Expand $\displaystyle (a+b)^2$ out into $\displaystyle a^2+2ab+b^2$.

Since $\displaystyle \frac{ab}{2}=6$, it can be seen that $\displaystyle 2ab=24$

$\displaystyle (a+b)^2 =34$

$\displaystyle a^2+b^2+24=34$

$\displaystyle a^2+b^2=10$

$\displaystyle 10< 11$ so Quantity B is greater.

To solve this problem, expand each function described by Quantities A and B:

Quantity A: $\displaystyle (a+5)(4a+2)=4a^2+22a+10$

Quantity B: $\displaystyle (4a+5)(a+2)=4a^2+13a+10$

Now note that Quantities A and B only differ in that Quantity A is greater by $\displaystyle 9a$.

Since we are told that $\displaystyle a$ is greater than $\displaystyle 0$ and thus always positive, Quantity A must be greater than Quantity B for all possible values of $\displaystyle a$.

Rather than manually finding common denominators and adding the fractions together, realize that

$\displaystyle \frac{1}{5}=\frac{1}{20}+\frac{1}{20}+\frac{1}{20}+\frac{1}{20}$

Since

$\displaystyle \frac{1}{16}>\frac{1}{17}>\frac{1}{18}>\frac{1}{19}>\frac{1}{20}$

Quantity A must be greater, and this can be seen without actually calculating its value.

Percent growth is given as:

$\displaystyle \left(\frac{Value_{new}-Value_{old}}{Value_{old}}\right)100percent$

For Beetleton, this can be expressed as (in terms of billions of US dollars):

$\displaystyle \left(\frac{14-12}{12}\right)100\%\simeq 17\%$

Let us write down what we are told in mathematical terms, designating the smaller integer as $\displaystyle s$ and the larger integer as $\displaystyle l$.

The sum of the two integers is $\displaystyle 22$:

$\displaystyle s+l=22$

And the larger integer is $\displaystyle 20$% greater than the smaller integer:

$\displaystyle l=1.2s$

Writing the first equation in terms of $\displaystyle s$ gives:

$\displaystyle s+1.2s=22$

$\displaystyle 2.2s=22$

$\displaystyle s=10$

Which allows us to find $\displaystyle l$:

$\displaystyle l=1.2s=1.2(10)=12$

Thus, the positive difference between the two is found as

$\displaystyle l-s=12-10=2$

The key to solving this problem is deciphering the language and translating it into a numerical representation. The first part can be written as an equaltiy as follows:

$\displaystyle 3n-4=4n+5$

Rearranging terms allows us to solve for this mystery number:

$\displaystyle -4-5=4n-3n$

$\displaystyle n=-9$

From there we can address the problem's question:

$\displaystyle 3n-3=3(-9)-3=-27-3=-30$

The definition of an arithmetic mean of a set of values is given as the sum of all the values divided by the total count of values:

$\displaystyle \sum_{i=1}^{n}\frac{x_{i}}{n}$

Where  $\displaystyle x_i$ represents the $\displaystyle i^{th}$ value in a set, and $\displaystyle n$ is the number of values in the set.

Quantity B can thus be defined as follows:

$\displaystyle \frac{(a+3b+2d)+(a-b+2c-48)}{2}$

Which simplifies to:

$\displaystyle \frac{2a+2b+2c+2d-48}{2}$

or, simplifying:

$\displaystyle a+b+c+d-24$

We are told that the mean of $\displaystyle a$, $\displaystyle b$, $\displaystyle c$, and $\displaystyle d$ is 14, which can be written as:

$\displaystyle \frac{a+b+c+d}{4}= 14$ 

and then as 

$\displaystyle a+b+c+d=56$

Plugging this value into our definition of Quantity B, we can find its numerical value:

$\displaystyle 56-24=32$

So 

$\displaystyle Quantity A=32=Quantity B$