Multiplying and Dividing Fractions?

Question

Solve and show answer in lowest terms:

1) (6y -12)/(8) times (4y)/( 3y^2)

2) (12x^2) divided by (4x)/(3)

Answer 1

1. [(6y-12)/8] X [4y/(3y^2)] simplify
=[(3y-6)/8] X (4/3y) times
=4(3y-6)/24 expand
=(12y-6)/24 simplify
=(2y-1)/4

2.[(12x^2) / (4x/3)] X3
=36x^2/4x simplify
=9x

Answer 2

Question 1
[ 6.(y - 2) x 4y ] / 24 y²
= (24 y² - 48 y) / 24 y²
= 24 y.(y - 2) / 24 y²
= y.(y - 2) / y²
= (1/y).(y - 2)

Question 2
= 12x² . (3/4x)
= 9 x

Answer 3

I'm not going to do your homework for you, but the process here is pretty straightforward. I'm going to cover all of the nitty-gritty details; if you feel you understand all of those things, you may want to skip to my summary at the end.

We solve these problems in 3 parts.
1) flip over the dividend (fraction we're dividing BY) in division problems.
2) Multiply the numerators (top part) of both fractions together to get the new numerator
3) Simplify the result by removing factors that are common to each 'term' in the expression.

Here's more detail:

1) When dividing one fraction by another, you need to first find the reciporical of the dividend. That's fancy way of saying you're going to take the fraction you're dividing BY (the dividend) and flip it over (reciporicate). This turns our division problem into a multiplication problem. If we started with a multiplication problem, we skip this step.
Example:
1/2 divided by 2/3 becomes 1/2 times 3/2 (after finding the reciporical of the dividend)
you then multiply: (1/2) * (3/2) = (1 * 3) / ( 2 * 2) = 3/6 = 1/2

2) When multiplying fractions, you multiply both numerators together (the top number or expression) to get the new numerator, then do the same with the denominators (bottom numbers or expressions) to get the new denominator.
Example:
1/2 times 2/3 = (1 * 2) / (2 * 3) = 2/6
This can later be simplified to 1/3.

It gets a little trickier when you're dealing with variables - but not much. We just need to remember our properties of multiplication (which are also true when multiplying regular numbers together).

First, Multiplication is commutative. This means that a * b = b * a, and that 6y * 4y = 4y * 6y. Or in English - when two things are multiplied together, we can change the order of multiplication and get the same result.

Second, Multiplication is Associative. This means that (a * b) * c = a * (b * c), and (12x * x) * 5x = 12x * (x * 5x) - Or in plain English: We can group things together however we'd like and get the same result.

Third, Multiplication is Distributive over addition or subtraction. Meaning: 4 * (3 + 2) = (4 * 3) + (4 * 2). In English: The sum of two numbers [ (3 + 2) in our example] times a third number (4 in our example) is equal each term in the sum of each term in the addend (3 and 2 in our example) multiplied by the third number.
It works with variables too: 4 * (3x + 5) = (4 * 3x) + (4 * 5).
CAREFUL: 4 * (3 + 2) DOES NOT equal (4 * 3) + 2

Order of operations sometimes comes into play.
Parentheses
exponents
multiplication
division
addition
subtraction.

Remember that with 'Please Excuse My Dear Aunt Sally', or 'Please Exhume My Dead Aunt Sally' if you're morbid like me.

Remember those things - The associative property, the commutative property, and the distributive property, and your order of operations. These are important - know how to use them, even if you don't remember what they're called.

So - how do we use that to solve an expression like (6y - 12) * 4y ? (from problem one)

First, we apply the distributive property. Since multiplication distributes over addition and subtraction, we have:

(6y * 4y) - (12 * 4y)

we can use factoring to break that down a little further if we'd like (making it clear that y^2 = y * y):

(6 * y * 4 * y) - (12 * 4 * y)

And then use the associative property to group things like so:

( (6 * 4 * y * y) - ( (12 * 4) * y ). Remember that y * y = y^2

So, working through the arithmetic, we get:

24y^2 - 24y. And that's the top term in problem one.

Now the bottom term: 8 * (3y^2)
There are only two terms ( 8 and 3y^2 ) so we don't need to distribute. We can factor 3y^2 if we want, so that it's more clear what's happening:

3y^2 = 3 * y * y

and we can apply the associative property when we're multiplying by 8:

(3 * 8) * (y * y) = 24 * y * y = 24 * y^2 = 24y^2

Now we have:

24y^2 - 24y
----------------
24y^2,

which we need to simplify.

