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How much more would $1,000 earn in 5 years in an account compounded continuously than an account compounded quarterly if the interest rate on both accounts is $3.7%

Respuesta :

[tex]\bf ~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^{rt}\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\to& \$1000\\ r=rate\to 3.7\%\to \frac{3.7}{100}\to &0.037\\ t=years\to &5 \end{cases} \\\\\\ A=1000e^{0.037\cdot 5}\implies A=1000e^{0.185}\implies A\approx 1203.21844\\\\ -------------------------------[/tex]

[tex]\bf ~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+\frac{r}{n}\right)^{nt} \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\to &\$1000\\ r=rate\to 3.7\%\to \frac{3.7}{100}\to &0.037\\ n= \begin{array}{llll} \textit{times it compounds per year}\\ \textit{quarterly, thus four} \end{array}\to &4\\ t=years\to &5 \end{cases} \\\\\\ A=1000\left(1+\frac{0.037}{4}\right)^{4\cdot 5}\implies A=1000(1.00925)^{20}\\\\\\ A \approx 1202.195676[/tex]

get the difference of the amounts.

Answer: There is a difference of $ 1.0228.

Explanation: Given, initial amount or principal = $ 1000,

Time= 5 years and given compound rate of interest = $3.7%

Now, Since the amount in compound continuously,

[tex]A= Pe^{rt}[/tex] , where, r is the rate of compound interest, P is the principal amount and t is the time.

Here, P=$ 1000, t=5 years and r= $3.7%,

Thus, amount in compound continuously ,  [tex]A=1000e^{3.7\times5/100}[/tex]

⇒[tex]A=1000e^{18.5}=1000\times 1.20321844013=1203.21844013[/tex]

Therefore, interest in this compound continuously rate =1203.21844013-1000=203.21844013

now, Since the amount in compound quarterly,

[tex]A=P(1+\frac{r/4}{100} )^{4t}[/tex], where, r is the rate of compound interest, P is the principal amount and t is the time.

Thus, amount in compound quarterly, [tex]A=1000(1+\frac{3.7/4}{100} )^{4\times5}[/tex]

⇒[tex]A=1000(1+\frac{3.7}{400} )^{20}[/tex]

⇒[tex]A=1000(1+\frac{3.7}{400} )^{20}[/tex]

⇒[tex]A= 1202.19567617[/tex]

Therefore, interest in this compound quarterly rate=1202.19567617-1000=202.19567617

So, the difference in these interests=203.21844013-202.19567617=1.02276396 ≈1.0228                                                  

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