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Multiplying fractions (newbie question)

  1. Aug 21, 2007 #1
    I'm having a bit of trouble with fraction multiplication. I understand the concept visually, but don't understand why the mechanical multiplication method works the way it does. Please bear with my descriptions, I'm just learning fractions (as well as basic math operations)and want to show how I work things out, hoping someone will be able to find the fault in my line of reasoning.

    For example, here's how I understand fraction multiplication through visualization. If we have 2/3 times 1/3, we're saying we want 1/3 to 'exist' 2/3 times. Because we're not multiplying 1/3 by a whole number (1,2,3, etc), we're going to end up with a fraction which represents less than 1/3. I then visualize a pizza (or any object/conceptual group) broken up into 3 equal groups. I realize that when we consider the symbol '1/3', we mean we're selecting 1 of the 3 visually separate divisions on our pizza/whatever. We're then saying we want that '1/3' of the pizza to 'exist' 2/3 times. Therefore, 2/3 of that section of the pizza will be selected. So then, I break up the third of the pizza into 3 equal portions, selecting 2 parts. To express the new 'selection' of parts, I break up the rest of the pizza into equal parts as well, resulting in 9 equal parts. Therefore, my new selection is 2/9. 2/9 is 1/3, 2/3 times.

    However, while this method works and makes visual sense to me, the common method of fraction multiplication doesn't. I'm watching TTC Basic Math training videos, and the teacher says the numerator is the digit, while the denominator is the place value -- and that this is why the traditional multiplication method of fractions works the way it does -- where we multiply numerator by numerator and denominator by denominator, getting our same answer -- 2/9.

    First of all, I don't understand why the numerator is the digit and why the denominator is the place value. I understand that in the case of a number like 1293, each of the numerical symbols are in specific place values, which represent 'amounts of 1's' (a thousand 1s, 200 1s, 90 1s and 3 1s). I understand the digit is just the numerical symbol in a specific place value, indicating said amounts of 1s (again, 1000's of 1s, etc). However, I've yet to understand how the numbers which comprise a fraction connect to the concepts of 'digit' and 'place value'. I understand fractions in this way -- that we have all the necessary symbols (1,2,3, etc) to complete various functions with what we consider '1' (whether '1' represents a group of 10 trees, a forest or a piece of pizza). These functions are those like multiplication, division, subtraction and addition. However, we want to be able to deal with parts of what we consider 'whole' or '1'. Therefore, we introduce the concept of the fraction, breaking up what we consider '1' into a set of equal parts. This set of parts is called the denominator of the fraction. The numerator tells us how many parts of the entire set of equal parts are being selected (or how many are left). The issue is that this is how I understand fractions -- that we simply needed a way to convey parts of what we consider '1', and therefore pulled the '/' symbol out of nowhere, placing one number on the top and another on the bottom. I understand the '/' symbol represents division, and understand that every proper fraction will result in a decimal, which curiously represents the same thing as the fraction. However, this is about as much as I understand, therefore, the traditional fraction multiplication method doesn't make one bit of sense (other than the fact that, somehow, quite magically, it replicates exactly what I did when I multiplied fractions in my long and tortuous visual method).

    This is the only reason I've been able to come up with for the traditional mechanical method of fraction multiplication is that, for some reason, it just works that way -- multiplying numerators by numerators and denominators by denominators somehow reproduces the exact set of steps I slowly took to determine the answer -- 2/9.

    Thanks for the help.
  2. jcsd
  3. Aug 22, 2007 #2
    You can look at the place value as 3 and the digit as 2 in 2/3. What does it say? Well, 3/3 = 1, so that it is also 1-1/3. Or that 2/3 is a number that requires 1 more in the numerator to be equal to 1.

    Now multiplying by 1/3 is just the same as dividing by 3. So: [tex] 1/3*2/3 =\frac{2/3}{3} [/tex]

    If this is problem, well its easy to see that 1/3 of 1/3 is 1/9. What is that? Because 1/9+1/9+1/9 = 1/3. So that using 2/3 gives 1/3*2/3 = 2/9.
    These things can be done by rote, or they can be thought out. Sometimes it is just as well to do them by rote, and then look at them later.

