Solved on Dec 13, 2023

Determine the periodic deposit to reach a $\$ 30,000$ financial goal over 5 years at a 3.5% quarterly compounded interest rate. Calculate how much of the goal comes from deposits vs. interest.
Periodic Deposit: $\$ ? \text{ every 3 months}$ Deposits: $\$ ?$ Interest: $\$ ?$

STEP 1

Assumptions
1. The financial goal is $30,000.<br />2. The interest rate is 3.5% compounded quarterly.<br />3. The time for achieving the financial goal is 5 years.<br />4. The periodic deposit is made at the end of every three months.<br />5. The formula for the future value of an annuity due to regular deposits is given by:$ $FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right)$ $where$ FV $is the future value,$ P $is the periodic deposit,$ r $is the interest rate per period, and$ n$ is the total number of deposits.

STEP 2

First, we need to determine the values of $r$ and $n$ for the formula. The interest rate per period (quarterly) is the annual rate divided by the number of periods per year.
$r = \frac{Annual\, interest\, rate}{Number\, of\, periods\, per\, year}$

STEP 3

Calculate the interest rate per period.
$r = \frac{3.5\%}{4} = \frac{0.035}{4} = 0.00875$

STEP 4

Determine the total number of deposits, $n$ , by multiplying the number of years by the number of periods per year.
$n = Years \times Periods\, per\, year$

STEP 5

Calculate the total number of deposits.
$n = 5 \times 4 = 20$

STEP 6

Now, we can use the future value of an annuity formula to solve for the periodic deposit $P$ .
$30,000 = P \times \left( \frac{(1 + 0.00875)^{20} - 1}{0.00875} \right)$

STEP 7

Calculate the factor by which the periodic deposit will be multiplied in the formula.
$\left( \frac{(1 + 0.00875)^{20} - 1}{0.00875} \right)$

STEP 8

First, calculate $(1 + 0.00875)^{20}$ .
$(1 + 0.00875)^{20} \approx 1.1916$

STEP 9

Subtract 1 from the result obtained in STEP_8.
$1.1916 - 1 = 0.1916$

STEP 10

Divide the result from STEP_9 by the interest rate per period.
$\frac{0.1916}{0.00875} \approx 21.8971$

STEP 11

Now, solve for the periodic deposit $P$ .
$P = \frac{30,000}{21.8971}$

STEP 12

Calculate the periodic deposit $P$ .
$P \approx \frac{30,000}{21.8971} \approx 1370.25$

STEP 13

Round up the periodic deposit to the nearest dollar as needed.
$P \approx \$1371$

STEP 14

Now that we have the periodic deposit, we can calculate how much of the financial goal comes from deposits and how much comes from interest.
$Total\, from\, deposits = P \times n$

STEP 15

Calculate the total from deposits.
$Total\, from\, deposits = \$1371 \times 20$

STEP 16

Calculate the total amount from deposits.
$Total\, from\, deposits = \$1371 \times 20 = \$27420$

STEP 17

To find the amount that comes from interest, subtract the total from deposits from the financial goal.
$Total\, from\, interest = Financial\, goal - Total\, from\, deposits$

STEP 18

Calculate the total from interest.
$Total\, from\, interest = \$30,000 - \$27,420$

STEP 19

Calculate the total amount from interest.
$Total\, from\, interest = \$30,000 - \$27,420 = \$2580$

STEP 20

Round the total from interest to the nearest dollar as needed.
$Total\, from\, interest \approx \$2580$
The periodic deposit is $1371. Of the$ 30,000 financial goal, $27,420 comes from deposits and$ 2,580 comes from interest.

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STEP 1

STEP 2

STEP 3

Calculate the interest rate per period.r=3.5%4=0.0354=0.00875 r = \frac{3.5\%}{4} = \frac{0.035}{4} = 0.00875 r=43.5%​=40.035​=0.00875

STEP 4

Determine the total number of deposits, nnn, by multiplying the number of years by the number of periods per year.n=Years×Periods per year n = Years \times Periods\, per\, year n=Years×Periodsperyear

STEP 5

Calculate the total number of deposits.n=5×4=20 n = 5 \times 4 = 20 n=5×4=20

STEP 6

Now, we can use the future value of an annuity formula to solve for the periodic deposit PPP.30,000=P×((1+0.00875)20−10.00875) 30,000 = P \times \left( \frac{(1 + 0.00875)^{20} - 1}{0.00875} \right) 30,000=P×(0.00875(1+0.00875)20−1​)

