S=p/(q+p(1-q))

Question

Solve for p.

Answer 1

S=p/(q+p(1-q))

by multiplying both the numerator and denominator by 1/(1-q)

S=(p*(1/(1-q)))/(1/(1-q))*q+p(1-q)*(1/(1-q))

You can now cancel out common terms to get 

S=(p/(1-q))/(q/(1-q))+(p(1-q))/(1(1-q))

(the common denominator is(1-q)

=(p/(1-q))/(q+p-pq)/(1-q)

S/1=(p/(1-q))*((1-q)/(q+p-pq))     (when you divide by a fraction, multiply by the reciprocal to cancel terms)

(1-q) cancels out in the numerator and the denominator to get

S/1=p/(q+p-pq)

p=s(q+p-pq)    (by cross multiplication)

p=sq+sp-spq(when you distribute the "s" to every term in parenthesis

Now, the p can be isolated by bringing all terms containing "p" to one side of the equation and all the terms without a "p," to the other side of the equation 

so for p=sq+sp-spq

        -sp      -sp

        +spq      +spq

This gives you p-sp+spq=sq (after terms are cancelled out)

pull out the common term for the left side of the equation which is "p"

Then, you will get p(1-s+sq)=sq

Finally, divide both sides by (1-s+sq) to get "p" by itself 

p=(sq)/(1-s+sq)

Answer 2

Hi Kirill;

Thank you for this great question.

We need to isolate p.

 

We begin with...

S=p/(q+p(1-q))

 

Let's spread it out...

S=p/(q+1p-pq)

 

1p=p  therefore...

S=p/(q+p-pq)

 

Let's get rid of the division...

S(q+p-pq)=p

Do you understand?  On the right side, the (q+p-pq) was on the bottom.  When moved to the left, it is on top.  It was originally a dividing factor, now it is a multiplier.

 

Let's merge...

Sq+Sp-Spq=p

 

Let's put all the ps on the right side...

 

Sq=p-Sp+Spq

 

On the left side, Sp was positive.  On the right side, it is negative.

On the left side, Spq was negative.  On the right side, it is positive.

 

Let's continue to isolate p...

Sq=p(1-S+Sq)

 

Let's continue to isolate p...

 

Sq/(1-S+Sq)=p

 

Do you understand?  On the right side (1-S+Sq) is a multiplier.  On the left side, it is a divider.  The issue of positive and negative is a non-issue because it is not being added or subtracted.

 

However, the p must be on the left...

-p=-Sq/(1-S+Sq)

 

Since both these are now negative numbers, we can make these positive.

 

p=Sq(1-S+Sq)

 

I hope this helps.  Pleaselet me know if I can be of further assistance.

 

Answer 3

S=p/(q+p*(1-q)) ⇔ S(q+p*(1-q))=p ⇔ q+p*(1-q)=p/S ⇔ q=p*(q-1+1/S) ⇔ p=q/(q-1+1/S) ⇔

p=qS/(1-S*(1-q))

 

Answer:

 

p=qS/(1-S(1-q))