    The whole system is simple for some people, thought I have found that many adults forget almost all they have learned about fractions, since they never use them--and maybe they never really thought it through anyway. Actually what concerns most people with fraction problems is understanding the common demoninator concept. Some of difficulity is seeing the difference between 2/5 + 1/3 and 2/7+1/7. (It may be that they confuse multiplication and addition.)

    This doesn't seem to be your problem, though. You want understanding. Now adding similar pieces of pie makes sense, multiplying them together is a bakery mess. Looking at this algebratically may help more.

    If 3X = 1, then X=1/3. If (3/2)X=`1/3, then X = 2/9. Why? (In this problem we are requiring a smaller answer since we have 3/2X and are looking for X.) Well, 1/3 was divided up 3 times then it would be 1/9, so that 2/3 of this division would be 2/9.

    You can also change these things to decimals. 1.5X = .33333..., then X = .222222.... There are many ways to skin a cat.
    Last edited: Aug 22, 2007
  4. Aug 22, 2007 #3
    Thank you. I'm going to have to muse over what you've just said. I'm still lost.
  5. Aug 22, 2007 #4


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    When you multiply numerator by numerator and denominator by denominator, here's what's happening (in your terms):

    The starting problem:
    2/3 * 1/3

    "Break up" the 1/3 into three pieces, making ninths:
    2 * 1/(3 * 3) = 2 * 1/9

    Take two of those ninths:

    You can see that I just multiplied the denominators as the first step and the numerators as the second.
  6. Aug 22, 2007 #5


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    Which is a perfectly okay method to look at it.
    Stick to it!

    Let's use that method to visualize the general result
    where "a" and "b" are two non-zero numbers.

    Suppose that you have broken your pizza into b pieces, a piece called P.

    Now, an a'th part of P (called p) means there are "a" p's in a single P.

    Since there are b P's in the whole pizza, it means there are a*b p's in the whole pizza.
    But therefore, a single p is an "a*b"'th part of the pizza, which is the result we wanted..
  7. Aug 22, 2007 #6
    I think it might help if you have some where to build from.

    Multiplying a integer by a fraction for example:

    [tex] 3 * \frac{2}{6} = \frac{3*2}{6} = \frac{6}{6} = 1[/tex]

    If that doesn't make sense remember that:
    [tex] 3 * \frac{2}{6} = \frac{2}{6}+\frac{2}{6}+\frac{2}{6} = \frac{6}{6} = 1[/tex]
    Which proves that it works.

    Now, isn't 3 = 3/1? We can look at the first think I mentioned in a slightly different (but in fact the same) way.

    [tex] \frac{3}{1} * \frac{2}{6} = \frac{3*2}{1*6} = \frac{6}{6} = 1[/tex]

    The general idea is:
    [tex]\frac{a}{c} * \frac{b}{d} = \frac{a*b}{c*d}[/tex]

    Does that make any sense at all?
  8. Aug 24, 2007 #7
    Thank you all. I'm musing over this. Starting to get it. I'll update with my latest understanding ASAP
  9. Aug 24, 2007 #8
    ^You seem very articulate to be just learning division. How old are you? I'm assuming you are at least 10.

    I could be way off though.
  10. Aug 24, 2007 #9


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    You can think of fractions as numbers with "units". For example, just like "2 meters" means you have two parts each a meter long, so "2/5" means you have 2 parts, each a "fifth" long. If you have a rectangle with length 2 m. and width 3 m., then the area is the product 6 m2 or 6 square meters. Just as you have to multiply the numbers, you also have to multiply the units. Same with fractions if you 2 "fifths" and 3 "fifths" then their product is 6 "fifths2" or 6/25.
  11. Aug 26, 2007 #10
    I'm 22. Just about to turn 23. I got through all high school math classes using rote memorization, and even ended up in the advanced math classes later on. I'm now forced to re-learn everything, seeing as I never visually understood how math related to geometrical concepts or even how concepts like fractions related visually to the whole number system. I'm trying to get to basic trigonometry in about a month (end of september). It's going slow though, seeing as I want to enforce visual understanding of every principle.

