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D/P =13/5, 
(D-32)/(P+32) =1, solve for D, P
D = 104 pounds - what Dan originally had
P = 40 pounds -what Paul originally had
104 - 32 =72 - what Dan has after giving Paul 32 P.
40 + 32   =72 - what Paul has after receiving 32 P.

Thanks Alan. 

That is interesting. It looked so complicated.  

"Solve the quadratic equation by completing the square.

2x^2+2=3x"

Divide through by 2

x^2 + 1 = 3x/2

Rearrange

x^2 - 3x/2 = -1

Add  (1/2*3/2)^2 to both sides:

x^2 - 3x/2 + (1/2*3/2)^2 = (1/2*3/2)^2 - 1  or  x^2 - 3x/2 +9/16 = -7/16

Equation can now be written as (x - 3/4)^2 = -7/16

Hence   x = 3/4 ± √(-7/16)    or  x = (3 ± i√7)/4

.

As follows:

This just reproduces the odd integers, so a₁₂₈ = 2*128 - 1 → 255

We know that Eli as already traveled 4 km   when Philip starts

And Eli has taken 4/3 hrs to do this

So....let  the time it takes Philip to reach the halway point = T   (in hrs)

So....the distance Philip travels = 7T 

And the distance that Eli travels  = the 4 km he has traveled plus his rate times the time it takes Philip to reach the halfway point =  4 km + 3T

So....the distance that Eli travels = the distance that Philip travels  ...so we have

4 + 3T = 7T

4 = 4T

T = 1  hr

So....Philip travels 7(1) = 7 km

And Eli travels  4 + 3(1)  = 7 km

So...each travels 7km

It takes Eli (4/3)hrs + 1 hr =  7/3 hrs  to travel halfway

And it takes Philip just 1 hour to travel halfway

Here's a pic :

Since MCFN is a rhombus....then FN is parallel to CM

But FN is also parallel to DC

And CE is a transversal cutting parallel segments which makes coresponding angles EFN and ECD equal

And angle FEN = angle CED

So....by AA congruency, triangle NFE is similar to triangle DCE

Then    FN / FE  = CD /CE

And FN = CF  = 8 

And FE = 12

And CE = CF + EF =  20

Therefore

8 /  12 =  CD / 20

2 / 3 = CD /20

CD = 40/3

Therefore DE  =  55 - CE - CD =  55 -  20 - 40/3  =  65/3

Hi Gibson:   

Before I start answering exact questions I want to work out what is happening. - I should have done it just by drawing the displacement graph. It is actually easier that way. I've done it both ways though.  :)

The biker starts 1 mile north of Pidgeon lake.  (the writing says so)

At first the biker is travelling north at 11mph.  

       after 1 hours he is 12km north of Pidgeon lake

At 1 hour he stops and changes direction and heads off south at 10mps 

      so he travels 2 hours south at 10mph that is 20miles south

      he was 12miles noth to start with so now he is 8 south

At the end of the 3rd hour, he stops and changes direction again and heads north at 9mph

After the end of 4 hours (one hour later) he is exhausted and he stops.

    In this last hour he has travelled 9miles north.

    He was 8 miles south in the beginning so now he is 1 mile north.

    So he is back to where he started at the beginning of the 4 hours.

It is actually EASIER to see this if you do the distance vs time graph FIRST.

I'll try and show you how.

It is much easier if you line the graphs up one under the other so that the times are in the same relative position.

You should be able to answer the questions now.

\(p_a + p_b+p_c = 1\\ \dfrac 1 2 + 2p_c + p_c = 1\\ 3p_c = \dfrac 1 2\\ p_c=\dfrac 1 6\\ p_b = \dfrac 1 3\)

\(P[a_3b_2c] = \dbinom{6}{3}\dbinom{3}{2} \left(\dfrac 1 2 \right)^3 \left(\dfrac 1 3 \right)^2\left(\dfrac 1 6 \right) = \dfrac{5}{36}\)

An angle bisector AC divides a trapezoid ABCD into two similar triangles △ABC and △ACD. Find the perimeter of this trapezoid if the leg AB=9 cm and the leg CD=12 cm.

Let AC bisect A

Then angle CAD = angle CAB

And angle ACD = angle BAC

So...triangle ACD is similar to  triangle CAB

So

CD/AB = AD/BC

12/9 = AD/BC

But angle ACD = angle CAB = angle CAD = angle ACB

So AD = CD

And BC = AB

So AD = 12

And BC = 9

So....the perimeter of the trapezoid is AB + BC + CD + AD = 9 + 9 + 12 + 12 =  42

(n - 6) + (n - 4) + (n - 2) + n + (n + 2) + (n + 4) + (n + 6)

7n =  182

n = 26

The smallest number is    26 - 6   =   20

You go into "MY PROFILE" from the top right when you click on your username. From there, you will have your current profile picture on the top left. You can upload a picture from your device there. 

12 and 21 aren't primes

Number of minutes in a week =

60 x 24 x 7   = 

( 5 * 3 * 2^2) * ( 2^3 * 3) * (7)  =   2^5 * 3^2 * 5 * 7

Greatest prime factor =  7

This is known as the pigeonhole principle

There are   (26)(26) =  676  possible sets of initials

Let's suppose one person has the initials  JG

We might have to go through a total of 676 more people to find another person with the same initials

And we might have to go through another set of 676 additional people to find the third person with the same initials

So  ....1 + 676 + 676  =  1353 people will guarantee three peope with the same initials

20 x 0.5 =10 mL
20 x 1     =20 mL
20 + 20   =40 shots
10 + 20   =30 mL
60 mL - 30 mL =30 mL left
30 mL / 2 mL = 15 shots
20 + 20 + 15 =55 shots - to get 60 mL

3^(336) * (9)^(336) * (27)^(336)  =

3^(336) * (3^2)^(336) * (3^3)^(336) =

3^336 * 3^672 * 3^1008

3^(336 + 672 + 1008) =

3^ (2016)

24* 3   =  12 * 6

A multiple of 3, 4 and 5   will be a multiple of 60

So

floor [ 10000/ 60 ]   = 16

(n - 10) + (n - 8) +( n - 6) + (n - 4) +( n - 2 ) +( n ) + (n + 2) +( n + 4) + (n + 6) =  1935

9n - 18 = 1935

9n = 1953

n = 217

The last odd number in the first series = 217 + 6 = 223

So....the next 9 odds are

The first term in the next series  = 225

And the last term = 225 + 8(2) =  241

And the sum of this series =  [ 225 + 241] * (9/20 =    2097

That's too many numbers for me , LOL

Let the Geometry book price = P

The Algebra book = 2P

And the Geometry book = (4/3)Calculus book

The geometry book costs (4/3)*21 = $28

The Algegra book costs 2(28) = $56

So

21 + 28 + 56  =  $105

270 pens   and 7 pencils

Well , to be able to solve those equations and explain it is amazing! Haha your funny lol