Answer: Rs 32000

Note: When interest is compounded annually for one year at a certain rate then, the interest becomes equal to the Simple interest calculated for the same time at the same rate.

Solution:

Given,

Time (T) = 1 year
Rate of interest (R%) = 5% per annum
Difference between compound interest compounded semi-annually and compounded annually (D) = Rs 20

To find: Principal (P) = ?


Compound Interest compounded semi annually,
A = $P[ ( 1 + \frac{R%}{200} )^{2T} - 1]

$= P [ ( 1 + \frac{5}{200} )^2 - 1]

$= P [ (\frac{205}{200})^2 -1]

$= P [ (frac{41}{40})^2 - 1]

$= P [ \frac{1681}{1600} - 1]$

$= P [\frac{1681-1600}{1600} ]$

$= \frac{81 P}{1600}$


Compound Interest compounded annually,
B = $P[ (1 + \frac{R%}{100} )^T - 1]$

$= P[(1 + \frac{5}{100})^1 - 1]$

$= P [1 + \frac{1}{20} - 1]$

$= P × \frac{1}{20}$

$= \frac{P}{20}$


Now,
$D = A - B$

$or, 20 = \frac{81P}{1600} - \frac{P}{20}$

$or, 20 = \frac{81P}{1600} - \frac{80P}{1600}$

$or, 20 = \frac{81P - 80P}{1600}$

$or, 20 × 1600 = P$

$\therefore P = Rs 32000$


Hence, the required sum of money is Rs 32000.

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