Question

Question #93e58

Answer 1

This is what I get

Explanation:

Price of car`=$23200`
Down payment `=$3000`
Loan financed`="Price"-"Down payment"` `=23200-3000=$20200`
For compound interest, let `n` installments of `$X` each and `r%` be rate of interest applicable for each installment. Then Principle `P` is given by

`P=X/(1+r/100)^n+X/(1+r/100)^(n-1).......... +X/((1+r/100))`

Given rate of interest `=8.4%" per annum compounded monthly"`
Effective rate for each installment `=8.4/12=0.7%`
Inserting given values we get

`20200=300/(1+0.7/100)^n+300/(1+0.7/100)^(n-1).......... +300/((1+0.7/100))`
`=>20200/300=1/(100.7/100)^n+1/(100.7/100)^(n-1).......... +1/(100.7/100)`
`=>202/3=(1.007)^-n+(1.007)^-(n-1).......... +(1.007)^-1`

RHS is a GP with first term `a=1/1.007`
Common ration `r=1/1.007`
Sum of `n` terms of a GP

`S_n=(a(r^n-1))/(r-1)`

Inserting in above expression we get

`202/3=(1/1.007(1/1.007^n-1))/(1/1.007-1)`
`202/3=((1-1/1.007^n))/(0.007)`
`=>1/1.007^n=1-202/3xx0.007`
`=>1/1.007^n=0.528bar6`

Taking log of both sides

`log(1/1.007^n)=log0.528bar6`
`-nlog1.007=log0.528bar6`
`=>n=-(log0.528bar6)/log1.007`
`=>n~~91.38=92` installments, as `n` is a positive integer, moreover installment can not be in fraction.