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Here’s the correct presentation. Previously, I transposed the number and modulus divisor.

An excess consumption of fermented bananas might be a major contributing factor in the cause.  I’ve left the previous presentation as a monument to my error (The math is fine). This is an extremely rare event: this is only the second time I’ve made a mistake.  The first time was when I thought I made a mistake, but I didn’t, so I was wrong in thinking I did. 

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\(\begin{array}{rcll} n &\equiv& {\color{red}231} \pmod {{\color{green}3331}} \\ n &\equiv& {\color{red}1247} \pmod {{\color{green}1361}} \\ \text{Set } m &=& 3331\cdot 1361 = 4533491\\ \\ \end{array} \)

\(\begin{array}{rcll} n &=& {\color{red}231} \cdot {\color{green}1361} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ {\color{green}1361}^{\varphi({\color{green}3331})-1} \pmod {{\color{green}3331}} ] }_{=\text{modulo inverse 1361 mod 3331} } }_{=1361^{3330-1} \mod {3331} }}_{=1361^{3329} \mod {3331}}}_{=1980} + {\color{red}1247} \cdot {\color{green}3331} \cdot \underbrace{ \underbrace{ \underbrace{ \underbrace{ [ {\color{green}3331}^{\varphi({\color{green}1361})-1} \pmod {{\color{green}1361}} ] }_{=\text{modulo inverse 3331 mod 1361} } }_{=3331^{1360-1} \mod {1361} }}_{=3331^{1359} \mod {1361}}}_{=552}\\\\ n &=& {\color{red}231} \cdot {\color{green}1361} \cdot [ 1980] + {\color{red}1247} \cdot {\color{green}3331} \cdot [552] \\ n &=& 622494180+ 2292873864\\ n &=& 2915368044\\\\ && n\pmod {m}\\ &=& 2915368044\pmod {4533491} \\ &=& 333331\\\\ n &=& 333331+ k\cdot 4533491\qquad k \in Z\\\\ \mathbf{n_{min}} & \mathbf{=}& \mathbf{333331} \end{array} \)

:)  

The pie store is having a 20% off sale on all of its pies.

If the pie you want regularly costs $18, how much would you save with the discount?

You'd save  20%  of  $18 .

20%  of  $18   =   0.2 * $18   =   $3.60

This is the formula you would use to calculate the FV of her deposits:

FV=P{[1 + R]^N - 1/ R}

FV=250 x {[1 + 0.0185/12]^(35*12) - 1 / (0.0185/12)}

FV=250 x {[1.0015416667]^420 - 1 / (0.0015416667)}

FV=250 x {[1.9098054874 - 1] / (0.001541667)}

FV=250 x {[0.9098054874 / 0.001541667]}

FV=250 x               590.144099920........... 

FV=$    147,536.02 - The amount of Sarah's retirement fund at 65. 

Her total deposits for the same period would be:

$250 x 420 months =$105,000. So, the difference would be:

$147,536.02  -  $105,000 =$42,536.02 interest earned @ 1.85% over 35 years.

Its Net Present Value =$77,251.86.

$147,536.02  -  $77,251.86 =$70,284.16 - Difference between FV and NPV.

It would be very very useful if you provide a graph. Could you provide a graph please, sir?

Loc mows (1/20) of the lawn in one minute and Reza mows 1/30 of the lawn in one minute....so

in one minute they both mow   (1/20 + 1/30)  =  50/600  =  1/12 of the lawn

Loc mows (5/20)  = 1/4 of the lawn in 5 minutes....so there is 3/4 left to mow

So

Fraction left to mow  / Fraction mowed per minute  = additional time required

(3/4) / (1/12)  =  36 / 4   =   9 min more......so....the lawn is finished at 1:14 

This triangle is a multiple of a 5-12-13 triangle

Its sides are    25 - 60 - 65

Its area is  (1/2) * product of the legs  =  (1/2)(25)(60)  =  750 units^2

Since ABC is isosceles, its altitude will  bisect BC.....  and can be found as :  

sqrt ( 50^2 - 14^2)  =  sqrt (2304) = 48

So......call the intersection point of the altitude  and BC, E

So  ED  =

sqrt (52^2 - 48^2 )  = sqrt (400)  = 20

And    CD  =  ED - (1/2)BC   =   20 - 14   =   6 units 

Thanks very much!

Since Loc can mow the lawn in 20 minutes, his rate is 1 lawn / 20 minutes or 1/20^th of the lawn per minute.