    Somehow, I managed to get through an entire year of high school trigonometry without knowing trig was related to triangles!!!!! This was something I realized just recently. I'm dead serious. :) Additionally, I passed the class all year with straight A's! Again, pure rote memorization and tests structured on homework where the homework varied very little from the tests.... just simple value changes. This resulted in horrid SAT test scores. It wasn't a great high school. I was able to get multiple 100.0 averages in a row there, if that's an indicator... and yet I don't understand fractions. :P

    Would you all mind if I come back and ask some more questions in the near future? You're all very helpful and I can't thank you enough. Additionally, is this the best place to ask my basic math questions?
    Last edited: Aug 26, 2007
  12. Aug 26, 2007 #11
    While visual insight can help (judgment via inspection can be quite useful if applied properly), I would (more strongly) advise pursuing an understanding of the defining logical structures fundamental to the concepts you seek to learn, even if a visual interpretation is not readily apparent

    I am not surprised at all!
    (Often a rather accurate indictment of American education :frown:)

    **Edit: you may want to read this (and possibly this)
    Most likely so...
    Unfortunately, "blaming the test/standards" is still more popular than "blaming the education"

    Sure, PF is a great place for help on your questions! :smile:
    (Just remember to show your work/effort on problems as needed :wink:)
    Last edited: Aug 26, 2007
  13. Apr 12, 2011 #12
    there is a visual way to look at it.
    it is based on areas of rectangles as a way of multiplication.
    it ends up just being counting the total number of cells and the number of cells that are 'filled', or in the intersection.

    I will just show a table illustration, and I will use an example of 2/3 * 3/7

    ` |x|x|x| | | | |
    _ ._._._._._._._.
    ` | | | | | | | |
    x |x|x|x|~|~|~|~|
    _ |_|_|_|_|_|_|_|
    ` | | | | | | | |
    x |x|x|x|~|~|~|~|
    _ |_|_|_|_|_|_|_|
    ` | | | | | | | |
    ` |~|~|~| | | | |
    _ |_|_|_|_|_|_|_|

    you can see that there are 2*3 filled cells, out of a total of 3*7 cells, which are supposed to be equal to a whole (its supposed to look like a square)
    so you have [tex]\frac{2*3}{3*7}[/tex]

    I remember kind of my math teacher in 5th grade sort of showing us this and I think I sort of got it visually, but I'm not sure whether I was listening to what she was saying.
    That year, I learned that xa * xb = xa+b, and that xp/q = [tex]\sqrt[q]{x}^{p}[/tex].
    I didn't learn this because they were accelerating us too much, our teacher had a system where students could form their own conjectures, and these were some of mine. the first one was something that appeared really obvious to me, something about like 6*15 = (2*3)*(3*5) = (2*5)*(3*3) = 10*9, whereas I think mine were pushing at the boundaries of what I was learning. having figured out rational exponents, I then got stuck on irrational and imaginary exponents until I taught myself calculus, and then complex analysis, from online.
    thanks to that teacher I love math so much...
  14. Apr 12, 2011 #13


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    You're about 4 years too late with that advice: originally posted on Aug21-07, 10:21 PM
  15. Apr 12, 2011 #14
    I realize that I am late, but if someone were to search google this site would pop up. others may have the same question. And I know this math stuff is super basic, but I was randomly reading a blog about how a lot of kids in elementary school are confused by fractions, and I wondered why fractions apparently are so confusing. I have almost always just taken for granted that a/b * c/d = a*c / b*d, but... what's the best explanation? other than using field theory or something like that.
  16. Apr 12, 2011 #15


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    Specifically, 5 out of 4 elementary students are confused by fractions. :wink:
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