At what rate percent per annum compound interest will 8000 amount to 8820 in 6 months the interest being compounded quarterly?

Correct Answer:

Description for Correct answer:
Principal = Rs. 8000,

Amount = Rs. 8820

Let Rate % = R

Time = 2 years

By using formula,

\( \Large 8820=8000 \left(1+\frac{R}{100}\right)^{2}\)

\( \Large \frac{8820}{8000}= \left(1+\frac{R}{100}\right)^{2}\)

\( \Large \frac{441}{400}= \left(1+\frac{R}{100}\right)^{2}\)

Taking square root of both sides,

\( \Large \frac{21}{20}= \left(1+\frac{R}{100}\right)\)

R= 5 %

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Correct Answer - Option 4 : 5%

GIVEN:

Principal = Rs.8000

Amount after 2 years = Rs.8820

CONCEPT:

As interest is compounded annually for 2 years, we have to square root both principal and amount and compare them to find the interest.

FORMULA USED:

Amount = Principal (1 + Rate/100)t

CALCULATION:

Let, the rate of interest be r%

Accordingly, 

8000 × (1 + r/100)2 = 8820

⇒ 8820/8000 = (1 + r/100)2

⇒ 441/400 = (1 + r/100)

⇒ 21/20 = 1 + r/100

⇒ r/100 = 1/20

⇒ r = 5

∴ Rate of Interest is 5%  per annum.

In what time will Rs. 8000 amount change to Rs. 8820 if it is compounded at a rate of 5 % annually.

Answer

Verified

Hint: The formula for compound interest is \[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}\] where A is the final amount, P is the principal amount, r is the interest rate per year, n is the number of years. Apply this to the given data of the question to get the answer.Complete step by step answer:-
Compound interest is the interest that is calculated on the principal amount along with the interest accumulated over the previous period or year.
The formula to calculate compound interest with a principal amount P, at an annual rate r for n years is given as follows:
\[A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}............(1)\]
In this problem, the principal amount P is given as Rs. 8000.
\[P = 8000...........(2)\]
The amount is compounded at a rate of 5 % annually.
\[r = 100...........(3)\]
We need to find the number of years n to accumulate an amount of Rs. 8820.
\[A = 8820..........(4)\]
Using equations (2), (3), and (4) in equation (1), we have:
\[8820 = 8000{\left( {1 + \dfrac{5}{{100}}} \right)^n}\]
Simplifying the terms in the bracket we get:
\[8820 = 8000{\left( {1.05} \right)^n}\]
Taking 8000 to the other side and dividing with 8820, we get:
\[\dfrac{{8820}}{{8000}} = {\left( {1.05} \right)^n}\]
\[1.1025 = {\left( {1.05} \right)^n}\]
Apply log to both the sides of the equation, to get as follows:
\[\log (1.1025) = \log ({1.05^n})\]
We know that the value of \[\log {x^n}\] is equal to \[n\log x\].
\[\log (1.1025) = n\log (1.05)\]
We know that the square of 1.05 is 1.1025, then, we have:
\[\log {\left( {1.05} \right)^2} = n\log (1.05)\]
\[2\log \left( {1.05} \right) = n\log (1.05)\]
Solving for n, we have:
\[n = 2\dfrac{{\log (1.05)}}{{\log (1.05)}}\]
\[n = 2\]
Hence, it takes 2 years for the amount Rs. 8000 to change to Rs. 8820 when it is compounded at the rate of 5 % annually.

At what rate of compound interest will a sum of Rs 8000 amount to Rs 8820 in two years?

What percentage is 8000 per annum?

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In what time will rupees 8000 amount to 8820 at 10% per annum interest compounded half yearly?