Chapter 12.3, Problem 94AYU

Sinking Fund For a child born in $2018$, the cost of a $4$- year college education at a public university is projected to be $\$185,000.$ Assuming a $4.75\%$ per annum rate of return compounded monthly, how much must be contributed to a college fund every month to have $\$185,000$ in $18$ years when the child begins the college?

To determine

To calculate: The amount which must be contributed to the college fund every month to have $\$185000$ in $18$ years when the child begins college. Assuming a $4.75\%$ per annum rate of return compounded monthly.

Answer to Problem 94AYU

Solution:

The amount must be contributed to a college fund every month to have $\$185000$ in $18$ years when child begins college is approximately $\$543.48$

Explanation of Solution

Given information:

For a child born in $2018$, the cost of $4$- years college education at a public university is projected to be $\$185000$. Assuming a $4.75\%$ per annum rate of return 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:

The cost of $4$- years college education at a public university is projected to be $\$185000$, so $A=\$185000$

Per annum rate of return is $4.75\%$ compounded monthly.

Per month rate of return is $\frac{0.0475}{12}$

The number of deposits $n=18\times 12=216$

By using the theorem of amount of an annuity, The amount should be placed in the saving account $A=P\frac{{\left(1+\frac{0.0475}{12}\right)}^{216}-1}{\left(\frac{0.0475}{12}\right)}$

$\Rightarrow 185000=P\frac{{\left(\frac{12.0475}{12}\right)}^{216}-1}{\left(\frac{0.0475}{12}\right)}$

$\Rightarrow 185000\left(\frac{0.0475}{12}\right)=P\left(2.34740923-1\right)$

$\Rightarrow 732.291667=P\left(1.34740923\right)$

$\Rightarrow P=\frac{732.291667}{1.34740923}\approx 543.48$.

Therefore, the amount must be contributed to a college fund every month to have $\$185000$ in $18$ years when child begins college is approximately $\$543.48$