What is the compound interest on Rs 5000 for 2 years at 10% per annum compounded annually?

What will be the compound interest on Rs. 5000 if it is compounded half-yearly for 1 year 6 months at 8 % per annum.

Answer

Hint: The amount can be calculated using the given data and the formula:
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}$ where,
A = Amount
P = Principal
R = Rate
T = Time
Remember to half the rate and double the time as the principal is compounded half-yearly.
Then, the compound interest can be calculated using the relationship:
Amount = Principal + Compound Interest

Complete step-by-step answer:
Given:
Principal (P) = Rs. 5000
Rate (R) = 8 %
As it is compounded half yearly, the rate will reduce to half
$R = \dfrac{8}{2}\% $
R = 4 %
Time (t) = 1 year 6 months
= $\left( {1 + \dfrac{1}{2}} \right)yrs$
 $\left(
 \because 12m = 1yr \\
 6m = \dfrac{1}{{12}} \times 6 \\
 6m = \dfrac{1}{2}yrs \\
 \right)$
= $\dfrac{3}{2}yrs$
As it is compounded half yearly, the time will be doubled:
$t = 2 \times \dfrac{3}{2}yrs$
T = 3 yrs.
Substituting these values in the formula of Amount, we get:
$A = P{\left( {1 + \dfrac{R}{{100}}} \right)^t}$
$A = 5000{\left( {1 + \dfrac{4}{{100}}} \right)^3}$
$A = 5000{\left( {\dfrac{{104}}{{100}}} \right)^3}$
$A = 5000 \times \dfrac{{104}}{{100}} \times \dfrac{{104}}{{100}} \times \dfrac{{104}}{{100}}$
A = 5624.32
The amount is equal to Rs. 5624.32
Now,
Amount = Principal + Compound Interest
Compound Interest = Amount – Principal
Substituting the values, we get:
Compound Interest = 5624.32 – 5000
Compound Interest = 624.32
Therefore, the compound interest is Rs. 624.32 on Rs. 5000 if it is compounded half yearly for 1 year 6 months at 8 % per annum.

Note: We make the respective changes when compounded half-yearly because:
Rate is halved. The rate is generally given for a year (per annum) but when we require for half-yearly, the per annum rate is also reduced to half.
Time is doubled.When we talk about half a year (6 months), it occurs twice every year (6 + 6 = 12) and hence the time is doubled.

Calculate the compound interest on Rs.5000 for 3 years at 8% per annum compounded annually.

Answer

Verified

Hint: To find the amount, we use the formula \[A = P{\left[ {1 + \dfrac{r}{{100}}} \right]^n}\] where, A is the amount, P is principal amount, r is the rate percent yearly and n is the number of years. Since, all the values are known and it is given, hence by substituting the values in the above formula we get the required amount.

Complete step-by-step solution:
Here in this question, we have to find the value of the amount where the number of years, principal amount and the rate is given.
The interest rate per annum, \[r = 8\]
The initial principal amount, \[P = Rs.5000\]
The number of years, \[n = 3\]
We have to find the amount to be paid in case of compound interest
To find the value of amount we have standard formula \[A = P{\left[ {1 + \dfrac{r}{{100}}} \right]^n}\]
Substituting the values to the formula we get,
\[A = 5000{\left[ {1 + \dfrac{8}{{100}}} \right]^3}\]
Take the L.C.M inside the bracket and simplify we get,
\[ \Rightarrow A = 5000{\left[ {\dfrac{{108}}{{100}}} \right]^3}\]
On further simplification.
\[ \Rightarrow A = 5000{\left[ {\dfrac{{27}}{{25}}} \right]^3}\]
On squaring, we get
\[ \Rightarrow A = 5000\left[ {\dfrac{{27 \times 27 \times 27}}{{25 \times 25 \times 25}}} \right]\]
On further simplification.
\[   \Rightarrow A = \dfrac{{5000 \times 27 \times 27 \times 27}}{{15625}} \]
\[  \Rightarrow A = Rs.6298.56 \]
Hence, the amount is 6298.56 rupees. That is, the amount Rs. 6298.56 to be paid at the end of 3 years on Rs. 5000 at 8% per annum compounded annually.
 Therefore the compound interest is determined by
\[C.I = A - P\]
By substituting the values we get
\[ \Rightarrow C.I = 6298.56 - 5000\]
On simplifying we get
\[ \Rightarrow C.I = Rs.1298.56\]

Thus the correct answer is \[ C.I = Rs.1298.56\]

Note: The compound interest is interest calculated on the amount that includes principal and accumulated interest of the previous period whereas simple interest is interest on the invested amount for the entire period. This is the difference between the simple interest and compound interest. To find the value of amount where principal amount, rate of interest and time is known we use the standard formula \[A = P{\left[ {1 + \dfrac{r}{{100}}} \right]^n}\] to determine the value of A. We can also determine the compound interest by subtracting the initial principal amount from the amount.

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