0.001, we're obviously going to get a much smaller answer, because it's only 0.001 of our original number.

    So using this logic, it should be clear why the answer is smaller, despite the multiplication. In rudimentary terms, if you want to divide a number a by a number b, you can simply multiply the number a by the reciprocal of the number b, so we have a\cdot \frac {1}{b} (and if you multiply it out, you get \frac {a}{b} which is obviously division). As you can see in the former problem, we're dividing by 2 and the reciprocal of 2 is \frac {1}{2}. Why the reciprocal? Well, by intuition if we want to divide something by say....3, the resulting number is one third ( \frac{1}{3}) of the original number. And how do we get multiples (by this I mean that the resulting number will be a multiple of the original number, because if you multiply it times 3, you get your original number back) of numbers? We multiply! So if we want a number that gives us our original number back if we multiply it by 3, it only makes sense to multiply it by \frac{1}{3}.

    I hope this clarifies it...somewhat. It's difficult to explain the fundamentals.
    Last edited by RobLikesBrunch; August 25th 2009 at 09:36 PM.
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    Quote Originally Posted by RobLikesBrunch View Post
    Yes, the answer is correct. We can see this intuitively because \frac {3}{5} =  \frac {6}{10} and half of \frac {6}{10} is simply  \frac {3}{10}.

    But I understand that you're unsure with the "mechanical" manipulations of fractions. In the former post (I forget who posted it), we saw that the problem was solved using multiplication, even though by intuition if you want one half of something, you divide. Well, here's the thing. Division is multiplication. Anything that you can write as a division, you can write as a multiplication as well.

    Revisiting the problem, we see that \frac {3}{5} \cdot \frac {1}{2} = \frac {3}{10}. If we multiply something by \frac {1}{2}, we will get an output of half that thing because we are multiplying by a number less than one. You know that if you multiply by 1, you get the same number. If we multiply by a number larger than 1 (as long as the number we're multiplying is also greater than 1), we get a larger number. But if we multiply it by less than 1, we will obviously get something smaller (for now, we are not considering negative numbers--0 is our endpoint). So if we multiply something by 0.001, we're obviously going to get a much smaller answer, because it's only 0.001 of our original number.

    So using this logic, it should be clear why the answer is smaller, despite the multiplication. In rudimentary terms, if you want to divide a number a by a number b, you can simply multiply the number a by the reciprocal of the number b, so we have a\cdot \frac {1}{b} (and if you multiply it out, you get \frac {a}{b} which is obviously division). As you can see in the former problem, we're dividing by 2 and the reciprocal of 2 is \frac {1}{2}. Why the reciprocal? Well, by intuition if we want to divide something by say....3, the resulting number is one third ( \frac{1}{3}) of the original number. And how do we get multiples (by this I mean that the resulting number will be a multiple of the original number, because if you multiply it times 3, you get your original number back) of numbers? We multiply! So if we want a number that gives us our original number back if we multiply it by 3, it only makes sense to multiply it by \frac{1}{3}.

    I hope this clarifies it...somewhat. It's difficult to explain the fundamentals.
    I think I get it. It's somewhat similar to decimal, if you multiply it, it would be smaller while dividing would be bigger.
    Last edited by mr fantastic; August 26th 2009 at 12:13 AM.
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  8. #8
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    Quote Originally Posted by BrilliantLegacy View Post
    I think I get it. It's somewhat similar to decimal, if you multiply it, it would be smaller while dividing would be bigger.
    That's only true (for both fractions and decimals) if the number you are dividing or multiplying by is larger than 1.
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  9. #9
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    Quote Originally Posted by RobLikesBrunch View Post
    We can see this intuitively...
    I never believe this. If you see it, you can see it intuitively. If you don't, you don't. Learn, Practice, Work. Don't worry so much about intuitive seeing.
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  10. #10
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    Quote Originally Posted by TKHunny View Post
    I never believe this. If you see it, you can see it intuitively. If you don't, you don't. Learn, Practice, Work. Don't worry so much about intuitive seeing.
    I agree that sometimes, people do not see it. But sometimes they do, so I think it's safe to include that, along with a more detailed explanation like I did. They might get it immediately, or they may have to carry on to the more detailed explanation.

    In more complex problems, I think providing an "intuitive" route to why something works is a good way to try to get someone to think about the problem, and understand why it works. But you're right--sometimes it doesn't work.
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