Question

A. Given ????=P(1+????t),
i. Solve for P.
ii. Solve for ????.

Answer 1

i)
`P=(A)/((1+r)^t)`

ii)
`r=1-((A)/(P))^{(1)/(t)}`

Explanation:

Given:
`A=P(1+r)^t`

(i) solve for P for above formula. We have to isolate P from formula.

`A=P(1+r)^t`

Divide both side by
`(1+r)^t`

`(A)/((1+r)^t)=(P(1+r)^t)/((1+r)^t)`

`P=(A)/((1+r)^t)`

(ii) solve for r. We have to isolate r from formula.

Taking log both sides

`\log A=\log (P(1+r)^t)`

`\log A=\log P+\log (1+r)^t`
`\because \log ab = \log a+\log b`

`\log A-\log P=t\log (1+r)`
`\because \log a^m=m\log a`

`(1)/(t)(\log A-\log P)=\log (1+r)`

`(1)/(t)(\log ((A)/(P)))=\log (1+r)`
`\because \log a-\log b=\log (a)/(b)`

`\log ((A)/(P))^{(1)/(t)}=\log (1+r)`

`1+r=((A)/(P))^{(1)/(t)}`

`r=1-((A)/(P))^{(1)/(t)}`