Maths abit of help

Will assume that 1 year has 52 weeks.And so, 1/2 a year would have 26 weeks. Since interest is compounded every 2nd week, then that gives us:26/2 =13 compounding periods:

FV=PV[1 + R ]^N

36,375=35,000[1 + R/13 ]^13 Divide both sides by 35,000,

1.0392857=[1 + R/13 ]^13 Take the log of both sides,

0.0167341...=13 x log[1 + R/13 ]

log[1 + R/13 ]=0.0167341/13=0.00128730

[1 + R/13 ] =10^0.00128730=1.00296852

R/13 =1.00296852 - 1 =0.00296852 Bi-weekly interest rate.

R=0.00296852 x 13 =0.0386 x 100 =3.86% compounded every 2 weeks for 6 months.

So the principal is $35 000      

 

the compounding period is every fortnight for 6 months    

There are approx 26 fortnights in a year and approx 13 fortnights in 6 months

 

the final value is $36 375 

 

what is the interest rate? with working out please?

 

\(FV= P(1+\frac{r}{26})^{13}\\ 36375= 35000(1+\frac{r}{26})^{13}\\ \frac{36375}{ 35000}=(1+\frac{r}{26})^{13}\\ \left(\frac{36375}{ 35000}\right)^{1/13}=1+\frac{r}{26}\\ \left(\frac{36375}{ 35000}\right)^{1/13}-1=\frac{r}{26}\\ 26\left[\left(\frac{36375}{ 35000}\right)^{1/13}-1\right]=r\\ r=26\left[\left(\frac{36375}{ 35000}\right)^{1/13}-1\right]\)

 

26((36375/35000)^(1/13)-1) = 0.07718165965354

 

so that is 7.7% p.a.  (that is the effective interest rate.)

The interest rate is=0.00296852 - Bi-weekly rate, or every 2 weeks.

0.00296852 x 13 =0.03859076 x 100 =3.859076% NOMINAL rate for 6 months ONLY.

0.00296852 x 26=0.07718152 x100 =7.718152% NOMINAL ANNUAL rate.

0.00296852 + 1 =1.00296852^26 =1.080115 - 1 x 100 =8.0115% EFFECTIVE ANNUAL RATE.