Question

How much would need to be deposited into an account earning 4.7%, compounded quarterly, so that the balance will be \[$1,000,000.00\] in 20 years?

Answer 1

Hint: Here we have to first figure out what we have to find. According to the question, we have to find the principal amount. For that we have to see whether we have the values according to the formula for finding the principal amount. After that we have to put the given values according to the formula and find the principal amount.
Formula used: \[A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}\]

Complete step by step solution:
We have to find the principal amount. The formula for finding the amount is:
 \[A = P{\left( {1 + \dfrac{r}{n}} \right)^{nt}}\]
Here, \[{\text{A = final}}\,{\text{amount}}\] , \[{\text{P = principal}}\,{\text{amount}}\] , \[{\text{r = rate}}\,{\text{of}}\,{\text{interest}}\] , \[{\text{t = number}}\,{\text{of}}\,{\text{years}}\] , \[{\text{n = times}}\,{\text{per}}\,{\text{year}}\] .
From the question, we have to see what values are given:
 \[A = \$ 1,000,000.00\]
 \[t = 20\,years\]
 \[r = 4.7\% \]
 \[n = 4\] (because compounded quarterly)
When we put these value in the formula, we get:
 \[1000000 = P{\left( {1 + \dfrac{{4.7\% }}{4}} \right)^{20 \times 4}}\]
We can write the rate of interest in the form of decimal number, and we get:
 \[ \Rightarrow 1000000 = P{\left( {1 + \dfrac{{0.047}}{4}} \right)^{20 \times 4}}\]
Now, we will try to simplify it. We will convert \[1\] into \[\dfrac{4}{4}\] , and we get:
 \[ \Rightarrow 1000000 = P{\left( {\dfrac{4}{4} + \dfrac{{0.047}}{4}} \right)^{20 \times 4}}\]
After solving this, we get:
 \[ \Rightarrow 1000000 = P{\left( {\dfrac{{4 + 0.047}}{4}} \right)^{80}}\]
 \[ \Rightarrow 1000000 = P{\left( {\dfrac{{4.047}}{4}} \right)^{80}}\]
Now, we will try to put \[P\] alone and try to solve, and we get:
 \[ \Rightarrow P = \dfrac{{1000000}}{{{{\left( {\dfrac{{4.047}}{4}} \right)}^{80}}}}\]
On solving this, we get that:
 \[ \Rightarrow P = 392774.20\]
So, our principal amount is \[392774.20\] . Therefore, we need to deposit \[\$ 392,774.20\] in the account.
So, the correct answer is “\[\$ 392,774.20\] ”.

Note: If we want to check whether the calculated result is correct or not, then we can simply put the value of the principal amount in the formula and calculate the final amount by keeping the rate of interest, number of years and times per year constant. We will get the final amount as \[\$ 1,000,000.00\] .