Here is my trying work

Given

Pv= $8000 M=4 t=2 years

j= 4%/100= 0.04

Ra= ?

Solution:

a. I= j/m= 0.04/4= 0.01

b. n= t*m= 2*4=8

c. Ra= Pv * I/ 1(1+i)-n

8000*0.01/1-(1+0.01)-8= 1045.52

Given:

Pv=1,000000 m=2 (not sure if this no. for semi annually)

j= 8%/100= 0.08

t= 15 years

Solution:

a. I= 0.08/2= 0.4

b. n= t*m= 5*2= 30

c. Ra= PV*I/1-(1+i)-n

1,000000*0.04 / 1-(1+0.04)-30 = 5783.09

Denis,

His notation is a bit strange, but not indecipherable.

\(\displaystyle p = \dfrac{ai}{1 - \left ( \dfrac{1}{1 + i} \right )^n}\)

In step a, he calculates the periodic interest rate from the annual rate, or i.

In step b, he calculates the number of periods, or n.

In step c, he apparently uses the correct formula but writes it down incorrectly in two different ways. Pieces are missing, and he uses a minus sign for exponentiation.

For the second problem, it gets weirder. In step b, he multiplies 5 times 2 and get 30, but he meant to write down 15 * 2, which does equal 30. Just a typo, I think. The correct formula is

\(\displaystyle \dfrac{fi}{(1 + i)^n - 1}.\)

Here, he again writes down the formula incorrectly and again uses a minus sign for exponentiation. This time he does not use the correct formula.

He really needs to memorize the correct formulas. Otherwise, he may not get full credit even when he gets the correct numeric answer and may easily miss the correct numeric answer by using some goofy formula.