Question

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%

Answer 1

`\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 (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\\\\ -------------------------------`


`\bf ~~~~~~ \textit{Compound Interest Earned Amount} \\\\ A=P\left(1+(r)/(n)\right)^(nt) \quad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\to &\$1000\\ r=rate\to 3.7\%\to (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+(0.037)/(4)\right)^(4\cdot 5)\implies A=1000(1.00925)^(20)\\\\\\ A \approx 1202.195676`

get the difference of the amounts.

Answer 2

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,

`A= Pe^(rt)` , 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 ,
`A=1000e^(3.7*5/100)`

⇒
`A=1000e^(18.5)=1000* 1.20321844013=1203.21844013`

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

now, Since the amount in compound quarterly,

`A=P(1+(r/4)/(100) )^(4t)`, where, r is the rate of compound interest, P is the principal amount and t is the time.

Thus, amount in compound quarterly,
`A=1000(1+(3.7/4)/(100) )^(4*5)`

⇒
`A=1000(1+(3.7)/(400) )^(20)`

⇒
`A=1000(1+(3.7)/(400) )^(20)`

⇒
`A= 1202.19567617`

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