Chapter 7.8, Problem 1CYU

a.

To determine

To write: A function to represent amount $A(t)$ that Tareq would earn after $t$ years through oak hill, then to write an equivalent function with exponent $4t$ .

Answer to Problem 1CYU

  $A(t)={(1.031)}^t$

  $A(t)={(1.00766150151)}^{4t}$

Explanation of Solution

Given Information: Initial investment is $\$1$ .

Calculation:

  $A(t)=P{(1+\frac{r}{n})}^{nt}(Formulaofcompound\mathrm{int}erest)$

Substituting the given values in the above stated equation,

  $\begin{array}{}$ A(t)=1{(1+\frac{0.031}{1})}^{1\times t}$$ A(t)={(1.031)}^t$\end{array}$

To keep the function equivalent but have an exponent of $4t$ , the value inside the brackets should be reduced as given below,

  $\begin{array}{}$ A(t)=1{(\sqrt[4]{1.031})}^{4\times t}$$ A(t)={(1.00766150151)}^{4t}$\end{array}$

Thus, functions are derived.

b.

To determine

The effective quarterly interest rate at Oak Hills Financial.

Answer to Problem 1CYU

Effective quarterly rate is $0.766\%$ .

Explanation of Solution

Given Information: Initial investment is $\$1$ . $A(t)={(1.031)}^t$(from previous part).

Calculation:

Annual effective annual rate should be $3.1\%$ .

  $\begin{array}{l}$ AnnualEffectiveRate={(1+\frac{No\min alRate}{n})}^n-1$$ 0.031={(1+\frac{NR}{4})}^4-1$$ {(1+\frac{NR}{4})}^4=1.031$& (1+\frac{NR}{4})=\sqrt[4]{1.031}\\ \\ \frac{NR}{4}=0.00766150151\\ \\ No\min alRate=0.030646\end{array}$

Effective Quarterly Rate $=\frac{AnnualNo\min alRate}{4}$

  $=0.007661510151$

Thus, effective quarterly rate is $0.766\%$ .

c.

To determine

The better plan out of the two.

Answer to Problem 1CYU

Thus, Oak Hills Financial is better plan.

Explanation of Solution

Given Information: First city bank has a compounded quarterly interest rate of $0.7\%$ .

As calculated in previous part, Oak Hills Financial provides compounded quarterly interest rate of $0.766\%$ while it is given that First city bank has a compounded quarterly interest rate of $0.7\%$ . Obviously, the plan that provides relatively more interest rate is considered better.

Thus, Oak Hills Financial is better plan.