STEP 7

Calculate the factor by which the periodic deposit will be multiplied in the formula.((1+0.00875)20−10.00875) \left( \frac{(1 + 0.00875)^{20} - 1}{0.00875} \right) (0.00875(1+0.00875)20−1​)

STEP 8

First, calculate (1+0.00875)20(1 + 0.00875)^{20}(1+0.00875)20.(1+0.00875)20≈1.1916 (1 + 0.00875)^{20} \approx 1.1916 (1+0.00875)20≈1.1916

STEP 9

Subtract 1 from the result obtained in STEP_8.1.1916−1=0.1916 1.1916 - 1 = 0.1916 1.1916−1=0.1916

STEP 10

Divide the result from STEP_9 by the interest rate per period.0.19160.00875≈21.8971 \frac{0.1916}{0.00875} \approx 21.8971 0.008750.1916​≈21.8971

STEP 11

Now, solve for the periodic deposit PPP.P=30,00021.8971 P = \frac{30,000}{21.8971} P=21.897130,000​

STEP 12

Calculate the periodic deposit PPP.P≈30,00021.8971≈1370.25 P \approx \frac{30,000}{21.8971} \approx 1370.25 P≈21.897130,000​≈1370.25

STEP 13

Round up the periodic deposit to the nearest dollar as needed.P≈$1371 P \approx \$1371 P≈$1371

STEP 14

Now that we have the periodic deposit, we can calculate how much of the financial goal comes from deposits and how much comes from interest.Total from deposits=P×n Total\, from\, deposits = P \times n Totalfromdeposits=P×n

STEP 15

Calculate the total from deposits.Total from deposits=$1371×20 Total\, from\, deposits = \$1371 \times 20 Totalfromdeposits=$1371×20

STEP 16

Calculate the total amount from deposits.Total from deposits=$1371×20=$27420 Total\, from\, deposits = \$1371 \times 20 = \$27420 Totalfromdeposits=$1371×20=$27420

STEP 17

To find the amount that comes from interest, subtract the total from deposits from the financial goal.Total from interest=Financial goal−Total from deposits Total\, from\, interest = Financial\, goal - Total\, from\, deposits Totalfrominterest=Financialgoal−Totalfromdeposits

STEP 18

Calculate the total from interest.Total from interest=$30,000−$27,420 Total\, from\, interest = \$30,000 - \$27,420 Totalfrominterest=$30,000−$27,420

STEP 19

Calculate the total amount from interest.Total from interest=$30,000−$27,420=$2580 Total\, from\, interest = \$30,000 - \$27,420 = \$2580 Totalfrominterest=$30,000−$27,420=$2580

STEP 20

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Calculate the interest rate per period.
$r = \frac{3.5\%}{4} = \frac{0.035}{4} = 0.00875$

Determine the total number of deposits, $n$ , by multiplying the number of years by the number of periods per year.
$n = Years \times Periods\, per\, year$

Calculate the total number of deposits.
$n = 5 \times 4 = 20$

Now, we can use the future value of an annuity formula to solve for the periodic deposit $P$ .
$30,000 = P \times \left( \frac{(1 + 0.00875)^{20} - 1}{0.00875} \right)$

Calculate the factor by which the periodic deposit will be multiplied in the formula.
$\left( \frac{(1 + 0.00875)^{20} - 1}{0.00875} \right)$

First, calculate $(1 + 0.00875)^{20}$ .
$(1 + 0.00875)^{20} \approx 1.1916$

Subtract 1 from the result obtained in STEP_8.
$1.1916 - 1 = 0.1916$

Divide the result from STEP_9 by the interest rate per period.
$\frac{0.1916}{0.00875} \approx 21.8971$

Now, solve for the periodic deposit $P$ .
$P = \frac{30,000}{21.8971}$

Calculate the periodic deposit $P$ .
$P \approx \frac{30,000}{21.8971} \approx 1370.25$

Round up the periodic deposit to the nearest dollar as needed.
$P \approx \$1371$

Now that we have the periodic deposit, we can calculate how much of the financial goal comes from deposits and how much comes from interest.
$Total\, from\, deposits = P \times n$

Calculate the total from deposits.
$Total\, from\, deposits = \$1371 \times 20$

Calculate the total amount from deposits.
$Total\, from\, deposits = \$1371 \times 20 = \$27420$

To find the amount that comes from interest, subtract the total from deposits from the financial goal.
$Total\, from\, interest = Financial\, goal - Total\, from\, deposits$

Calculate the total from interest.
$Total\, from\, interest = \$30,000 - \$27,420$

Calculate the total amount from interest.
$Total\, from\, interest = \$30,000 - \$27,420 = \$2580$