What is this the simplest way to solve this problem (and the key way to avoid making mistakes with the algebra) is to simply plug in your own numbers for a, r and y. if we keep it simple, let us say that the loan amount a is 100 dollars, the interest rate r is 0.1, and the length of the loan y is 2 years. now we can find our initial p.

p=
0.5ary+a
12
y

p=
0.5(100)(0.1)(2)+100
12(2)

p=
110
24

p=4.58

now if we leave everything else intact, but double our loan amount (a value), we get:

p=
0.5ary+a
12
y

p=
0.5(200)(0.1)(2)+200
12(2)

p=
220
24

p=9.16

when we doubled our a value, our p value also doubled.