At what rate percent will 6000 amount to 6615 in 2 years when interest is compounded annually?

If the principal is Rs. 6000 at the rate of 5% per annum compounded annually for 2 years. Find the compound interest.

1. Rs. 600
2. Rs. 610
3. Rs. 615
4. Rs. 620

Answer (Detailed Solution Below)

Option 3 : Rs. 615

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Given:

P = 6000, R = 5% and Time = 2 year

Formula used:

Amount = Principal amount × [1 + (r/100)]t

Compound interest = Amount - Pricipal value

Calculation:

Amount = 6000 × [1 + (5/100)]2

⇒ Amount = 6000 × (21/20)2

⇒ Amount = 6000 × (21/20) × (21/20)

⇒ Amount = 6615

Compound ineterest = Amount - principal amount 

C.I. = 6615 - 6000

C.I. = 615

∴ Compound interest is 615.  

We know the successive increase formula

Net increase = [x+y+ (xy/100)]%

Rate of compound interest for 2 years= 10.25%

Required C.I. = 6000 × 10.25% = 615 

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Let's discuss the concepts related to Interest and Compound Interest. Explore more from Quantitative Aptitude here. Learn now!

At what rate percent per annum will Rs $6000$ amount to Rs $6615$ in $2$ year when interest is compounded annually?

Answer

Verified

Hint: The addition of interest to the principal sum of deposit So, that the interest in the next period is then earned on the principal sum plus previously accumulated interest is Compound interest. To solve this we have to use the formula of compound interest.

Complete step-by-step solution:
We have A= $6615$ , P= $6000$ and T= $2yr$
Putting the values in the formula
 $A = P{\left(1 + \dfrac{R}{{100}}\right)^T}$
 $6615 = 6000{\left(1 + \dfrac{R}{{100}}\right)^2}$
 $6000$ will shift towards left side in division
 $\dfrac{{6615}}{{6000}} = {\left(1 + \dfrac{R}{{100}}\right)^2}$
When we divide $\dfrac{{6615}}{{6000}}$ we get $1.025$
$1.025 = {\left(1 + \dfrac{R}{{100}}\right)^2}$
Taking square root both the side
$1.05 = \left(1 + \dfrac{R}{{100}}\right)$
1 will shift towards left in subtraction
$1.05 - 1 = \dfrac{R}{{100}}$
 $0.05 = \dfrac{R}{{100}}$
$100$ will shift towards left in multiplication
$5\% = R$
Hence the rate of compound interest will be $5\%$.

Note: Compound interest (or compounding interest) is the interest on a loan or deposit calculated based on both the initial principal and the accumulated interest from previous periods. Simple interest is a quick and easy method of calculating the interest charge on a loan. Simple interest is determined by multiplying the daily interest rate by the principal by the number of days between payments.

At what rate per cent per annum will $Rs\,6000$ amount to $Rs\,6615$ in two years when interest is compounded annually?

Answer

Verified

Hint: The problem can be solved easily with the concept of compound interest. Compound interest is the interest calculated on the principal and the interest of the previous period. The amount in compound interest to be cumulated depends on the initial principal amount, rate of interest and number of time periods elapsed. The amount A after a certain number of time periods T on a given principal amount P at a specified rate R compounded annually is calculated by the formula: $A = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$ .

Complete step-by-step answer:
In the given problem,
Principal $ = P = Rs\,6,000$
Rate of interest $ = R\% $
Time Duration $ = 2\,years$
In the question, the period after which the compound interest is compounded or evaluated is given as a year.
So, Number of time periods $ = n = 2$
Now, The amount A to be paid after a certain number of time periods n on a given principal amount P at a specified rate R compounded annually is calculated by the formula: $A = P{(1 + \dfrac{R}{{100}})^T}$ .
Hence, Amount $ = P{\left( {1 + \dfrac{R}{{100}}} \right)^T}$
Substituting the values of known quantities, we get,
$ \Rightarrow 6615 = 6000{\left( {1 + \dfrac{R}{{100}}} \right)^2}$
Shifting the terms in the equation, we get,
$ \Rightarrow {\left( {1 + \dfrac{R}{{100}}} \right)^2} = \dfrac{{6615}}{{6000}}$
Cancelling common factors in numerator and denominator, we get,
 $ \Rightarrow {\left( {1 + \dfrac{R}{{100}}} \right)^2} = \dfrac{{441}}{{400}}$
Taking square root on both sides of equation, we get,
$ \Rightarrow \left( {1 + \dfrac{R}{{100}}} \right) = \sqrt {\dfrac{{441}}{{400}}} $
We know that square roots of $441$ and $400$ are $21$ and $20$ respectively. So, we get,
$ \Rightarrow \left( {1 + \dfrac{R}{{100}}} \right) = \dfrac{{21}}{{20}}$
Isolating the variable R. we get,
$ \Rightarrow \dfrac{R}{{100}} = \dfrac{{21}}{{20}} - 1$
Taking LCM of fractions,
$ \Rightarrow \dfrac{R}{{100}} = \dfrac{{21 - 20}}{{20}}$
Multiplying both sides of equation by $100$,
$ \Rightarrow R = \dfrac{1}{{20}} \times 100$
Simplifying the calculations, we get,
$ \Rightarrow R = 5$
So, the rate of interest per annum for which $Rs\,6000$ amounts to $Rs\,6615$ in two years is $5\% $.
So, the correct answer is “ $5\% $”.

Note: Time duration is not always equal to the number of time periods. The equality holds only when the compound interest is compounded annually. If the compound interest is compounded half yearly, then the number of time periods doubles in the given time duration and the rate of interest in each time period becomes half of the specified rate of interest. Care should be taken while doing calculations.

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