Find the compound interest on Rs 1000 for two years at the rate of 3 compounded half yearly

Compute the compound interest on Rs.1000 for 2years at 10% per annum when compounded half yearly.(write answer in the nearest integer).

Answer

Verified

Hint: According to given in the question we have to determine the compound interest on Rs.1000 for 2years at 10% per annum when compounded half yearly but first of all we have to understand about the compound interest which is explained below:
Compound interest: Compound interest is when the interest we have earned on the balance amount in a saving or investing amount which is reinvested, earning your more interest. Compound interest accelerates the growth of your savings and investments over the time.

Formula used: Now, to find the compound interest on the principal amount now we have to use the formula as given below:
$ = P{\left( {1 + \dfrac{R}{{200}}} \right)^{2n}}........................(a)$
Where, P is the principal amount, R is the rate of interest and n is the time of investment.
Now, to obtain the compound interest we have to subtract the principal amount with the obtained amount.

Complete step-by-step answer:
Given,
Principal amount (P) = 1000Rs,
Rate (R) = 10%, and
Time in years (n) = 2 years
Step 1: First of all we have to use the formula (A) to find the amount that will be obtained after 2 years as mentioned in the solution hint. Hence, on substituting all the values in formula (A),
$ = 1000{\left( {1 + \dfrac{{10}}{{200}}} \right)^{2 \times 2}}$…………………..(1)
Step 2: On solving the expression (1) as obtained in the solution step 1.
\[
   = 1000{\left( {1 + \dfrac{1}{{20}}} \right)^4} \\
   = 1000{\left( {\dfrac{{21}}{{20}}} \right)^4} \\
   = 1000 \times \dfrac{{21}}{{20}} \times \dfrac{{21}}{{20}} \times \dfrac{{21}}{{20}} \times \dfrac{{21}}{{20}} \\
   = 1215.50Rs
 \]
Step 3: Now, to calculate the value of compound interest we have to subtract the principal amount with the obtained amount as mentioned in the solution hint.
$
   = 1215.50 - 1000 \\
   = 215.50Rs
 $

Hence, with the help of formula (A) compound interest on Rs.1000 for 2years at 10% per annum when compounded half yearly is $ = 215.50Rs$

Note: Compound interest is the interest that is calculated on the initial principal amount, which also includes the accumulated interest from the previous period on the deposit or on loan amount.
Compound interest is the interest calculated on the principal and is accumulated over the previous period.

Solution

Principal (P)= Rs 1000
Rate(R)=20%
Time(T)= 1.5yrs
n= 2 ( as compounded half yearly)
Amount(A) = P(1+R/n)^(n*t)
=1000(1+ 0.20/2)^(2*3/2)
=1000(1+1/10)^3
=1000(11/10)^3
=1000* (1331/1000)
=1331

Therefore,
Amount (A) = Rs 1331

Principal (P) =Rs.1000
Rate (R) =8% p.a
Period (T) =112years=3 half-years
∴ Interest for the first half-year =PRT100
=1000×8×1100×2=Rs. 40

Amound after one half-year =Rs.1040

or, Principal for the second half-year =Rs.1040

Interest for the second half-year

=1040×8×1100×2=Rs.4160100=Rs.41.60

Amount after second half-year
=Rs.1040+41.60=Rs.1081.6

or, Principal for the third half-year =Rs.1081.60
Interest for the third half-year
=1081.60×8×1100×2=Rs.4326.40100=Rs.43.264

∴ Compound interest for the third half-year
or 112 years =Rs.40+Rs.41.60+Rs.43.264=Rs.124.864

Alternate way:

Interest is compounded half yearly

∴ Interest @ half year =82=4%

n = number of times interest added in 112 years =3

Amount (A) is given by

A=P[1+r@ half year100]n

=1000[1+4100]3

=1000×104100×104100×104100

=104×104×10410×100

=1124.864

Compound Interest = Amount − Principal

=1124.864−1000=Rs.124.864