Respuesta :
Answer:
d. 0.0463% points.
Step-by-step explanation:
We have been given that Anna's bank gives her a loan with a stated interest rate of 10.22%.
To find the how mush the interest rate is effective, while compounded daily than compounded monthly, we will find the difference between interest compounded daily from interest compounded monthly.
[tex]\text{The effectiveness of interest rate compounded daily}=(1+\frac{r}{365})^{365}-(1+\frac{r}{12})^{12}[/tex]
Let us convert our given rate in decimal form.
[tex]10.22\%=\frac{10.22}{100}=0.1022[/tex]
Upon substituting our given interest rate in above equation we will get,
[tex]\text{The effectiveness of interest rate compounded daily}=(1+\frac{0.1022}{365})^{365}-(1+\frac{0.1022}{12})^{12}[/tex]
[tex]\text{The effectiveness of interest rate compounded daily}=(1+0.00028)^{365}-(1+0.008516666)^{12}[/tex]
[tex]\text{The effectiveness of interest rate compounded daily}=(1.00028)^{365}-(1.008516666)^{12}[/tex]
[tex]\text{The effectiveness of interest rate compounded daily}=1.1075891260304368-1.1071257622419648[/tex]
[tex]\text{The effectiveness of interest rate compounded daily}=0.000463363788[/tex]
Let us convert our rate in percentage by multiplying our answer by 100.
[tex]\text{The effectiveness of interest rate compounded daily}=0.000463363788*100[/tex]
[tex]\text{The effectiveness of interest rate compounded daily}=0.0463363788\%[/tex]
Therefore, the Anna's effective interest rate will be 0.04633% points and option d is the correct choice.
[tex]\boxed{0.0463{\text{ percentage}}}[/tex] greater will Anna’s effective interest rate be if the interest is compounded daily, rather than compounded monthly. Option (d) is correct.
Further explanation:
Given:
The options are as follows,
(a). 0.5389 percentage points.
(b). 0.1373 percentage points.
(c). 0.4926 percentage points.
(d). 0.0463 percentage points.
Explanation:
The relationship between the effective interest rate for one year and for effective interest rate for n times in a year can be calculated as follows,
[tex]\boxed{\left( {1 + i} \right) = {{\left( {1 + \frac{{{i^n}}}{n}} \right)}^n}}[/tex]
The difference between the effective interest rate be if the interest is compounded daily, rather than compounded monthly can be calculated as follows,
[tex]\begin{aligned}{\text{Difference}} &= {\left( {1 + \frac{{0.1022}}{{365}}} \right)^{365}} - {\left( {1 +\frac{{0.1022}}{{12}}} \right)^{12}}\\&= 1.107589 - 1.107126\\&= 0.000463\\&= 0.0463\%\\\end{aligned}[/tex]
[tex]\boxed{0.0463{\text{ percentage}}}[/tex] greater will Anna’s effective interest rate be if the interest is compounded daily, rather than compounded monthly. Option (d) is correct.
Option (a) is not correct.
Option (b) is not correct.
Option (c) is not correct.
Option (d) is correct.
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Simple interest
Keywords: nominal rate, compounded daily, effective rate, compounded monthly, Anna’s bank, Anna, bank, loan, percentage, interest rate, Principal, invested, interest rate, account, effective interest rate, total interest, amount.
