Question

किसी धनराशि का 3 वर्षों में 10% वार्षिक ब्याज की दर से साधारण एवं चक्रवृद्धि ब्याज में अंतर 93 है तो मूलधन कितना होगा?​

Answer 1

AnswEr :

☯$\:\:\underline{\textsf{Difference b/w CI and SI for 3 years :}}$

$:\implies\tt Difference = P \times r^2\bigg[\dfrac{r + 300}{(100)^3}\bigg]\\\\\\:\implies\tt 93 = P \times (10)^2\bigg[\dfrac{10 + 300}{(100)^3}\bigg]\\\\\\:\implies\tt 93 = P \times 100 \times \dfrac{310}{100 \times (100)^2}\\\\\\:\implies\tt 93 = P \times\dfrac{310}{(100)^2}\\\\\\:\implies\tt \dfrac{93 \times 100 \times 100}{310} = P\\\\\\:\implies\underline{\boxed{\red{\tt Principal = Rs.\: 3000}}}$

⠀

$\therefore\:\underline{\textsf{Therefore Principal will be \textbf{Rs. 3000}}}.$

$\rule{200}{2}$

☯$\:\:\underline{\textsf{Derivation of Formula for 3 years :}}$

$\dashrightarrow\tt\:\:Difference = Compound\:Interest - Simple \:Interest\\\\\\\dashrightarrow\tt\:\:Diff. = \bigg[P \bigg(1 + \dfrac{r}{100} \bigg)^{t} - 1\bigg]- \bigg[\dfrac{Prt}{100}\bigg]\\\\\\\dashrightarrow\tt\:\:Diff. = \bigg[P \bigg(1 + \dfrac{r}{100} \bigg)^3 - 1 \bigg]- \bigg[\dfrac{Pr\times 3}{100}\bigg]\\\\\\\dashrightarrow\tt\:\:Diff. = \bigg[P \bigg(\dfrac{100 + r}{100} \bigg)^{3} - 1 \bigg]- \bigg[\dfrac{Pr \times 3}{100}\bigg]\\\\\\\dashrightarrow\tt\:\:Diff. = \bigg[P \bigg(\dfrac{(100 + r)^3}{(100)^3} - 1\bigg)\bigg]- \bigg[\dfrac{3Pr}{100}\bigg]\\\\\\\dashrightarrow\tt\:\:Diff. = \bigg[P \bigg(\dfrac{(100 + r)^3 - (100)^3}{(100)^3}\bigg)\bigg]- \bigg[\dfrac{3Pr}{100}\bigg]\\\\\\\dashrightarrow\tt\:\:Diff. = \bigg[P \bigg(\dfrac{(100)^{3} +r^3+ 300r(100 + r) -(100)^3}{(100)^3}\bigg)\bigg]- \bigg[\dfrac{3Pr}{100}\bigg]\\\\\\\dashrightarrow\tt\:\:Diff. =\bigg[P \bigg(\dfrac{r^3+ 300r(100 + r)}{(100)^3}\bigg)\bigg]- \bigg[\dfrac{3Pr}{100}\bigg]\\\\\\\dashrightarrow\tt\:\:Diff. = P\bigg[ \bigg(\dfrac{r^3 + 30000r + 300{r}^{2}}{(100)^3}\bigg)-\dfrac{3r}{100}\bigg]\\\\\\\dashrightarrow\tt\:\:Diff.= P\bigg(\dfrac{r^3+ 30000r + 300r^2 - (3r \times100^2)}{(100)^3}\bigg)\\\\\\\dashrightarrow\tt\:\:Diff.= P\bigg(\dfrac{r^3 + 30000r + 300r^2 - 30000R}{(100)^3}\bigg)\\\\\\\dashrightarrow\tt\:\:Diff.= P\bigg( \dfrac{r^3+300r^2}{(100)^3} \bigg)\\\\\\\dashrightarrow\:\:\underline{\boxed{\orange{\tt Diff.= Pr^2\bigg[\dfrac{r + 300}{(100)^3}\bigg]}}}$

Answer 2

Answer:

$\underline{\bigstar\:\:\textsf{According to the Question :}}$

$\longrightarrow\sf Difference = P\bigg(\dfrac{r}{100}\bigg)^2\bigg(\dfrac{r}{100}+3\bigg)\\\\\\\longrightarrow\sf 93 = P\bigg(\dfrac{10}{100}\bigg)^2\bigg(\dfrac{10}{100}+3\bigg)\\\\\\\longrightarrow\sf 93 = P\bigg(\dfrac{1}{10}\bigg)^2\bigg(\dfrac{10 + 300}{100}\bigg)\\\\\\\longrightarrow\sf 93 = P \times \dfrac{1}{100} \times \dfrac{310}{100}\\\\\\\longrightarrow\sf 93 \times 100 \times \dfrac{100}{310} =P \\\\\\\longrightarrow\sf30 \times 100 =P \\\\\\\longrightarrow\underline{\boxed{\textsf{\textbf{Principal = Rs. 3,000}}}}$