How to simplify complex fractions?

Question
Note: I had a difficulty understanding the equation. Below is my understanding of the equation.






(m + 2)* (m - 2)
--------------------
(2m + 4) * (m^2 - 4)






Solution
First, breakdown the equation to its simplest form. Hence

(m + 2)* (m - 2) (m + 2)* (m - 2)
-------------------- = ------------------------------
(2m + 4) * (m^2 - 4) 2 * (m + 2) * (m -2) * (m-2)

Now, you can cancel like numbers. Meaning if a number is in the numerator AND denominator, then they cancel each other. Hence, (m + 2) and (m -2) are canceled in the numerator and denominator.

Therefore,
1
------------------
2* (m-2)

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How to solve for and find a missing fractions?

Question
If a certain fraction is multiplied by 4 and the result subtracted from twice the reciprocal of the original fraction there remains 1/2 of the original fraction. find the original fraction.




Solution
Let x = missing fractions
Then, the above word problem can be states as 2*(1/x) - 4x = (1/2)*x
Solve for x by isolating all variables with x.
Hence, 2/x - 4x - x/2 = 0.
Simplify 2/x - 4x - x/2.
(2/x - 4x - x/2)*2x = 0
(4 - 8x^2 - x^2)/2x = 0
(4 - 9x^2) = 0
(-3x + 2) * (3x +2) = 0
Then, look for x
(-3x + 2)= 0 =====> -3x = -2 =====> x = 2/3
(3x +2) = 0 =====> 3x = -2 =====> x = -2/3

Now, test the validity of the two solutions above by substituting them in the formula 2*(1/x) - 4x = (1/2)*x.

1. x = 2/3 ======> 2*(3/2) - 4*(2/3) = (1/2)*(2/3) ====> 2/6 = 2/6
2. x = -2/3 ======> 2*(-3/2) - 4*(-2/3) = (1/2)*(-2/3) ====> -2/6 = -2/6

Now, there is actually only ONE answer the above problem. The answer is 2/3 BECAUSE fractions are usually POSITIVE.

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The Easy Guide: How to write fractions

Question
In Mark's collection of antique bottles, 4/9 of the bottles are dark green. Write three equivalent fractions for 4/9.




Solution
First, we need to write 4/9 in three different forms.
Second, multiple the fraction with 1 or its equivalent, such as 1/1, 2/2, 3/3, 4/4 and so on, to arrive at three different fractions which, when simplified, are still equal to 4/9.

1. 4/9 * 2/2 = 4*2/9*2 = 8/18
2. 4/9 * 3/3 = 4*3/9*3 = 12/27
3. 4/9 * 4/4 = 4*4/9*4 = 16/36

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