Question

Nathan receives a lump sum inheritance of $55 000 and invests the money into a savings account with an annual interest rate of 7.5%, compounded quarterly.(a) Calculate the value of Nathan's investment after 5 years, rounding your answer to thenearest dollar. After m months, the amount in the savings account has increased to more than $70000.(b) Find the minimum value of m, where me N.Nathan is saving to purchase a property. The price of the property is $150 000. Nathan puts down a 15% deposit and takes out a loan for the remaining amount.(c) Write down the loan amount.The loan duration is for eight years, compounded monthly, with equal monthly payments of$1500 made by Nathan at the end of each month.(d) For this lonn, find(i) the amount of interest paid by Nathan over the life of the loan.(i) the annual interest rate of the loan, correct to two decimal places After 5 years of paying this locu, Nathan decides to pay the outstanding loan amount in onefinal payment. (e) Find the amount of the final payment after 5 years, rounding your answer to the nearestdollarif Find the amount Nathan saved by making this final payment after 5 years, roundingyour answer to the nearst dollar.

Answer 1

The Solution:

Given:

$\begin{gathered} P=\text{ \ }55000 \\ r=7.5\text{ \% compounded quarterly}=\frac{7.5}{400}=0.01875 \\ t=5\text{ years}=5\times4=20\text{ periods} \end{gathered}$

Required:

Find the value of the investment after 5 years.

The Formula:

$V=P(1+\frac{r}{n})^{nt}$

In this case,

$\begin{gathered} V=Value\text{ of the investment}=? \\ P=\text{ \ 55000} \\ r=0.075 \\ n=\text{ number of periods in a year}=4 \\ t=5\text{ years} \end{gathered}$

Substitute:

$V=55000(1+\frac{0.075}{4})^{(5\times4)}$$V=55000(1+0.01875)^{20}=55000(1.01875)^{20}$$V=79747.1414\approx\text{ \ }79747.14$

Answer:

(a) $79,747.14

Find the number of months it will take the account to increase to more than $70,000

Solve for n in the equation below:

$55000(1.01875)^n>70000$$\begin{gathered} (1.01875)^n>\frac{70000}{55000} \\ \\ (1.01875)^n>\frac{14}{11} \end{gathered}$$ln(1.01875)^n>ln(\frac{14}{11})$$n=\frac{ln(\frac{14}{11})}{ln(1.01875)}>12.982\approx13\text{ months}$

Answer:

13 months or more