Thread: Need help with fractions. Get basics.

Hi i am a new member and i do apologise if i have posted in the wrong section. However to the point.

I have fractions problem. The problem is well understanding it. I can do the manipulations but understanding is important. If someone can help or direct me to a website i will be glad. I just need to understand so i can be at the same level as other students.

you could try Khan Academy

you haven't given enough information to identify the topic you should look at, but maybe this one? Decimals and Fractions | Developmental Math | Khan Academy

Originally Posted by Thinx

Hi i am a new member and i do apologise if i have posted in the wrong section. However to the point.

I have fractions problem. The problem is well understanding it. I can do the manipulations but understanding is important. If someone can help or direct me to a website i will be glad. I just need to understand so i can be at the same level as other students.

well, one of the keys to understanding fractions, is to understand why we need them in the first place.

suppose i have an equation:

2x = 3

i'd like to solve for x. well, to "unmultiply", we divide.

now 2x/2 doesn't pose much of a problem, 2 divides evenly into 2, and we have just x. but what is 3/2? it's certainly not a number in terms of "counting things", it's a ratio: 3 on top, for every 2 on the bottom. well, if we know what "1/2" is, then it seems logical that 3/2 ought to be 3 of these "1/2" things. so everything really boils down to what 1/2 means.

and 1/2 is what we need to multiply 2 by, to divide it in 2, so as to get unity, or 1. in symbols:

(1/a)a = a(1/a) = 1.

so fractions are made to "unmultiply" things. so it comes as not much of a surprise that multiplying fractions is very easy:

(a/b)(c/d) = (ac)/(bd), we simply multply "straight-across" top times top, over bottom times bottom.

but adding fractions...that is tough. what is 1/2 + 1/3? it's hard to say, off the top of one's head, because halves and thirds are two different kinds of things.

so we have to get halves and thirds expressed in some "common language" that will allow us to combine them in a way that makes sense.

so what can we do? we need to find "a common denominator", something that both 2 and 3 go into evenly. so what we do is "convert to 6ths".

why 6ths? naively, if we want something that a and b both divide, then ab certainly fits the bill: ab/b = a, and ab/a = b, so these "come out nice".

so how do we convert to 6ths? we multiply everything by "what is missing" over itself. for example, to get 6 from 2, we multiply by 3:

so 1/2 = (1/2)(3/3) (because 3/3 is just another name for 1, and 1 times anything is itself).

but we know how to multiply fractions, that's EASY:

(1/2)(3/3) = (1*3)/(2*3) = 3/6.

well, with 3rds, what's "missing" to get to 6ths? a factor of 2. so we multiply 1/3 by 2/2 (= 1):

(1/3)(2/2) = (1*2)/(3*2) = 2/6.

now, we have a "common language", we're just adding 3 of something, and 2 of something, which makes 5 somethings:

1/2 + 1/3 = 3/6 + 2/6 = 5/6. that is: when the denominators are the same, keep the bottom the same, and add the tops.

so this is just what we do, in "the general formula":

a/b + c/d = (a/b)(d/d) + (c/d)(b/b)
= (ad)/(bd) + (bc)/(bd) <---bottoms the same, now add the tops
= [ad + bc]/(bd).

*******

now, while all of the above is perfectly true, one must realize that fractions aren't unique, like numbers are. for example:

1/2 = 2/4 = 3/6 = 4/8 = 50/100 = 17/34 = 25/50 = 5/10 =.....there are LOTS of different ways for writing this one fraction.

so, often, one wants "the reduced form" a/b, where a and b don't have "any common factors". for example:

3/12 = (3*1)/(3*4).

the top and bottom have a common factor of 3. so we "split those out":

(3*1)/(3*4) = (3/3)(1/4) = (1)(1/4) = 1/4. since 1 and 4 have no common factors besides 1, we're done.