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irene had a total of 1686 red,blue and green balloons for sale.the ratio of red to blue was 2:3.after irene sold 3/4 blue and 1/2 red.she had 922 balloons left.hiw many blue balloons does irene have at first ?

 $$\small{\text{$
\begin{array}{lrcl}
(1) & r + b + g &=& 1686 \\\\
(2) & \dfrac{r}{b} &=& \dfrac23 \qquad \text{ so } \qquad r = \dfrac23 \cdot b\\\\
\hline
\\
(2) \text{ in } (1): \quad & \dfrac23 \cdot b + b+ g &=& 1686 \\\\
(I) & \dfrac53 \cdot b + g &=& 1686 \\\\
\hline
\\
(3) & \dfrac14 \cdot b + \dfrac12 \cdot r +g &=& 922\\\\
(2) \text{ in } (3): \quad & \dfrac14 \cdot b +\dfrac12 \cdot \dfrac23 \cdot b + g &=& 922 \\\\
& \dfrac14 \cdot b +\dfrac13 \cdot b + g &=& 922 \\\\
& \dfrac3{12} \cdot b +\dfrac4{12} \cdot b + g &=& 922 \\\\
(II) & \dfrac7{12} \cdot b + g &=& 922 \\\\
\hline
\\
(I)-(II): & \dfrac53 \cdot b - \dfrac7{12} \cdot b &=& 1686 - 922 \\\\
& \dfrac53 \cdot b - \dfrac7{12} \cdot b &=& 764 \\\\
& \dfrac{20}{12} \cdot b - \dfrac7{12} \cdot b &=& 764 \\\\
& \dfrac{13}{12} \cdot b &=& 764 \\\\
& b &=& \dfrac{ 764\cdot 12 } {13} \\\\
&\mathbf{ b }& \mathbf{=}& \mathbf{705}
\end{array}
$}}$$

Irene does  have at first 705 blue balloons.

check: 176 ( blue ) + 235 ( red ) + 511 ( green ) = 922

          705 ( blue )  + 470 ( red ) + 511 ( green ) = 1686

          176 / 705 = 1 / 4

          235 / 470 = 1 / 2

          705 * ( 2 / 3 ) = 470

Perhaps I've gone wrong somewhere or misinterpreted part of the question!

Edit:  Of course, I made a numerical mistake towards the end.  I should have had 13b/12 = 764 (giving b = 705.231), not 13b/4 = 764.  Never-the-less, this still doesn't result in integer numbers of balloons!

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Two math students erect a sun "shade" on the beach (similar to a lean too). This "shade" is rectangular in shape with dimensions 1.5 m long and 2 m wide, and it makes an angle of 60° with the ground. When the sun's rays cast down on this "shade" there will be an area of shade made on the ground (also a rectangle). What is the area of shade that the students will have to sit in at 12 noon ? (that is, what is the projection of the shade onto the ground)? (Assume the sun’s rays are shining directly down).

$$\vec{a}=1.5\cdot
\begin{pmatrix} \cos{ (60\ensurement{^{\circ}} ) \\ \sin{ (60\ensurement{^{\circ}} ) }\\ \end{pmatrix}\qquad
\vec{b} = \begin{pmatrix} 0\\2 \end{pmatrix}\\\\
\begin{array}{lll}
\text{area}=\left|\vec{a}_{projection}\times\vec{b} \right|
& \qquad \vec{a}_{projection} = \left(\vec{a}\cdot \vec{e}_x\right)\cdot \vec{e}_x
& \qquad \vec{e}_x = \begin{pmatrix} 1\\0 \end{pmatrix}\\\\
& \qquad \vec{a}_{projection} =\left( 1.5\cdot \begin{pmatrix} \cos{ (60\ensurement{^{\circ}} ) \\ \sin{ (60\ensurement{^{\circ}} ) }\\ \end{pmatrix}\cdot \begin{pmatrix} 1\\0 \end{pmatrix}\right)\cdot \begin{pmatrix} 1\\0 \end{pmatrix} \\\\
& \qquad \vec{a}_{projection} = \left( 1.5\cdot \cos{(60\ensurement{^{\circ}} )} +0\right) \cdot \begin{pmatrix} 1\\0 \end{pmatrix} \\\\
& \qquad \vec{a}_{projection} = \begin{pmatrix} 1.5\cdot \cos{(60\ensurement{^{\circ}} )}\\0 \end{pmatrix} \\\\
\text{area}=\left|\begin{pmatrix} 1.5\cdot \cos{(60\ensurement{^{\circ}} )}\\0 \end{pmatrix}\times\begin{pmatrix} 0\\2 \end{pmatrix} \right| \\\\
\text{area}=2\cdot 1.5\cdot \cos{ (60\ensurement{^{\circ}} ) }
\qquad | \qquad \cos{ (60\ensurement{^{\circ}} ) } = \dfrac{1}{2}\\\\
\text{area}=2\cdot 1.5\cdot \dfrac{1}{2}\\\\
\mathbf{\text{area}=1.5 ~\mathrm{m^2}}\\\\
\end{array}$$

Depends how it is used.  It could just be a separator; it could be part of an absolute value sign ...

