Could someone check my answer (compound interest - math)

The question is: What annual interest rate, compounded monthly, would be necessary in order for $5000 to be worth $6000 in 4 years?

So, the general equation is A = P (1 + r)^n

I figured A=$6000, P=$5000, n=48, and r=?

6000 = 5000 (1 + r)^48
6000 = 5000 (1r)^48
6000 = 5000 (1r)
6000 = 5000 r

6000 / 5000 = 1.2 % = r


Does this look right? I got confused on whether I should have combined the 1 and the r, or whether they should be seperate.

Thanks!

A = P (1 + r)^n

Remember that the interest rate is always per year!

If it is compounded monthly, the rate is r/12, or
quarterly r/4, or semiannually r/2.

(and you would not combine the 1 and the r)

6000 = 5000 (1 + r/12)^48 divide by 5000

1.2 = (1 + r/12)^48 use exponent (1/48) so that the r term has exponent 1

1.2^(1/48) = (1 + r/12)^48(1/48) simplify

1.0038 = 1 + r/12 subtract 1

0.0038 = r/12 multipliy by 12

0.0456 = r

4.56% = r

You should almost always work with the growth or accumulation factor, which is 1+r. That's usually the most important factor mathematically. E.g., you can't find the interest earned with compound interest directly -- you have to find the accumulated value and subtract the initial amount.

To solve 6000 = 5000 (1 + r)^48
1+r = (6/5)^(1/48) = 1.2^(1/48) = 1.0038055887000079006297308771219

Only at this point can you separate the 1 and the r:
r = .0038055
The nominal annual rate is 12r = 0.04566706440009480755677052546223 or 4.566%