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Actually, your idea also is pretty neat! I learnt something from it :D!

Thanks guest, you have a much smarter way of algebraically solving it --- I just plugged in random values of x and hoped for the best...

I got it! The answer is 9!

How I did it:

First I plugged in x as 0, f(0) = 0 / -1 = 0

(0p + q)/(0r + 1) = f(5 * 0) = 0 => q/1 = 0 => q = 0

Next I plugged in x as -1, f(-1) = 1/2

(p/2 + 0)/(r/2 + 1) = f(-5) = 5/6

p/(r + 2) = 5/6

6p = 5r + 10  (1)

Next I plugged in x as -2, f(-2) = 2/3

(2p/3)/(2r/3 + 1) = f(-10) = 10/11

2p/(2r + 3) = 10/11

22p = 20r + 30

11p = 10r + 15   (2)

Take equation (1) and multiply by 2 => 12p = 10r + 20 (1b)

Take equation 1b - equation 2 => p = 5

Plug it back into equation 1 => 30 = 5r + 10 => r = 4

p + q + r = 5 + 0 + 4 = 9

Hi proyaop!

\(f(x)=\dfrac{x}{x-1}\)

So: \(f(5x)=\dfrac{5x}{5x-1}\)

Now equate the given with this:
That is, 

\(\dfrac{5x}{5x-1}=\dfrac{pf(x)+q}{rf(x)+1}\)

But, we know \(f(x)\):

\(\dfrac{5x}{5x-1}=\dfrac{p(\dfrac{x}{x-1})+q}{r(\dfrac{x}{x-1})+1}\)

Multiply the right with (x-1)/(x-1):

\(\dfrac{5x}{5x-1}=\dfrac{p*x+q(x-1)}{r*x+(x-1)}\)

Expand and equate.

\(\dfrac{5x}{5x-1}=\dfrac{(p+q)x-q}{(r+1)x-1}\)

Well by comparing we get:

 \(p+q=5 \\ q=0 \\ r+1=5 \\\)

Hence, \(p=5,q=0,r=4\)

So, \(p+q+r=5+0+4=9\)

I hope this helps!

You can't have 0 reds.

You can have 1 red, and there are 2 scenarios for that.

You can have 2 reds, and there are 8 scenarios for that.

You can't have 3+ reds.

In total, there are 10 scenarios that work. There are 3^5 - 3 total cases, so the probability is 10 / 240 = 1/24

1) [already known number] = y

If GCD (y, x) = y, then x is divisible by y. 

2) 

If LCM (y, x) = y, then x is a factor of y.

If you are asking for an integer number --- no... Sextillion has 21 zeroes.

The square root of 9.5e+21 = 97467943448.1

This is relatively the same problem as posted above --- read the one above this to see how to graph:

Points: (5, 65) and (8, 99) --- also if you have a typo you can edit the problem (I know this is probably your typo-ed version)

Slope = (99-65)/(8-5) = 34/3 = 11.3 (infinite trailing 3s).

First draw a graph, label the x-axis as "amount of uniforms ordered", and the y-axis as "amount of dollars"

Then you can plot the points (5, 66) and (8, 99), and make sure you make the y-axis 0, 33, 66, 99 so the graph isn't too tall.

Slope = (change in y)/(change in x) => (99 - 66)/(8-5) = 33/3 = 11.

If a line is parallel to another line, they must have the same slope:

y = -x/4 + b is the equation of the line.

Plug in -4, -7 as (x, y)

-7 = 4/4 + b

-7 = b + 1

b = -8

Equation of the line: y = -x/4 - 8

4) If angle DCB is x, then angle BDC is also x.

Angle DBC = 180 - 2x

Angle DBE = x - (180 - 2x) = 3x - 180 degrees

Angle EDB = 3x - 180

Angle BED = 180 - 2(3x - 180) = 180 - 6x + 360 = 540 - 6x degrees

Angle AED = 180 - (540 - 6x) = 6x - 360 degrees

Angle A = 6x - 360 degrees.

Angle B + Angle C + Angle A = 180 

x + x + 6x - 360 = 180

8x = 540

Angle BAC = 67.5 degrees

For number 1 the question is: The region between the graph of y=f(x) and the x-axis, shaded in this figure, has an area of 10 square units. What is the area between the graph of y=f(x+2) and the x-axis?

And yes the negative sign is in the correct place for number 2

it is 2 

2 times 9 equals 18 

and 2 times 10 equals 20

Z = cis(2pi/3).

We have a calculator...

31f(6) = -119

f(6) = -119/31

119 + 31 = 150

Thanks melody!!!

Here's an algebraic routine that gets you to the complete solution.

Let x = u + v and y = u - v, so that u = (x + y)/2 and v = (x - y)/2.

Then,

\(\displaystyle x^{3} - y^{3} = (u + v)^{3} - (u - v)^{3} \\ = u^{3}+3u^{2}v+3uv^{2}+v^{3}-(u^{3}-3u^{2}v+3uv^{2}-v^{3})\\ =6u^{2}v +2v^{2}.\)

\(\displaystyle u^{2} = (x+y)^{2}/4=100/4=25,\)

so

\(\displaystyle 6.25.v+2v^{3}=504,\\ v^{3}+75v-252 = 0.\)

That has \(v=3\)

as an obvious solution from which

\(\displaystyle (v-3)(v^{2} +3v+84)=0.\)

So,

 \(v=(x-y)/2=3,\\ x-y=6.\)

The other two possibles are complex.

Suppose that the equation of the tangent is y = ax+b.

Its slope, (and therefore the slope of the curve), will be a.

The line y = ax + b intersects the curve y = x^2 - 5x + 4 at points with x coordinate(s) given by

x^2 - 5x + 4 = ax + b,

or 

x^2 - ( 5 + a)x  + (4 - b) = 0. 

If the line is a tangent to the curve then the single point will be x = (5 + a)/2, (that's from the formula for the solution of a quadratic with the b^2 - 4ac part equal to zero).

From that, a = 2x - 5, and that's the slope.

Oh! Thank you!

P.E. = mgh

P.E. = 6 x  9.8 x 100 ==5,880 Joules

Thank you for translating it !

The smallest positive integer ==73

1 / 73 = 0.01369863  01369863  01369863  01369863.........and so on.

I will simplify if for you

What is the smallest positive integer n such that 1/n is equal to a decimal number with an 8-digit repeating block?

1 - The smallest positive integer n ==10

10! =3,628,800

(10 + 1)! =11! ==39,916,800

2 - There is a mistake in this question. Check your numbers carefully.

3 - N_100 =2^100 =1,267,650,600,228,229,401,496,703,205,376

N_k =3 - Number of monsters

N_k  *  N_100 =3 x  1,267,650,600,228,229,401,496,703,205,376=3,802,951,800,684,688,204,490,109,616,128

Let the number of pounds of peanuts=P
Then, the number of pounds of cashews=5 - P

1.7P + 4.55(5 - P)==5 * 2.80
1.7P +  22.75 - 4.55P==14
1.7P - 4.55P ==14 - 22.75== - 8.75
- 2.85P == - 8.75
P == - 8.75 / - 2.85

P==~3.07 pounds of peanuts
5 - 3.07 ==~1.93 pounds of cashews

Sorry, don't understand your question with all those slashes and $ signs !