Can someone explain this equation?

Question

p/(1+p)=r
p=r(p+1)
p=rp+r
p-rp=r
p(1-r)=r
p=r/(1-r)

Answer 1

Actually, that's a series of equivalences. There's really only one equation. You've multiplied both sides by the same value, and used the distributive law both of which change the form but not the meaning or solutions of the equation.


As a simple example, consider

x=2
and
2x=4

Those are equivalent equations.

There is something to keep in mind. Anytime you use division in manipulating an equation, you must make sure that you're not dividing by zero.

If I say
x^2=x
then
x^2-x=0
x(x-1)=0
solutions x=0, or 1

but if I'm not careful:
x^2=x
x=1
see, in dividing by x I've implicitly assumed that x cannot equal zero and have 'lost' a solution.

Answer 2

It's simply walking through the steps to go form the first equation (r in terms of p) to the last equation (p in terms of r).

p/(1+p)=r ... multiply both sides by (1+p)

p=r(1+p) ... multiple r through on the right side

p=r+rp ... subtract rp from both sides

p-rp=r ... divide p out on the left side

p(1-r)=r ... divide both sides by (1-r)

p=r/(1-r)

Answer 3

OK, the first step was multiplying both sides by (1+p). On the left, that cancelled out the division by (1+p), leaving p. On the right, it gave r(1+p).

The second step was multiplying out the parentheses. You multiply the number or variable outside the parentheses (r) by everything inside the parentheses (1+p), giving rp + r. The left side is unchanged.

The third step was subtracting rp from each side. On the right side that cancelled out the addition of rp, leaving r. On the left it gave p - rp.

The fourth step was factoring out the p on the left side. Both the terms there (p and rp) have p in them, so you divide them both by p. Put the two results in parentheses and put the p outside the parentheses. It's just the opposite of the second step (multiplying out parentheses).

The fifth step was dividing both sides by (1-r). On the left, this cancelled out the multiplication by (1-r), leaving p. On the right, it gave r/(1-r).

The overall effect of this transformation was to go from a system in which r was defined in terms of p, to one in which p was defined in terms of r.

Answer 4

What's there to explain? All the steps are fine.

By the way, if you set r=1/p, your steps show off some neat properties of the Golden Ratio = 1.618... for example
p=rp+r ---> p=1 + 1/p, i.e. the Golden Ratio is equal to one + its inverse.

Answer 5

no count what proportion motorcycles you have, you alway choose one greater ! (n+a million), yet you're able to provide up procuring at one under the variety which will carry approximately a divorce (s-a million). besides the fact that some might say the divorce shouldn't are available in the way of procuring for greater motorcycles !

Answer 6

What is there to explain? It works.