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

Recall that the exponent of the denominator is sutracted from the exponent of the numerator. 

Therefore Quantity A is equivalent to x^5–y, and because we would then be comparing an arithmetic operation to a geometric operation, it does not matter whether y is a multiple of 5.  The two quantities cannot be compared.

To begin, notice that using your calculator, you can find:

Now, the closest even multiple of  that is less than  is .  Therefore, you know that your number is:

This is the same as:

, or simply, .  This is your mixed fraction.

Although there are many ways to convert improper fractions into mixed fractions, the easiest way is to use your calculator to your advantage.  Begin by dividing  by .  This gives you . Therefore, you can eliminate all the options that have do not have  for their first portion. Next, multiply  by the denominator (), and get .  This means that you have  and , or .  Thus, your answer is .

Though there are several ways you could solve this, let's convert the improper fraction into a mixed one so we can compare them. Start by dividing  by .  This gives you  

Now, since we know that the two numbers have the same whole-number value, we need to compare their decimal portions. Compare  to . The latter is indeed   Therefore, the two values are equal.

To find the improper fraction value, we must effectively add together 71 and ⁵/₇. To do this, we will give 71 a denominator of 7; therefore, we are transforming ⁷¹/₁ to ^x/₇.  The shortest way to do this is to multiply by ⁷/₇ (which really is 1); therefore, 71 = 71 * (⁷/₇) = ⁴⁹⁷/₇.

Now add them: ^{(497 + 5)}/₇ = ⁵⁰²/₇