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The speed of the boat upstream =14 - 5 =9 mph
Distance =Speed x Time
35 =9 x T
T =35 / 9
T =3 8/9 - hours - or about 3 hours and 53 1/3 minutes.

Let the sales of Person C = S

The sales of Person B      =1.5S

The sales of Person A      =2 x 1.5S =3S

S + 1.5S + 3S =$220,000

5.5S = $220,000

S =$220,000 / 5.5

S =$40,000 - total earning of salesperson C over a 3-year period.

$40,000 x 1.5 =$60,000 - total earning of salesperson B over a 3-year period.

$60,000 / 3 =$20,000 - average earning of salesperson B per year for 3 years.

Edited: Sorry-Took the amount as $200,000 instead of $220,000

Equations: A=2B, B=0.5C+C. You can also say A=2(0.5C+C), A=C+2C, A=3C

Now, we have A+B+C=$220

Plugging in the values, we have 3C+0.5C+C+C=220

5C+0.5C=220

5.5C=220

C=40

And, B= 0.5(40)+40=20+40=\(\boxed{60}\).

Or, is that a comma after that 220?

We have: \(b=4a, c=5a-17, a=a\).

Since the interior angles in a triangle sum to \(180^\circ\), we have \(4a+5a-17+a=180, 10a-17=180, 10a=197, a=19.7, b=19.7*4=78.8\) and \(c=81.5\). 

Yes, it sums up to what Rom got! Your mistake was summing up 6r^2 - 32r. It is 6r^2 + 32r !!.

Yeah, EP....I finally realized that my first answer was correct....

[ Thanks for your reinforcement  ...LOL!! ]

.....BUT it says that you push 5 buttons AT THE SAME TIME ....at ONCE, so there is no sequencing involved.....SO I think YOUR ANSWER is the correct one.    ~EP

Good.....!!!

Since triangle ABC  is similar to triangle PAQ

Then angle BAC  = angle APQ

But angle BAC  = 70

So angle APQ  = 70

And angle QPC  is supplemental to angle APQ

So angle QPC  = 110

And triangle ABQ  is simialr to triangle QCP

So  angle ABQ = angle QCP  

But angle ABQ  = angle ABC = angle QCP = angle ACB

So...triangle ABC  isosceles  with angles ABC, ACB  being equal

So...their measures  =  [ 180  - m angle BAC ] / 2  =   [180  - 70] / 2  = 110 / 2  = 55

And since m angle ACB  = m angle QCP...then m angle QCP  = 55

So...in triangle QPC......m angle PQC  = [ 180  - m angle QPC - m angle QCP ] =

180  - 110  - 55  = 180 - 165  =   15°  =  m angle PQC

CPhill is right. This is excellent. Not your math, but your amazing skills for shoving 4 variables of bullshitt in a 1 variable bag.

If you were in Texas and wanted to go to Mexico, you first go to Russia, and then to Hong Kong, then swing over to equatorial Africa then to Mexico.  That is the scenic route.  

For the direct rout, just divide 4/3 and multiply by 5. Or divide 5/3 and multiply by 4.  

293_12 + 468_12 =73B_12

B_12 =11_10

What is 73B_a?

Is there a multiply sign between the 3 and the B in which case is both the B and the 73 in base a

OR is B just the last digit of the base a number.

two formulas are of use

\(\sum \limits_{r=1}^n~r^2 = \dfrac{1}{6} n (n+1) (2 n+1) \\ \sum \limits_{r=1}^n~r = \dfrac 1 2 n(n+1) \\ \text{applying these we get, after some algebra}\\ \sum \limits_{r=1}^n~6r^2 + 32r = n (n+1) (2 n+17) \\ \text{plugging }n=1000 \text{ into the rightmost factor we see that }\\ 2(1000)+17 = 2017 \\ \text{and thus the overall sum is a multiple of 2017}\)

Fiona's answer is correct

See the diagram :

Since  in triangle XBY...angle XBY  = 60

And since  BX  and BY  are tangents to the incircle  drawn from B....then  BX  = BY

And the angles opposite these sides are also equal  = [ 180 - angle XBY ] / 2  =

[ 180 - 60 ] / 2  = 120 / 2  = 60

So angle BYX  = angle BXY  = 60

But angle  BYX  is supplemental to angle AYX  =  180 - 60  = 120 (degrees)

Mmmmm...I  see what you mean, guest....but....if " at the same time" is true...then my original answer is correct

We don't have to consider any permutations because all we are worried about  is  choosing any 5 numbers at one time from a set of 6

Since  C(6,5)   = 6      and only one of these sets is correct

So    ..  1/6   

What do you think???

