Question

the difference between the compound interest compounded annually and simple interest on the certain sum at rate of 6% per annum for 2 years is 54 find the sum​

Answer 1

Answer:-

Given:

Time period (T) = 2 years

Rate of interest (R) = 6 %

Difference between CI and SI = 54

Let the sum be x.

We know that,

$\boxed{ \large{\sf \: CI = P\bigg( 1 + \frac{R}{100} \bigg) ^{T} - P}}$

And,

SI = PTR / 100

[ Where P is the sum or principle ]

According to the question,

$\implies \sf \: P \bigg( 1 + \frac{R}{100} \bigg) ^{T} - P - \frac{PTR}{100} = 54 \\ \\ \\ \implies \sf \:x \bigg( 1 + \frac{6}{100} \bigg) ^{2} - x - \frac{(x)(2)(6)}{100} = 54 \\ \\ \\ \implies \sf \:x \bigg( \frac{100 + 6}{100} \bigg) ^{2} - x - \frac{12x}{100} = 54 \\ \\ \\ \implies \sf \: \frac{x \times 106 \times 106}{100 \times 100} - x - \frac{3x}{25} = 54 \\ \\ \\ \implies \sf \: \frac{2809x}{2500} - x - \frac{3x}{25} = 54 \\ \\ \\ \implies \sf \: \frac{2809x - 2500x - 300x}{2500} = 54 \\ \\ \\ \implies \sf \:9x = 54 \times 2500 \\ \\ \\ \implies \sf \:x = \frac{54 \times 2500}{9} \\ \\ \\ \implies \boxed{ \sf \:x = 15000}$

∴ The sum is Rs. 15000.

Answer 2

Step-by-step explanation:

Given :

• Difference between the Compound Interest and Simple Interest = Rs. 54

• At rate = 6% p.a

• Time = 2 years

To find :

• The sum of money

Solution :

Simple Interest -

• Rate = 6%
• Time = 2 years
• Principal = P

$\bigstar \: \boxed{\tt{SI = \frac{Principal \times Rate \times Time}{100}}} \: </p><p>$

Substitute all values :

$\tt{\implies} \:SI = \dfrac{P \times 6 \times 2}{100} \\ \\ \\ \: \tt{\implies} \:SI = \frac{12p}{100}$

Compound Interest -

$\bigstar \: {\boxed{\tt{CI= P\left( 1 + \frac{R}{100}\right)^{N}} -P }} \:$

Substitute all Values :

$\begin{lgathered}\tt{\longrightarrow} \: {CI= P\left( 1 + \dfrac{6}{100}\right)^{2}} -1 \\ \\ \tt{\longrightarrow} \: {CI=P\left(\dfrac{106}{100}\right)^{2}} -1\\ \\ \tt{\longrightarrow} \: {CI=P\left(\dfrac{53}{50}\right)^{2}} -1 \\ \\ \tt{\longrightarrow} \: {CI=P \times \frac{2809}{2500} -1} \\ \\ \tt{\longrightarrow} \: {CI= \frac{309P}{100}}\end{lgathered} \:$

Difference between the CI and SI

$\sf \: Compound \: Interest = \tt{\frac{309P}{100}}$

$\sf \: Simple \: Interest = \tt{\frac{12P}{100}}$

$\begin{lgathered}\tt{\longrightarrow} \: \dfrac{309P}{2500} - \dfrac{12P}{100} = 54 \\ \\ \tt{\longrightarrow} \: \dfrac{309P - 300P}{2500} = 54 \\ \\ \tt{\longrightarrow} \: \dfrac{9P}{2500} = 54 \\ \\ \tt{\longrightarrow} \: 9P = 54 \times 2500 \\ \\ \tt{\longrightarrow} \: 9P = 135000 \\ \\ \tt{\longrightarrow} \: P = \dfrac{135000}{9} \\ \\ \tt{\longrightarrow} \: P = 15000\end{lgathered} \:$

Principal = Rs. 15,000

∴ The sum of money is Rs. 15,000.