Chapter 5.7, Problem 53AYU

Collage Costs The average annual cost of college at $4-$ year private colleges was $\$34,740$ in the $2017-2018$ academic year. This was a $3.6\%$ increase from the previous year.

If the cost of college increases by $3.6\%$ each year, what will be the average cost of college at a $4-$ year private college for the $2037-2038$ academic year?

College savings plans, such as a $529$ plan, allow individuals to put money aside now to help pay for college later. If one such plan offers a rate of $2\%$ compounded continuously, how much should be put in a college savings plan in $2019$ to pay for $1$ year of the cost of college at a $4-$ year private college for an incoming freshmen in 2037?

Source: The College Board

To determine

To calculate: The average cost of college at a $4$-year private college for the $2037-2038$ academic year, if the average annual cost is $\$34,740$ and the cost of college increased by $3.6\%$ each year.

Answer to Problem 53AYU

Solution:

The average cost of college at a $4$-year private college for the $2037-2038$ academic year is $\$70,473.35$

Explanation of Solution

Given information:

The average annual cost of college in the academic year $2017-2018$ is $\$34,740$ increased by $3.6\%$ each year.

Formula used:

The amount $A$ after $t$ years due to a principal $P$ invested at an annual interest rate $r$, compounded per year is $A=P{(1+r)}^t$

Calculation:

Total number of academic years from $2017-2018$ to $2037-2038$ is $20$

The cost of college increased by $3.6\%$

The average cost of college in the $2017-2018$ academic year is $\$34,740$.

Use the compound interest formula with $P=34740$, $r=0.036$ and $t=20$ to find $A$.

$A=P{(1+r)}^t$

$A=34740{\left(1+0.036\right)}^{20}$

$A=70473.35$

Hence, the average cost of college at a $4$-year private college for the $2037-2038$ academic years is $\$70,473.35$.

To determine

To calculate: The amount should be put in a college savings plan in the year $2019$ at a rate of $2\%$ compounded continuously to pay for one year of the cost of college at a $4$-year private college for an incoming freshman in year $2037$

Answer to Problem 53AYU

Solution:

The amount to put in a college savings plan in year $2019$ is $\$49167.59$

Explanation of Solution

Given information:

A college savings plan in $2019$ offers a rate of $2\%$ compounded continuously.

Formula used:

The present value formula is $P=A{e}^{-rt}$

Calculation:

Total number of years from $2019$ to $2037$ is $18$

From part (a) cost of college in $2037$ is $\$70,473.35$.

Use the present value formula with $A=70473.35$, $r=0.02$ and $t=18$ to find $P$.

$P=A{e}^{-rt}$

$P=70473.35\times e^{-0.02(18)}$

$P=49167.59$

Hence, the amount to put in a college savings plan in year $2019$ is $\$49167.59$