Detailed Solution

\(\mathrm{Line\:up\:the\:numbers}\)

\(\begin{matrix}\:\:&2&7\\ +&1&4\end{matrix}\)

\(\mathrm{Add\:each\:column\:of\:digits,\:starting\:on\:the\:right\:and\:working\:left.}\)

\(If\:the\:sum\:of\:a\:column\:is\:more\:than\:ten,\:'carry'\:digits\:to\:the\:next\:column\:on\:the\:left.\)

\(\mathrm{Add\:the\:digits\:of\:the\:bolded\:column}:\quad \:7+4=11\)

\(\begin{matrix}\:\:&2&\textbf{7}\\ +&1&\textbf{4}\end{matrix}\)

\(\mathrm{Carry\:}1\mathrm{\:to\:the\:column\:on\:the\:left\:and\:write\:}1\mathrm{\:in\:the\:bolded\:column}\)

\(\frac{\begin{matrix}\:\:&1&\textbf{\:\:}\\ \:\:&2&\textbf{7}\\ +&1&\textbf{4}\end{matrix}}{\begin{matrix}\:\:&\:\:&\textbf{1}\end{matrix}}\)

\(\mathrm{Add\:the\:digits\:of\:the\:bolded\:column}:\quad \:1+2+1=4\)

\(\frac{\begin{matrix}\:\:&\textbf{1}&\:\:\\ \:\:&\textbf{2}&7\\ +&\textbf{1}&4\end{matrix}}{\begin{matrix}\:\:&\textbf{4}&1\end{matrix}}\)

= 41

what is a fraction that has a quotient 3 1/5 

     16    

   ––––    that'll do it ... usually it wouldn't be called a quotient; it'd be called a reduction

      5  

Let the number of problems in November ==N

Let the number of problems in March ==M

1/3N  +  2/7M ==15........................(1)

N  +  M == 28.................................(2), solve for N, M

N==147     and    M== - 119  !!!!!!  Do these numbers make sense to you ??

I think there is a major mistake in your question. Check it carefully.

You don't know what a "quotient" is,  do you ??!

8 / 4 ==2  -   8 is called the "DIVIDEND". 4 is called the "DIVISOR". 2 is called the "QUOTIENT"

Copy the above and memorize it.

x == 2y

y ==1/2x

[x  +  ( 6x - 5*1/2x) +  (5x  +  3*1/2x) ]==180, solve for x

11x == 180

x=180 / 11 == 16.364 degrees

1 yard == 3 feet

6 yards  x  3 feet ==18 feet

$16.38 / 18 feet ==$0.91 - price per foot.

[7  x  80]  -  [70 + 72 +  89 + 92 + 89 + 80]==?

I will let you get the answer.

Well, how many square feet is the size of her garden ?!

Consider

\(\displaystyle \left(\frac{x^{2}y}{z}\right)^{m}.\left(\frac{y^{\,4}z}{x}\right)^{n}=24^{m}.30^{n}\)

Simplify the lhs and determine the values of m and n so that it is identical with \(\displaystyle x^{8}/(yz)^{5}.\)

Substitute m and n into the rhs.

\(\)

LCM[25, 40] ==200 minutes after 7:00 am

200 / 60 ==3 + 1/3 hours after 7:00 am

7:00 + 3 1/3 ==10:20 am - when both buses will depart the dpot at the same time.

Wow, the class has half a student!

Maybe the class has a halfwit ...or a midget. 

GA 

--. .- 

8  x  11 ==88 pounds - total weight of the 8 bowling balls.

11 + 12 + 7 + 11 + 9 + 7 + 11==68 pounds - total weight of first 7 bowling balls

88  -  68 ==20 pounds - the weight of the 8th and last bowling ball !!

This ratio is part:part, meaning that its 3 parts to 2 parts.

12/3=4

2*4=8

12:8 is similar to 3:2

The width is 8.

Yay!

\(3x+\frac{2}{x}-4=2(x+9)-(7-\frac{1}{x})\)

\(3x+\frac{2}{x}-4=2x + 18-7+\frac{1}{x}\)

\(3x + {1 \over x} - 4 = 2x + 11\)

\(x + {1 \over x} - 4 = 11\)

\(x + {1 \over x} = \color{brown}\boxed{15}\)

Note that \(x^2 + y^2 = (x+y)^2 - 2xy\)

So, \(17^2 - 2xy = 145\)

Now, solve for xy.

3x+2/x-4=2x+18-7+1/x

3x+2/x-4=2x+1/x+11

3x+2/x+7=2x+1/x

3x+1/x+7=2x

[(3x+1/x+8)/2]/(3x+1/x+7)/2

This simplifies down to 

(3x^2+8x+1)/(3x^2+7x+1)

Yay!

...41

(I checked my work 3 times because if I mess this one up I will never live this down)

Yay?

Just multiply the two numbers together:

8.834 moles  x   [6.022  x  10^23] ==5.32  x  10^24 atoms

$40,000

\(\lim _{\:h\rightarrow 0\:}\frac{\left(\sqrt{6+h}-7\right)-\left(\sqrt{6}-7\right)}{h}\)

\(=\lim _{h\to \:0}\left(\frac{\sqrt{h+6}-\sqrt{6}}{h}\right)\)

\(=\lim _{h\to \:0}\left(\frac{1}{\sqrt{h+6}+\sqrt{6}}\right)\)

\(=\frac{1}{2\sqrt{6}}\) = 0.20412