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

**Given :**

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.**

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