Chapter 12.3, Problem 93AYU

Sinking Fund Scott and Alice want to purchase a vacation home in $10$ years and need $\$50000$ for a down payment. How much should they place in a savings account each month if the per annum rate of return is assumed to be $3.5\%$ compounded monthly?

To determine

To calculate: The amount which Scott and Alice should place in the saving account each month when they want to purchase a vacation home in 10 years and need $\$50000$ for a down payment and the per annum rate of return is assumed to be $3.5\%$ compounded monthly.

Answer to Problem 93AYU

Solution:

Alice and Scott should place approximately $\$348.60$ in the saving account each month.

Explanation of Solution

Given information:

Scott and Alice want to purchase a vacation home in 10 years and need $\$50000$ for a down payment and the per annum rate of return is assumed to be $3.5\%$ compounded monthly.

Formula used:

Theorem of amount of an annuity:

Suppose that $P$ is the deposit in dollars made at the end of each payment period for an annuity paying $i$ percent interest per payment period. The amount $A$ of the annuity after $n$ deposits is

$A=P\frac{{\left(1+i\right)}^n-1}{i}$

Calculation:

Scott and Alice want to purchase a vacation home in $10$ years and need $\$50000$ for a down payment, so $A=\$50000$

Per annum rate of return is $3.5\%$

Per month rate of return is $\frac{3.5\%}{12}=0.002917$

The number of deposits $n=10\times 12=120$

By using the theorem of amount of an annuity, The amount should be placed in the saving account $A=P\frac{{\left(1+0.002917\right)}^{120}-1}{0.002917}$.

$\Rightarrow 50000=P\frac{{\left(1.002917\right)}^{120}-1}{0.002917}$

$\Rightarrow 50000=P\frac{1.418345-1}{0.002917}$

$\Rightarrow 50000=P\frac{0.418345}{0.002917}$

$\Rightarrow P=\frac{50000\cdot 0.002917}{0.418345}\approx 348.60$

Therefore, Alice and Scott should place approximately $\$348.60$ in the saving account each month.