eng math F.1

One important formula is:
Cost * (1 +/- profit/lose %) = selling price

if no cost price and just have lose% and the selling price
how can I find the lose?

Let C be the cost, r be the lose % and S be the selling price.
　C * (1-r) = S
　C = S/(1-r)
The lose is C - S
　　　　　= S/(1-r) - S
　　　　　= S [1/(1-r) - 1]
　　　　　= S [(1-1+r)/(1-r)]
　　　　　= rS/(1-r)

A is sold at a profit of 20%.
if the cost price is decrease by4%
and the selling price is still the same ,
then the owner will gain how many percent more?

Let C be the original cost, and S be the selling price.
　C * (1+20%) = S
　1.2C = S　--- [1]
Let D be the new cost, and r be the new profit %.
　D = C*(1-4%)
　　= 0.96C　--- [2]

　D * (1+r) = S　--- [3]

Put [1] & [2] into [3]
　0.96C * (1+r) = 1.2C
　　0.96 + 0.96r = 1.2
　　　　　　　r = 0.25 = 25%

A sells his goods to B at a profit of 25%.
B sells it to C at a loss f 25%.
if C buy it at75,find the difference of A's cost price and C's cost price.

Let X be the cost of the goods.

As A sold the goods to B at a profit of 25%,
the price that A sold his goods to B = X (1+25%) = 1.25X

Then B sold it to C at a loss of 25%,
the price that B sold his goods to C = 1.25X (1-25%) = 0.9375X

Since C but it at 75, therefore　0.9375X = 75
　　　　　　　　　　　　　　　　　X = 80

The difference of A's cost price and C's cost price = 80 - 75 = $5

Let X be the A's cost price
Then (X*1.25)*0.75=75
......... X*1.25=100
......... X=80

The difference of A's cost price and C's cost price is $5