We have the following equation :

[ 36B  +   6B  +  A }  + [ 4*36  +  6  +  B  ] + [ 36A  + 6 + 5 ] =  [ 216A  + 36 + 30 + 2  ]

[ 42B + A ]   + [ 150 + B ] + [ 36A + 11 ] = 216A + 68

43B + 37A   + 161  = 216A + 68

43B + 93  = 179A

This equation is true when B  = 2  and A  = 1

And the absolute value  of their difference is  l A  - B  l  = l 1  - 2  l  =  l -1  l  =  1

If that is true, then instead of having a fucked up lock, we have a fucked up question!

The question says “The lock is opened by pushing down 5 buttons at the same time.”

It doesn’t say “The lock is opened by pushing down 5 buttons in a sequence.”

Buttons that are pressed at the same time do not have a sequence or a permutation. Buttons that are pressed at the same time can only have combinations. 

B=2 A=1

Look at hundred place

B+4+A <=14 for 0<=A,B<=5

So, A =0 or 1 

Look at the ones place

Even B=1, the tens place is increment by one 

So let's assume B>=1 ,then we ahve 1+B+1+1=5 \(\Rightarrow\)B=2

Then look at the left parts, 2+4+A=6*A+1\(\Rightarrow\)A=1

Verify it ,it works.

Assume Circle \(\Gamma \) is center at point  \(\Gamma \). Connect \(\Gamma \) to point X and Y

Angle \(\Gamma \)XB = Angle \(\Gamma \)YB =90 degrees.

Angle X\(\Gamma \)Y= 360 - 60 - 90 - 90 =120 degree

And segment \(\Gamma \)X = segment \(\Gamma \)Y \(\Rightarrow \) angle \(\Gamma \)XY =  angel \(\Gamma \)YX =1/2 *(180 degree - angel X\(\Gamma \)Y)=1/2(180-120）=30 degrees.

Angel AYX= angel AY\(\Gamma \) +angle \(\Gamma \)YX=90 degrees + 30 degrees= 120 degrees.

edited: to understand my answer, you need know the properties of inscrided circle(incircle) and angles. I urge you to draw picture to better understand my answer. You should check my answer ,because it might not be correct.

Thanks, fiora  !!!

BTW....good to see you  !!!

Excellent, Knockout  !!!!

BTW....welcome aboard    !!!!

( 1 + i)^2  =  1 + 2i  + i^2  =  1  + 2i  - 1  = 2i

(1 - i)^2  =  1 - 2i  + i^2  =  1  - 2i  - 1  =  -2i

So

( 1 + i)^2016  - (1 - i)^2016  =

[ (1 + i)^2 ] ^1008  -  [ (1 - i)^2 ]^1008  =

(2i)^1008  - (-2i)^1008  -

(2i)^1008  -  (-2)^1008 * (i)^1008  =           [ (-2)^even power  = 2^even power]

(2i)^1008  -  (2)^1008 * i^1008  =

(2i)^1008  -  (2i)^1008  =

0

CPhill last answer is right.

The general formula for permutation is \(nPr=\frac{n!}{(n-r)!}\)

In this case \({6}^{P}{5}=\frac{6!}{(6-5)!}=720\).possible combination,that is Jocab has 1/720 chance to open the lock at one time.

A rule of thumb to dealing with permutaion and combination is 

permutation is nPr read as "n pick r", and order does matter.

combination is nCr read as "n choose r", and order does not matter.

See the following online etextbook of probability chapter 3 and 5 for more informations about permutation and combination: