The difference between the compound interest for a year payable half-yearly

The difference between the compound interest for 1 year, compounded half-yearly and the simple interest for 1 year on a certain sum of money at 10% per annum is Rs. 360.

To do :

We have to find the sum.

Solution :

Let the sum be x

Rate of interest $= 10$% $= \frac{10}{100} = 0.1$.

Time $= 1$ year

Number of compounds per year $= n = 2$.

Amount $= P(1+\frac{r}{n})^{nt} $

               $= x(1+\frac{0.1}{2})^{2 \times 1}$

               $= x (\frac{2.1}{2})^2$

               $= x (1.05)^2 = 1.1025 x$

Compound Interest $=$ Amount $-$ Principal

$ = 1.1025x - x = 0.1025x$

Simple interest $= \frac{Ptr}{100} $

                          $= \frac{x \times 1 \times 10}{100} = 0.1x$

The difference between compound interest and simple interest $=$ Rs.360

Therefore,

$0.1025x - 0.1x=360$

$0.0025x=360$

$x =\frac{360}{0.0025} = \frac{360}{25} \times 10000 = 360 \times 400 = 144000 $

$x=144000$

Therefore, the sum is Rs.144000.

Let sum of money be Rs. y
To calculate S.I.
S.I. = `[ "P" xx "R" xx "T"]/100 = [ y xx 10 xx 1]/100 = Rs. y/10`

To calculate C.I.(compounded half-yearly)
∴ C.I. = P`[( 1 + r/[ 2 xx 100 ])^( n xx 2) - 1] = y[( 1 + 10/[ 2 x 100])^(1 xx 2) - 1 ]`

= `y[ (21/20)^2 - 1 ] = (41/400)y`

Given : C.I. - S.I. = Rs. 15

⇒ `(41/400)y - y/10 = 15`

⇒ `y/400 = 15 ⇒ y = Rs. 6,000.

What is the difference between the compound interest in compounded yearly and compounded half yearly for 18 months at 20% per annum on a sum of ₹ 12000?

What is the difference between compound interest and compound annually?

What is payable half yearly?

What is the difference between the compound interest on 5000 for 1 and half years and 4% per annum compounded yearly and half yearly?