Question

Suppose you'd like to save enough money to pay cash for your next car. The goal is to save an extra $28,000 over the next 6 years. What amount must be deposited quarterly into an account that earns 5.3% interest, compounded quarterly, in order to reach your goal? Round your answer to the nearest cent, if necessary

Answer 1

To reach a savings goal of $28,000 over 6 years with a 5.3% interest compounded quarterly, approximately $4,100.51 must be deposited quarterly into the account.

Step-by-step explanation:

To calculate the amount that must be deposited quarterly into an account to reach the savings goal of $28,000 over 6 years with a 5.3% interest compounded quarterly, we can use the formula for compound interest:

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

Where:

• A is the total amount after the specified time
• P is the principal amount (the initial deposit)
• r is the annual interest rate (as a decimal)
• n is the number of times the interest is compounded per year
• t is the number of years

Plugging in the given values, we get:

A = P(1 + 0.053/4)^(4*6)

To solve for P, we rearrange the equation:

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

Substituting in the values, we have:

P = $28,000 / (1 + 0.053/4)^(4*6)

Using a calculator, we find that P is approximately $4,100.51.

Answer 2

Given:

• Amount to save, A = $28,000

,

• Time, t = 6 years

,

• Interest rate, r = 5.3% ==> 0.053

,

• Number of times compounded = quarterly = 4 times

Let's find the amount that must be deposited into the account quarterly.

Apply the formula:

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

Where:

FV is the future value = $28,000

r = 0.053

n = 4

t = 6 years

Thus, we have:

`28000=P(((1+(0.053)/(4))^(4*6)-1))/((0.053)/(4))`

Let's solve for P.

We have:

`\begin{gathered} 28000=P(((1+0.01325)^(24)-1)/(0.01325)) \\ \\ 28000=P(\frac{(1.01325)^(24)-1)^{}}{0.01325}) \\ \\ 28000=P((1.371509114-1)/(0.01325)) \\ \\ 28000=P((0.371509114)/(0.01325)) \end{gathered}`

Solving further:

`28000=P(28.0384237)`

Divide both sides by 28.0384237:

`\begin{gathered} (28000)/(28.0384237)=(P(28.0384237))/(28.0384237) \\ \\ 998.6=P \\ \\ P=998.6 \end{gathered}`

Therefore, the amount that must be deposited quarterly into the account is $998.60

ANSWER:

$998.60