Question

एक विशेष धन को चकरवृधि ब्याज पर निवेसित किया जाता है। प्रथम दो वर्ष मे अर्जित किया गया ब्याज 272 रुपया है,जबकि प्रथम तीन वर्ष में अर्जित किया गया ब्याज 434 रुपया हैं। दर क्या हैं​

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Answer 1

Question in English:-

A certain amount of money is invested at an interest compounded annually. The interest earned in the first 2 years is Rs.272, while the interest earned in the first 3 years is Rs.434. Find the interest rate.

Solution:-

The amount obtained in $\sf{n}$ years of investing a principal amount $\sf{P}$ at an interest rate $\sf{r\,\%}$ compounded annually is,

$\sf{\longrightarrow P'=P\left[1+\dfrac{r\,\%}{100}\right]^n}$

From this, interest rate is given by,

$\sf{\longrightarrow r\,\%=\left[\left(\dfrac{P'}{P}\right)^{\frac{1}{n}}-1\right]100}$

The interest earned in first 2 years is Rs.272. So the amount obtained will be $\sf{P+272.}$

Then, interest rate will be,

$\sf{\longrightarrow r\,\%=\left[\left(\dfrac{P+272}{P}\right)^{\frac{1}{2}}-1\right]100\quad\quad\dots(1)}$

The interest earned in first 3 years is Rs.434. So the amount obtained will be $\sf{P+434.}$

Then, interest rate will be,

$\sf{\longrightarrow r\,\%=\left[\left(\dfrac{P+434}{P}\right)^{\frac{1}{3}}-1\right]100\quad\quad\dots(2)}$

Equating (1) and (2),

$\sf{\longrightarrow\left[\left(\dfrac{P+272}{P}\right)^{\frac{1}{2}}-1\right]100=\left[\left(\dfrac{P+434}{P}\right)^{\frac{1}{3}}-1\right]100}$

$\sf{\longrightarrow\left(\dfrac{P+272}{P}\right)^{\frac{1}{2}}=\left(\dfrac{P+434}{P}\right)^{\frac{1}{3}}}$

Raising both sides to the power 6,

$\sf{\longrightarrow\left(\dfrac{P+272}{P}\right)^3=\left(\dfrac{P+434}{P}\right)^2}$

$\sf{\longrightarrow\dfrac{(P+272)^3}{P}=(P+434)^2}$

$\sf{\longrightarrow(P+272)^3=P(P+434)^2}$

$\sf{\longrightarrow P^3+3\cdot 272P^2+3\cdot272^2P+272^3=P^3+2\cdot434P^2+434^2P}$

$\sf{\longrightarrow (3\cdot 272-2\cdot434)P^2+(3\cdot272^2-434^2)P+272^3=0}$

$\sf{\longrightarrow52P^2-33596P-20123648=0}$

Since $\sf{P>0,}$

$\sf{\longrightarrow P=\dfrac{33596+\sqrt{33596^2-4\times52\times-20123648}}{2\times52}}$

$\sf{\longrightarrow P=\dfrac{33596+72900}{104}}$

$\sf{\longrightarrow P=1024}$

Then, from (1),

$\sf{\longrightarrow r\,\%=\left[\left(\dfrac{1024+272}{1024}\right)^{\frac{1}{2}}-1\right]100}$

$\sf{\longrightarrow\underline{\underline{r\,\%=12.5\,\%}}}$

Answer 2

$\huge{\underline{\bf\red{Questi {\mathbb{O}} n} :}}$

• एक विशेष धन को चकरवृधि ब्याज पर निवेसित किया जाता है। प्रथम दो वर्ष मे अर्जित किया गया ब्याज 272 रुपया है,जबकि प्रथम तीन वर्ष में अर्जित किया गया ब्याज 434 रुपया हैं। दर क्या हैं ?

$\huge{\underline{\: \bf\green{Answ {\mathbb{E}} r}\ \: :}}$

• $\boxed{ \quad12.5\% \quad}$

$\Large{\underline{\bf\pink{Giv {\mathbb{E}} n}\ :}}$

• प्रथम दो वर्ष मे अर्जित किया गया ब्याज = ₹ 272

• प्रथम तीन वर्ष में अर्जित किया गया ब्याज = ₹ 434

$\Large{\underline{\bf\pink{To \ Fi {\mathbb{N}} d}\ :}}$

• ब्याज दर

$\huge{\underline{\bf\blue{Soluti {\mathbb{O}} n}\ :}}$

The amount obtained in $\sf{n}$ years of investing a principal amount $\sf{P}$ at an interest rate $\sf{r\,\%}$ compounded annually is,

$\sf{\implies P'=P \bigg[1+\dfrac{r\,\%}{100} \bigg]^n}$

From this, interest rate is given by,

$\sf{\implies r\,\%= \bigg[\bigg(\dfrac{P'}{P}\bigg)^{\frac{1}{n}}-1 \bigg]100}$

The interest earned in first 2 years is Rs.272. So the amount obtained will be \sf{P+272.}[/tex]

Then, interest rate will be,

$\sf{\implies r\,\%= \bigg[\bigg(\dfrac{P+272}{P}\bigg)^{\frac{1}{2}}-1 \bigg]100\quad\quad\dots(1)}$

The interest earned in first 3 years is Rs.434. So the amount obtained will be \sf{P+434.}[/tex]

Then, interest rate will be,

$\sf{\implies r\,\%= \bigg[\bigg(\dfrac{P+434}{P}\bigg)^{\frac{1}{3}}-1 \bigg]100\quad\quad\dots(2)}$

Equating (1) and (2),

$\sf{\implies \bigg[\bigg(\dfrac{P+272}{P}\bigg)^{\frac{1}{2}}-1\bigg]100=\bigg[\bigg(\dfrac{P+434}{P}\bigg)^{\frac{1}{3}}-1 \bigg]100}$

$\sf{\implies\bigg(\dfrac{P+272}{P}\bigg)^{\frac{1}{2}}=\bigg(\dfrac{P+434}{P}\bigg)^{\frac{1}{3}}}$

Raising both sides to the power 6,

$\sf{\implies\bigg(\dfrac{P+272}{P}\bigg)^3=\bigg(\dfrac{P+434}{P}\bigg)^2}$

$\sf{\implies\dfrac{(P+272)^3}{P}=(P+434)^2}$

$\sf{\implies(P+272)^3=P(P+434)^2}$

$\sf{\implies P^3+3\cdot 272P^2+3\cdot272^2P+272^3=P^3+2\cdot434P^2+434^2P}$

$\sf{\implies (3\cdot 272-2\cdot434)P^2+(3\cdot272^2-434^2)P+272^3=0}$

$\sf{\implies52P^2-33596P-20123648=0}$

Since $\sf{P > 0,}$

$\sf{\implies P=\dfrac{33596+\sqrt{33596^2-4\times52\times-20123648}}{2\times52}}$

$\sf{\implies P=\dfrac{33596+72900}{104}}$

$\sf{\implies P=1024}$

Then, from (1),

$\sf{\implies r\,\%= \bigg[\bigg(\dfrac{1024+272}{1024}\bigg)^{\frac{1}{2}}- 1 \bigg]100}$

$\sf{\implies\underline{\boxed{r\,\%=12.5\,\%}}}$

$\red{\rule{5.5cm}{0.02cm}}$

$\Large{ \mid {\underline{\underline{\bf\green{BrainLiest \ AnswEr}}}} \mid }$