Tanisha is saving money and plans on making quarterly contributions into an account earning an annual interest rate of $6.7 \%$ compounded quarterly. If Tanisha would like to end up with $\$ 96,000$ after $11$ years, how much does she need to contribute to the account every quarter, to the nearest dollar? Use the following formula to determine your answer. $\newline$ $A=d\left(\frac{(1+i)^{n}-1}{i}\right)$ $\newline$ $A=$ the future value of the account after $n$ periods $\newline$ $d=$ the amount invested at the end of each period $\newline$ $i=$ the interest rate per period $\newline$ $n=$ the number of periods $\newline$ Answer:

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Grade 7

Compound interest

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Q. Tanisha is saving money and plans on making quarterly contributions into an account earning an annual interest rate of

6.7 \%

compounded quarterly. If Tanisha would like to end up with

\$ 96,000

after

11

years, how much does she need to contribute to the account every quarter, to the nearest dollar? Use the following formula to determine your answer.

\newline

A=d\left(\frac{(1+i)^{n}-1}{i}\right)

\newline

A=

the future value of the account after

n

periods

\newline

d=

the amount invested at the end of each period

\newline

i=

the interest rate per period

\newline

n=

the number of periods

\newline

Answer:

Identify Given Values: Identify the given values from the problem. $\newline$ $A$ (future value of the account) = $\$96,000$ $\newline$ $i$ (interest rate per period) = $6.7\%$ annual interest rate compounded quarterly, which means $i = \frac{6.7\%}{4} = \frac{0.067}{4}$ per quarter $\newline$ $n$ (number of periods) = $11$ years $* 4$ quarters/year = $44$ quarters $\newline$ Now, we can use these values to find $d$ (the amount invested at the end of each period).
Convert Interest Rate: Convert the annual interest rate to a quarterly interest rate. $\newline$ $i = \frac{0.067}{4} = 0.01675$ $\newline$ This is the interest rate per quarter.
Calculate Number of Periods: Calculate the number of periods $n$ . $n = 11 \text{ years} \times 4 \text{ quarters/year} = 44 \text{ quarters}$
Calculate $(1+i)^n$ : Use the formula $A = d\left(\frac{(1+i)^{n}-1}{i}\right)$ to find $d$ . First, calculate $(1+i)^n$ . $(1+i)^n = (1 + 0.01675)^{44}$
Calculate $(1+i)^n$ : Calculate $(1+i)^n$ using a calculator. $\newline$ $(1 + 0.01675)^{44} \approx 2.03009$
Calculate $((1+i)^n - 1)$ : Calculate $((1+i)^n - 1)$ . $\newline$ $((1+i)^n - 1) \approx 2.03009 - 1 = 1.03009$
Calculate Denominator: Calculate the denominator of the formula, which is $i$ . $i = 0.01675$
Calculate Right Side: Calculate the entire right side of the formula, which is $\left(\left(1+i\right)^{n}-1\right)/i$ . $\left(\left(1+i\right)^{n}-1\right)/i \approx \frac{1.03009}{0.01675} \approx 61.5493$
Solve for d: Solve for d using the formula $A = d\left(\frac{(1+i)^{n}-1}{i}\right)$ .
$\$96,000 = d \times 61.5493$
Now, divide both sides by $61.5493$ to find $d$ .
$d \approx \frac{\$96,000}{61.5493}$
Calculate d: Calculate d using a calculator. $\newline$ $d \approx \$(96,000) / 61.5493 \approx \$(1559.68)$ $\newline$ Since we need to round to the nearest dollar, $d \approx \$(1560)$ .