Question

Maths : Time Value of Money​

Answer 1

ANSWER:

Given:

• Difference between SI and CI on a certain sum is Rs 228.75
• Rate = 5% p.a.
• Time = 3 years

To Find:

• CI on the sum for 2 years at 5% p.a.

Solution:

We are given that,

$\implies \sf Compound\:Interest - Simple\:Interest=Rs228.75$

First, we will find Compound Interest.

We know that,

$\hookrightarrow \sf Amount_{C.I.}=P\left(1-\dfrac{R}{100}\right)^T$

So,

$\implies \sf Compound\:Interest = Amount-Principal$

$\implies \sf Compound\:Interest = P\left(1+\dfrac{R}{100}\right)^T -P$

Substituting the values of Rate and Time,

$\implies \sf Compound\:Interest = P\left(1+\dfrac{5}{100}\right)^3 -P$

$\implies \sf Compound\:Interest = P\left(\dfrac{105}{100}\right)^3 -P$

$\implies \sf Compound\:Interest = P\left(\dfrac{(105)^3}{(100)^3}-1\right)$

$\implies \sf Compound\:Interest = P\left(\dfrac{(105)^3-1000000}{1000000}\right)$

$\implies \sf Compound\:Interest = P\left(\dfrac{1157625-1000000}{1000000}\right)$

$\implies \sf Compound\:Interest = P\left(\dfrac{157625}{1000000}\right)$

Hence,

$\implies \sf Compound\:Interest = \dfrac{157625P}{1000000} - - - -(1)$

Now, we will find Simple Interest.

We know that,

$\hookrightarrow \sf Simple\:Interest =\dfrac{P\times R\times T}{100}$

Substituting the values of Rate and Time,

$\implies \sf Simple\:Interest =\dfrac{P\times 5\times 3}{100}$

$\implies \sf Simple\:Interest =\dfrac{15P}{100} - - - -(2)$

We had,

$\implies \sf Compound\:Interest - Simple\:Interest=Rs228.75$

Substituting the values of CI & SI,

$\implies \sf \dfrac{157625P}{1000000} - \dfrac{15P}{100} =Rs228.75$

Taking LCM,

$\implies \sf \dfrac{157625P-150000P}{1000000} =Rs228.75$

$\implies \sf \dfrac{7625P}{1000000} =Rs228.75$

$\implies \sf 7625P = 228750000$

So,

$\implies \sf P = \dfrac{228750000}{7625}$

Hence,

$\implies \sf P = Rs30,000$

Now, we need to find, the Compound Interest.

We have, Time = 2years, Rate = 5% p.a. and Principal = Rs30,000

So,

$\implies \sf Compound\:Interest = P\left(1+\dfrac{R}{100}\right)^T -P$

$\implies \sf Compound\:Interest = 30000\left(1+\dfrac{5}{100}\right)^2 -30000$

$\implies \sf Compound\:Interest = 30000\left(\dfrac{105}{100}\right)^2 -30000$

$\implies \sf Compound\:Interest = 30000\left(\dfrac{105\times105}{10000}\right)-30000$

$\implies \sf Compound\:Interest = 3\times11025 - 30000$

$\implies \sf Compound\:Interest = 33075-30000$

Hence,

$\implies \bf Compound\:Interest = Rs 3075$

Therefore, the Compound Interest is Rs 3075.