Can you give an example of where you have seen it used?

.

In general  (x^a)^b = x^a*b so [(H/J)⁹]¹⁰ = (H/J)⁹⁰

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Use cot^-1(-1.952)  (or acot(-1.952), which is the same thing)

$${acot}{\left(-{\mathtt{1.952}}\right)} = -{\mathtt{27.125\: \!840\: \!246\: \!299^{\circ}}}$$

This is the fourth quadrant solution.  There is also the second quadrant solution obtained by adding 180° to the above result.

.

what is 5 C 4 ?

$$\small{
\boxed{~~ \binom{n}k=\binom{n}{n-k}\qquad \text{ and } \qquad \binom{n}1 = n ~~}\qquad
_5C_4 = \binom54 = \binom51 = 5
}$$

$$\frac{60}{14}\rightarrow\frac{30}{7}\rightarrow4\frac{2}{7}$$

.

Happy birthday, cousin!

Youre the best person in world!!!!!,

thanks formall you did with and for me.

THAAAANKS!!!!!!!

The 6th President of the United States is known as:

John Quincy Adams, son of the 2nd U.S. President, John Adams

Just divide 5 into 9     9/5  =  9 divided by 5  =  1.8      Ta Da!

In a bell curve  (normal distribution)  1 standard deviation + or - from the mean covers 34% (each direction for a total of 68%) .   The second standard deviation covers 13.5% (each direction from the mean for 27%)

   The MEAN will be exactly between 90 and 62    =   76   (the peak of the bell curve)

95%  (34 + 34 + 13.5 +13.5)   covers 2 positive and 2 negative standard deviations or FOUR standard deviations.     90-62 = 28     Each Standard deviation would be 28/4 = 7

Sorry, this is clearer with a picture which I cannot post here, but here is a link: (I think)

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0CAcQjRw&url=http%3A%2F%2Fnationalpainreport.com%2Fliving-well-bell-curve-8821914.html&ei=tdmZVc7EPMXOsAWpnZP4DQ&bvm=bv.96952980,d.aWw&psig=AFQjCNHhdNu6pdDh8HBtaf9XHiVCqvpBvQ&ust=1436232474284197

Let's take it step-by-step......

[3y - 5] / 6 = [2y - 1]/ 3 + 2       multiply through by the common denominator of 6.....so we have

3y - 5   = 2[2y - 1]  + 12   simplify

3y - 5  =  4y - 2 + 12

3y - 5   = 4y  + 10         subtract 3y, 10 from both sides

-15 = y

[2x] / [ x + 3]  = 7     multiply both sides by x + 3

2x =  7 [x + 3]     simplify

2x = 7x + 21        subtract 7x from both sides

-5x  = 21       divide both sides by  -5

x = -21/5

Let x be the amount James had this morning

Then his sister had 2x  and his brother had x + 27

After his sister pays James and his brother 12 dollars each, she has 2x - 24 dollars and his brother has x + 27 + 12   dollars......and his brother has twice as much as his sister....so we have.....

2(2x - 24)  =  x + 27 + 12   simplify

4x - 48 =   x + 39    simplify

3x  = 87       divide both sides by 3

x = 29     this is what James had before his sister pays him $12.....so he has             $[29 + 12]  = $41, now

Check :

James' sister starts with $2(29) =  $58 

And his brother starts with $[ 29 + 27 ]=  $56

After the sister pays both brothers $12......

She has  $[58 - 24 ]  = $ 34   and James's brother has  $[56 + 12 ] =  $68  ......  and his brother has twice as much as his sister......!!!

[ 3x + 12 ] /  [ x + 4 ] =  [4x^2 + 8x + 8x^2] / [ 2x]     simplify and combine terms

3[ x + 4] / [ x + 4 ]  =  [ 12x^2 + 8x] . [2x] 

3   =  4x [ 3x + 2 ] ] [2x]

3  = 2[3x + 2 ]

3  = 6x + 4      subtract 4 from both sides

-1   = 6x        divide by 6 on both sides

-1/6 =  x

18% of 100=18

so

$${\mathtt{100}}{\mathtt{\,-\,}}{\mathtt{18}} = {\mathtt{82}}$$

Overall answer=82

Wow XD

unlike some of you, I take the advantage of the quietness and attack/tackle some questions before RoseThrower and other people, just like Martin Luther before the Holy Roman Emperor and his Catholic nobles at Vorms, or, as some of you better know this as, the Diet Of Worms

i take advantage of it, and it has allowed me to even further reach the leaderboards

(top answerers)

i am so close to reaching 1,000, and ill keep climbing, until i reach the Top Of The World like Imagine Dragons reaching the moon

am i the only U.S. Mate on the forums? I wonder what place I'd be in the Leaderboards if it was just US

i think it would be cool with a update where you can view the leaderboards WorldWide and local nation

7/4/1776-the day the United States Of America signed the Declaration of Independence and broke free from the forces of the British

5/1/1776-the day Adam Weishaupt discovered and found Illuminati (**** the illuminatian triforce)