Begin by finding the reciprocal of , the reciprocal of , and the reciprocal of :

, , 

In order to add the four terms together, we will need to find a common denominator. Find the common denominator by multiplying the three different terms in the denominator (x, y, and z). Our common denominator will be . Put each term into terms of this common denominator:

Now, add each term together:

Look for the answer choice that matches this.

To find the reciprocal of a rational expression--any rational expression--all you need to do is flip the numerator and the denominator.  

Therefore, the correct answer is:

You can check this by multiplying the two expressions together and ensuring that your answer is .

To find the reciprocal of any fraction, you merely need to invert the numerator and denominator.  

In this case, first you have to convert the mixed number into a fraction:

Therefore, the answer is:

To find the least common denominator of two numbers, it is easiest first to factor them into prime factors:

Now, you need to compare each number and choose the case in which the prime factor has the highest power.  Therefore, since  and  are found only in , you will select those.  You can take "either" .  Finally, the  in  is the largest factor of .  Your LCD is found by multiplying all of these together:

To begin to solve this, you need to find the least common denominator of  and .  The easiest way to do this is to begin by factoring them into prime factors:

The LCD is found by selecting the largest power for each factor across the two values.  Therefore, you will take ,, and  from  and the  from the .  Your LCD is therefore:

.

Now, apply this to your fractions:

First, find the smallest number that both  and  will factor into, which is .

This means that the fist fraction should be multiplied by  and the second should be multiplied by .

Therefore,

 .

Then, add the numerators while the denomenators stay the same: 

.

Then, reduce the fraction to its lowest terms: 

.

If  is even, then  is not in lowest terms, since both  and 4 are divisible by 2. Therefore, the only possibilities are 27,29,31,33, and 35. But , , and , so only two integers satisfy the given condition: 29 and 31.

In order to find out the probability of choosing a boy, you must first find out how many boys there are. Since there are  girls out of  students,  students should be boys.  

Therefore, the probability of choosing a boy is,

 .

 However, this is not one of the answer choices; therefore, you must reduce the fraction.  

In order to reduce a fraction, you have to find their GCM, or greatest common multiple. This is the biggest number that will go into both the numerator  and denominator .  The largest number is . Divide both the top and bottom by , and you will get the answer:

To simplify the fraction

It may be helpful to write it in terms of scientific notation:

To simplify the fraction

Begin by factoring out common terms in the numerator and denominator:

Now notice that the coefficients for the polynomials in the numerator and denominator match the form of Pascal's Triangle:

The fraction can be rewritten as:

And finally reduced to