Verified Answer :
Answer:
The answer is 8.24%.
Step-by-step explanation:
The rate r = 8% is for 1 year or 52 weeks.
As given there are 13 weeks term so total there will be = 4 terms.
So, n = 4
The effective interest rate R is given by:
R = \((1+r/n)^n-1\)
= \((1+0.08/4)^4-1\)
= \((1.02)^4-1\)
= 0.0824
And in percentage, it is 8.24%.
The effective interest rate is \(\boxed{8.24\% }\) and the interest is \(\boxed{200}.\)
Further explanation:
The compound interest rate formula can be expressed as follows,
\(\boxed{A = P{{\left( {1 + i}\right)}^n}}\)
Here, A represents the amount, P represents the principal amount, i represents the interest rate and n represents the time.
The interest rate in a year is known as the effective interest rate.
The relationship between effective interest rate and nominal interest rate can be expressed as follows,
\(\boxed{i = {{\left( {1 + \dfrac{{{i^n}}}{n}}\right)}^n} - 1}\)
Here, i represents the effective interest rate, \({i^n}\) represents the nominal interest rate, n represents the period of interest rate.
Given:
The nominal interest rate is \(8\%\) in a 13 week treasury bill.
The amount of Treasury bill is \(\$ 10000.\)
Explanation:
There are 52 weeks in a year.
The value of n can be calculated as follows,
\(\begin{aligned}n&= \frac{{52}}{{13}}\\&= 4\\\end{aligned}\)
The effective interest rate can be calculated as follows,
\(\begin{aligned}i&= {\left( {1 + \frac{{0.08}}{4}} \right)^4} - 1\\&= {\left( {1.02} \right)^4} - 1\\&= 0.0824\\\end{aligned}\)
The interest can be calculated as follows,
\(\begin{aligned}{\text{Interest}}&= P\times i \times t\\&= 10000\times 0.08 \times \frac{{13}}{{52}}\\&= 200\\\end{aligned}\)
The effective interest rate is \(\boxed{8.24\% }\) and the interest is \(\boxed{200}.\)
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Compound interest
Keywords: Principal, invested, interest rate, account, effective interest rate, total interest, 8\% Treasury bill, nearest hundredth percent, amount.