3) Simplifying algebraic fractions is the same as simplifying numerical fractions: It's all about removing factors that are common to both the numerator (top) and denominator (bottom). One way to do that is to find the largest factor that's common to every term in the expression (there are three terms here - 24y^2, 24y and 24y^2 again), and dividing each term by that greatest common factor (GCF). In this case, the largest thing common to each term is 24y - every term contains at least ONE instance of 24y. So we can divide each term by 24y:

(24y^2 / 24y) - (24y / 24y) y - 1
---------------------------------- = ------
(24y^2 / 24y) y

And yes, that's you're final answer.
But finding the GCF is sometimes a little difficult, especially when you're first learning this stuff. So, I sometimes find it useful to factor every term into it's prime components, and remove like terms. I use these rules:
a) If I see any number or variable that exists in every term, I cross it out of every term.
b)If I cross out every single number or variable from any one term, That term becomes a '1'.
c) If any term becomes a '1', we've found the simplest form, because the only remaining common factor is '1'.
d) Likewise, if there are no more prime numbers or variables that are common to every term, we're also done.

That process is somewhat tedious, but if you realize that by crossing out a number or variable from the term you're essentially performing a division operation on every term, you can how your expression is being simplified at a very simple level.

Here's a simple example with numbers:

26 2 * 13
---- -> factoring = ----------
16 2 * 2 * 2 * 2

Two occurs in each expression at least once, so we cross out a two from each expression, and get:

13 13
------------- = ----
2 * 2 * 2 8

A more complex example with variables, which resembles our problem above:

12x^2 - 14x (2 * 2 * 3 * x * x) - ( 2 * 7 * x)
---------------- = ---------------------------------------
14x^2 2 * 7 * x * x

every expression has at least one 2 and one 'x'. So we cross one of each out of every expression:

(2 * 3 * x) - ( 7 ) 6x - 7
--------------------- = --------
7x 7x

Notice that If we multiplied together all of the common factors, we would get the GCF: 2 * x = 2x - and dividing each term by 2x yields the same result as above.

Lastly, YOUR problem simplified using that method:
24y^2 - 24y (2 * 2 * 2 * 3 * y * y) - (2 * 2 * 2 * 3 * y)
---------------- = ----------------------------------------------------
24y^2 2 * 2 * 2 * 3 * y * y

2, 2, 2, 3, and y are common to each term, so we cross those out. Notice that since we cross everything out of the second term in the numerator, it becomes a '1':
y - 1
-----
y
and again, notice that if we multiply together the common primes, we have 24y, the GCF we identified in the first method.

A third method is sort of a hybrid of the previous two. It's less tedious than the last but easier than the first (GCD) method. First remove the largest common factor that you see immediately, and continue doing that until the problem is in simplest form. In our your problem, it's easy: every term has the coefficient 24, so we can divide each term by 24, to get:

y^2 - y
--------
y^2

We then easily notice that each term has a 'y' factor in it, and divide each term by 'y', to get:

y - 1
-----
y

That's a lot of info, and goes into probably a lot more detail than you need - but I think it demonstrates every part involved in solving these problems, beyond the simplest arithmetic.

To summarize:
1) - If division is involved, flip over the fraction that you're dividing BY.
2) - multiply together numerators, and multiply together denominators to get the new result. Use your properties for multiplication:
a) Distributive: a ( b + c ) = ab + ac ; a ( b - c ) = ab - ac
b) Associative: (ab) c = a (bc); (a + b) + c = a + (b + c)
c) Commutative: ab = ba; a + b = b + a
d) Identity: 1a = a; a + 0 = a
e) reflection: a = a (everything is equal to itself)

The last two are important aspects of algebra, but not terribly useful for your purposes. Thankfully, they're also very obvious.

3) simplify the result by eliminating factors that are common to each term in the final expression. Do this by identifying the GCD, by brute factorization, or simply by removing the largest common factor that you can easily identify.

I hope that helps in your studies!

Answer 4

1) (6y -12)/(8) times (4y)/( 3y^2) =
4y(6y - 12)/24y^2 =
(6y - 12)/6y =
(y - 2)/y


2) (12x^2) divided by (4x)/(3)
3(12x^2)/4x =
9x

Answer 5

1. In multiplying factions it's Top * top and bottom * bottom. Thus we have...
4y(6y-12)
-------------
8(3y^2)

Distribute:
24y^2 - 48y
----------------
24y^2

Factor out a 24y...GCF: Greatest common factor.
24y(y^2 - 2)
----------------
24y(y)

24y/24y cancels out to 1....so all that's left is (y^2 - 2)
------------
y.

2. Don't divide fractions...flip and multiply.
so....12x^2 * 3/4x

Cross cancel with GCF of 4x....3x * 3/1
Final answer 9x/1 or just 9x.

Answer 6

1) [6y(4y)-12] / [8(3y^2)]
(24y-12) / 24y^2
(24)/(24) * y/(y^2) * (-12)/1
1/1 * 1/y * -12/1
-12/y

Answer 7

1) (y-2)/(24-y)

2) 9x



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