Solution

\[ \begin{aligned} & \color{magenta}\textbf{Given}: \\[6pt] P &= \text{₹ }5000 \\[6pt] \text{R} &= \frac{16\%}{2} \implies 8\% \ \text{(Half-yearly)} \\[6pt] \text{T} & = 1\dfrac{1}{2} \text{(years)} \implies 3 \ \text{(Half years)} \end{aligned} \] \[ \begin{aligned} & \color{magenta}\textbf{First Half-year} \\[6pt] P &= \text{₹ }5000,\quad R = 8\%,\quad T = 1 \\[6pt] \text{S.I.} &= \frac{P \times R \times T}{100} \\[6pt] &= \frac{5000 \times 8 \times 1}{100} \\[6pt] &= 50 \times 8 \\[6pt] &= \text{₹ }400 \\[6pt] \text{Amount} &= 5000 + 400 \\ &= \text{₹ }5400 \\ \\ & \color{magenta}\textbf{Second Half-year} \\[6pt] P &= \text{₹ }5400 \\[6pt] \text{S.I.} &= \frac{5400 \times 8 \times 1}{100} \\[6pt] &= 54 \times 8 \\[6pt] &= \text{₹ }432 \\[6pt] \text{Amount} &= 5400 + 432 \\[6pt] &= \text{₹ }5832 \\ \\ & \color{magenta}\textbf{Third Half-year} \\[6pt] P &= \text{₹ }5832 \\[6pt] \text{S.I.} &= \frac{5832 \times 8 \times 1}{100} \\[6pt] &= 58.32 \times 8 \\[6pt] &= \text{₹ }466.56 \\[6pt] \text{Amount} &= 5832 + 466.56 \\ &= \text{₹ }6298.56 \end{aligned} \] \[ \begin{aligned} \text{Compound Interest } &= \text{Amount} - \text{Principal} \\ &= 6298.56 - 5000 \\ &= \text{₹ }1298.56 \end{aligned} \] \[ \begin{aligned} & \color{brown} \boxed{\textbf{Method - } 2} \\[6pt] P &= \text{₹ }5000 \\[6pt] R &= 16 \% \\[6pt] n &= 1\frac{1}{2} \text{ years } \implies \frac{3}{2} \text{ years } \\[6pt] & \color{magenta} \textbf{CI - half yearly} \\[6pt] A &= P\left(1 + \frac{R}{200}\right)^{2n} \\[6pt] &= 5000\left(1 + \frac{16}{200}\right)^{2 \times \frac{3}{2}} \\[6pt] &= 5000\left( \frac{200+16}{200}\right)^{3} \\[6pt] &= 5000\left( \frac{216}{200}\right)^{3} \\[6pt] &= 5000\left(\frac{108}{100}\right)^3 \\[6pt] &= \cancel{5000}^{\cancel{\ 50}^1} \times \frac{\cancel{108}^{\ 54}}{\cancel{100}_1} \times \frac{108}{\cancel{100}_{\cancel 2_1}} \times \frac{108}{100} \\[6pt] &= \frac{54 \times 108 \times 108}{100} \\[6pt] &= \frac{629856}{100} \\[6pt] A &= 6298.56 \\[8pt] \text{C.I.} &= A - P \\ &= 6298.56 - 5000 \\ &= \text{₹ }1298.56 \end{aligned} \]

Answer Compound Interest \(= \color{red}{\text{₹ }1{,}298.56}\)

Solution

\[ \begin{aligned} & \color{magenta}\textbf{Given}: \\[6pt] P &= \text{₹ }15625 \\[6pt] \text{R} &= \frac{16\%}{4} \implies 4\% \\[6pt] \text{T} &= 9 \text{ months} \implies 3 \text{ quarters} \end{aligned} \] \[ \begin{aligned} & \color{magenta}\textbf{First Quarter} \\[6pt] P &= \text{₹ }15625,\ R=4\%,\ T=1 \\[6pt] \text{S.I.} &= \frac{15625 \times 4 \times 1}{100} \\[6pt] &= 156.25 \times 4 \\ &=\text{₹ }625 \\[6pt] \text{Amount} &= 15625+625 \\ & =\text{₹ }16250 \\ \\ & \color{magenta}\textbf{Second Quarter} \\[6pt] P &= \text{₹ }16250 \\[6pt] \text{S.I.} &= \frac{16250 \times 4\times 1}{100} \\[6pt] &= 162.50 \times 4 \\ &=\text{₹ }650 \\[6pt] \text{Amount} &= 16250+650\\ &=\text{₹ }16900 \\ \\ & \color{magenta}\textbf{Third Quarter} \\[6pt] P &= \text{₹ }16900 \\[6pt] \text{S.I.} &= \frac{16900\times4\times1}{100} \\[6pt] &= 169 \times 4 \\ &=\text{₹ }676 \\[6pt] \text{Amount} &= 16900+676 \\ &=\text{₹ }17576 \end{aligned} \] \[ \begin{aligned} \text{Compound Interest} &= \text{Amount} - \text{Principal} \\ &= 17576-15625\\ &=\text{₹ }1951 \end{aligned} \]\[ \begin{aligned} & \color{brown} \boxed{\textbf{Method - } 2} \\[6pt] P &= \text{₹ }15625 \\[6pt] R &= 16 \% \\[6pt] n &= 9 \text{ months } =\frac{9}{12} \implies \frac{3}{4} \ years \\[6pt] & \color{magenta} \textbf{CI - quarterly} \\[6pt] A &= P\left(1 + \frac{R}{400}\right)^{4n} \\[6pt] &= 15625\left(1 + \frac{16}{400}\right)^{4 \times \frac{3}{4}} \\[6pt] &= 15625\left(1 + \frac{4}{100}\right)^3 \\[6pt] &= 15625\left(\frac{104}{100}\right)^3 \\[6pt] &= 15625\left(\frac{26}{25}\right)^3 \\[6pt] &= 15625 \times \frac{17576}{15625} \\[6pt] &= 17576 \\[8pt] \text{C.I.} &= A - P \\ &= 17576 - 15625 \\ &= \text{₹ }1951 \end{aligned} \]

