Question #280806

bellamy borrowed 65,000 at 7% interest compounded quarterly and agreed to repay the loan in quarterly payments of 5,000 each . the first payment is due in two years. Find the number of payments


1
Expert's answer
2021-12-21T12:39:29-0500

Given,

Quarterly Payments= 5,000

Principle Amount = 65,000

Rate = 7% p.a

principle=payment

[1(1+rn)ntrn]65000=5000[1(1+0.074)4t0.074]65000=5000[1(1+0.0175)4t0.074]650005000=[1(10.0175)4t0.175]12×0.0175=[1(1.0175)4t]0.2275=10.01754t0.7725=1.01754t[\frac{1-(1+\frac{r}{n})^{-nt}}{\frac{r}{n}}]\\ 65 000=5000 [\frac{1-(1+\frac{0.07}{4})^{-4t}}{\frac{0.07}{4}}]\\ 65 000=5000 [\frac{1-(1+0.0175)^{-4t}}{\frac{0.07}{4}}]\\ \frac{65 000}{5 000}=[\frac{1-(1-0.0175)^{-4t}}{0.175}]\\ 12\times 0.0175=[1-(1.0175)^{-4t}]\\0.2275=1-0.0175^{-4t}\\0.7725=1.0175^{-4t}

Now taking log on both sides,

log(0.7725)=log(1.0175)4tlog(0.7725)=4tlog(1.0175)t=log(0.7725)4tlog(1.0175)t=3.7196log (0.7725)=log(1.0175)^{-4t}\\ log (0.7725)=-4t log(1.0175)\\t=\frac{log (0.7725)}{-4t log(1.0175)}\\t=3.7196


Therefore, total number of payments if made quarterly will be:

number of payments

=3.7196×4=14.8784around 15 payments=3.7196\times 4\\=14.8784\\ around \space 15 \space payments




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