Question

the compound interest on certain sum of money at a certain rate per annum for 2 years is ₹4100 and the simple intrest on the same amount of money at the same rate for 3 years is ₹6000 . then what is the sum of money
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Answer 1

Answer:

• SI for 3 years = Rs. 6000
• CI for 2 years = Rs. 4100

$\underline{\bigstar\:\textbf{Simple Interest :}}$

$:\implies\sf SI=\dfrac{Principal \times Rate \times Time}{100}\\\\\\:\implies\sf 6000=\dfrac{P \times r \times 3}{100}\\\\\\:\implies\sf \dfrac{6000 \times 100}{3}=Pr\\\\\\:\implies\sf 2000 \times 100=Pr\\\\\\:\implies\sf Pr=2 \times 10^{5}\qquad\dfrac{\quad}{}\:\bf{eq.\:i}$

$\rule{130}{1}$

⠀⠀⋆ we know that Simple Interest for ⠀⠀⠀every year is equal. Hence ;

⠀⠀⋆ ${\small\:\:\sf3\:yrs=Rs.\:6000}\\ {\small\:\:\sf1\:yr=\frac{Rs.\:6000}{3}}\\{\small\:\:\sf2\:yrs=Rs.\:2000 \times 2}\\{\small\:\:\sf2\:yrs=Rs.\:4000}$

$\underline{\bigstar\:\textbf{Difference b/w CI \& SI for 2 years :}}$

$:\implies\sf Difference=P \times \bigg\lgroup\dfrac{r}{100}\bigg\rgroup^2\\\\\\:\implies\sf Rs.\:4100-Rs.\:4000=P \times \dfrac{r^2}{100^2}\\\\\\:\implies\sf 100=\dfrac{Pr^2}{10^4}\\\\\\:\implies\sf 100 \times 10^4=Pr^2\\\\\\:\implies\sf Pr^2=10^6\qquad\dfrac{\quad}{}\:\bf{eq.\:ii}$

$\rule{150}{1}$

$\underline{\bigstar\:\textbf{Dividing eq. ii by eq. i:}}$

$\dashrightarrow\sf\:\:\dfrac{Pr^2}{Pr}=\dfrac{10^6}{2\times10^5}\\\\\\\dashrightarrow\sf\:\:r^{(2 - 1)} = \dfrac{10^{(6 - 5)}}{2}\\\\\\:\implies\sf r = \dfrac{10}{2}\\\\\\:\implies\sf r = 5\% \:p.a$

$\rule{200}{2}$

$\underline{\bigstar\:\textbf{Using value of r in eq. ii:}}$

$\dashrightarrow\sf\:\:Pr^2=10^6\\\\\\\dashrightarrow\sf\:\:P \times (5)^2=10^4\times100\\\\\\\dashrightarrow\sf\:\:25P=10^4\times100\\\\\\\dashrightarrow\sf\:\:P=\dfrac{10^4 \times 100}{25}\\\\\\\dashrightarrow\sf\:\:P=10000\times 4\\\\\\\dashrightarrow\:\:\underline{\boxed{\textsf{P = Rs.\:40,000}}}$

⠀

$\therefore\:\underline{\textsf{Hence, Sum of the money is \textbf{Rs. 40,000}}}.$