Answer:
i) [tex]P=\dfrac{A}{(1+r)^t}[/tex]
ii) [tex]r=1-(\dfrac{A}{P})^{\frac{1}{t}}[/tex]
Step-by-step explanation:
Given: [tex]A=P(1+r)^t[/tex]
(i) solve for P for above formula. We have to isolate P from formula.
[tex]A=P(1+r)^t[/tex]
Divide both side by [tex](1+r)^t[/tex]
[tex]\dfrac{A}{(1+r)^t}=\dfrac{P(1+r)^t}{(1+r)^t}[/tex]
[tex]P=\dfrac{A}{(1+r)^t}[/tex]
(ii) solve for r. We have to isolate r from formula.
Taking log both sides
[tex]\log A=\log (P(1+r)^t)[/tex]
[tex]\log A=\log P+\log (1+r)^t[/tex] [tex] \because \log ab = \log a+\log b[/tex]
[tex]\log A-\log P=t\log (1+r)[/tex] [tex]\because \log a^m=m\log a[/tex]
[tex]\frac{1}{t}(\log A-\log P)=\log (1+r)[/tex]
[tex]\frac{1}{t}(\log (\dfrac{A}{P}))=\log (1+r)[/tex] [tex]\because \log a-\log b=\log \frac{a}{b}[/tex]
[tex]\log (\dfrac{A}{P})^{\frac{1}{t}}=\log (1+r)[/tex]
[tex]1+r=(\dfrac{A}{P})^{\frac{1}{t}}[/tex]
[tex]r=1-(\dfrac{A}{P})^{\frac{1}{t}}[/tex]
