Question

Sue decides to start saving money for a new car. She knows she can invest money into an account which will earn 7.5% APR, compounded monthly, and would like to have saved $15,000 after 6 years. How much money will she need to invest into the account now so that she has $15,000 after 6 years?Determine the APY (Annual Percent Yield) for the account. Determine the 6-year percent change for the account.

Answer 1

Given:

The rate of interest is, r = 7.5% = 0.075.

The required total amount is, A = $15,000.

The number of years, t = 6 years.

The objective is to find the principal amount required to invest.

Step-by-step explanation:

The general formula of compound interest is,

`A=P(1+(r)/(n))^(nt)`

Here, n represents the number of times the interest is compounded.

It is given that the interest is compounded monthly, so the value of n = 12.

To find principal amount:

Substitute the given values in the above formula.

`\begin{gathered} 15000=P(1+(0.075)/(12))^(12(6)) \\ P=(15000)/((1+0.00625)^(72)) \\ P=9577.83 \end{gathered}`

To find APY:

The annual principal yield can be calculated as,

`\begin{gathered} \text{APY}=(1+(r)/(n))^n-1 \\ =(1+(0.075)/(12))^(12)-1 \\ =0.0776 \end{gathered}`

To find percent change:

The percent change can be calculated as,

`\begin{gathered} \text{\%change}=(A-P)/(P)*100 \\ =(15000-9577.83)/(9577.83)*100 \\ =(5422.17)/(9577.83)*100 \\ =0.5661167\ldots.*100 \\ =56.61\text{ \%} \end{gathered}`

Hence,

The principal money to be invested in the account is $ 9577.83.

The Annual Percent Yield is 0.0776.

The percent change for 6 year is 56.6%.