Since Reza can mow the lawn in 30 minutes, his rate is 1 lawn / 30 minutes or 1/30^th of the lawn per minute.

Loc mows for 5 minutes before he his joined by Reza. In these 5 minutes, he can mow (1/20)·(5) = 5/20^th = 1/4^th of the lawn.

Now, there is only 3/4^th of the lawn to finish.

Assuming that the time they mow together is x, we can create this equation: 

     Amount done by Loc  +  Amount done by Reza  =  Total Amount

              (1/20)(x)              +         (1/30)(x)               =          3/4

Multiplying both sides by 60:

           60(1/20)(x)     +     60(1/30)(x)     =     60(3/4)

                  3x            +             2x           =         45

                                                 5x           =         45

                                                   x           =          9          [It will take them 9 minutes, working together, to finish the lawn.]

Adding 9 to the 5 minutes that Loc mows alone   =  14 minutes after Loc starts     --->     1:14 pm

You can integrate them separately thus:

integral_(-1)^1 x^z dx = ((-1)^z + 1)/(z + 1) for Re(z)>-1

integral_(-1)^1 y^z dy = ((-1)^z + 1)/(z + 1) for Re(z)>-1

gene3141's answer is correct in the following sense:

Since Loc has already cut 1/4 of the lawn, then there is 3/4 left to cut.

Since Reza takes 30 minutes to cut the whole lawn, it, therefore, will take him: 30 x 3/4=22.5 minutes to cut the remaining 3/4 of the lawn.

1/15 + 1/22.5 =1/9. Take the reciprocal of this:

=9 minutes for both working together to finish the lawn

5 +  9 = 14 minutes after 1 pm, when they will finish

Or 1 + 14 =1:14 pm

Makes sense...

11 2/3   =   35/3   =   280/24

7 7/8   =   63/8   =   189/24

The GCF of  280  and  189  is  7 , so....

We can divide the side that is  280/24 m long into  40  equal parts, each  7/24 m long.

We can divide the side that is  189/24 m long into  27  equal parts, each  7/24 m  long.

So the maximum size of a square tile that will fit the room exactly is  7/24 m  by  7/24 m  .

For the lines, x = a and x = -a, with a > 0:  the graph will be a pair of vertical lines, one passing through the point (a,0) and the other passing through the point (-a,0). There will be no 'solution' beccause there is no point of intersection.

For the lines, y = b and y = -b, with b > 0:  the graph will be a pair of horizontal lines, one passing through the point (0,b) and the other passing through the point (0, -b). There will be no 'solution' because there is no point of intersection.

acetone n = 1.36 v = c/n v = 3.0 x 10^8 / 1.36 = 2.2 x 10^8 m/s. So light travels 2.2 x 10^8 m/s in acetone. Source: http://hudeckijrsci.weebly.com/uploads/3/7/0/2/3702339/chapter_12_homework_answers.pdf

An explanation, along with a calculator, can be found at http://scienceprimer.com/snells-law-refraction-calculator

A line segment starting at point A(2, -2) extends through point B(14, 4) to point C.

If the length from B to C is 1/3^rd the length from A to B, what are the coordinates of point C?

The x-distance from A to B is 12 because 14 - 2 = 12.

The y-distance from A to B is 6 because 4 - -2 = 6.

Since 1/3^rd of 12 is 4, the x-value of point C must be 4 more than the x-value of point B  --->   14 + 4 = 18.

Since 1/3^rd of 6 is 2, the y-value of point C must be 2 more than the y-value of point B  --->   4 + 2 = 6.

Therefore, the coordinates of point C are (18, 6). 

Uh...that's why it's "off-topic."

Some interesting coding by GingerAle and Heureka:  Thanks to both of you :)

It is from this question:

The time it took for you to type in all those numbers deserves a thumbs up :)

Let  s  be the number of grams of salt in the solution.

We want....

number of grams of salt / number of grams of solution   =   10 / 100

s / (810 + s)   =   10 / 100

                                               Multiply both sides of the equation by  810 + s .

s   =   (810 + s) * 10 / 100

                                               Distribute the  10/100 .

s   =   81 + 0.1s

                                               Subtract  0.1s  from both sides.

0.9s   =   81

                                               Divide both sides by  0.9 .

s   =   90

So Ilya needs to add  90  grams of salt.

That makes  90 + 810  =  900 grams of solution.