Answer Compound Interest for nine months \(= \color{red}{\text{₹ }1{,}951}\)

Solution

\[ \begin{aligned} & \color{magenta}\textbf{Given}: \\[6pt] P &= \text{₹ }10000 \\[6pt] R &= \frac{12\%}{4} \implies 3\% \\[6pt] T &= 6 \text{ months } \implies 2 \text{ quarters} \end{aligned} \] \[ \begin{aligned} & \color{magenta}\textbf{First Quarter} \\[6pt] P &= \text{₹ }10000 \\[6pt] \text{S.I.} &= \frac{10000\times3\times1}{100} \\[6pt] &= 100 \times 3 \\ &= \text{₹ }300 \\[6pt] \text{Amount} &= 10000+300 \\ &=\text{₹ }10300 \\ \\ & \color{magenta}\textbf{Second Quarter} \\[6pt] P &= \text{₹ }10300 \\[6pt] \text{S.I.} &= \frac{10300\times3\times1}{100} \\[6pt] &= 103 \times 3 \\ &=\text{₹ }309 \\[6pt] \text{Amount} &= 10300+309 \\ &=\text{₹ }10609 \end{aligned} \] \[ \begin{aligned} & \color{brown} \boxed{\textbf{Method - } 2} \\[6pt] P &= \text{₹ }10000 \\[6pt] R &= 12 \% \\[6pt] n &= 6 \text{ months } = \frac{6}{12} \implies \frac{1}{2} \text{ years} \\[6pt] & \color{magenta} \textbf{CI - quarterly} \\[6pt] A &= P\left(1 + \frac{R}{400}\right)^{4n} \\[6pt] A &= P\left(1 + \frac{12}{400}\right)^{4 \times \frac{1}{2}} \\[6pt] &= 10000\left(1 + \frac{3}{100}\right)^2 \\[4pt] &= 10000\left(\frac{103}{100}\right)^2 \\[6pt] &= 10000 \times \frac{103 \times 103}{100 \times 100} \\[4pt] &= \frac{\cancel{10000} \times 10609}{\cancel{10000}} \\[4pt] &= 10609 \end{aligned} \]

Answer Amount on maturity \(= \color{red}{\text{₹ }10{,}609}\)

Solution

\[ \begin{aligned} & \color{magenta}\textbf{Case I: Half-yearly} \\[6pt] P &= \text{₹ }25000 \\[6pt] R & =\frac{16\%}{2} \implies 8\% \\[6pt] T&= 6 \text{ months} \implies 1 \text{ (Half year)} \\[6pt] \text{S.I.} &= \frac{25000\times8\times1}{100} \\[6pt] &=\text{₹ }2000 \\[6pt] \text{Amount} &= 25000+2000 \\ &=\text{₹ }27000 \\[6pt] \text{C.I.} & = 27000 - 25000 \\ &= \color{green} \text{₹ }2000 \end{aligned} \]\[ \begin{aligned} & \color{magenta}\textbf{Case II: Quarterly} \\[6pt] P &= \text{₹ }25000 \\[6pt] R & =\frac{16\%}{4} \implies 4\% \\[6pt] T & = 6 \text{ months} \implies 2 \text{ quarters} \\ \\ &\color{magenta}{\text{First Quarter:}} \\[6pt] \text{S.I.} &=\frac{25000 \times 4 \times 1}{100} \\[6pt] &= 250 \times 4 \\ &=\text{₹ }1000 \\[6pt] \text{Amount} &= 25000 + 1000 \\ & = \text{₹ }26000 \\ \\ & \color{magenta}{\text{Second Quarter:}} \\[6pt] \text{S.I.} & =\frac{26000 \times 4 \times 1}{100} \\[6pt] &= 260 \times 4 \\ &=\text{₹ }1040 \\[6pt] \text{Amount} &= 26000 + 1040 \\ &= \text{₹ }27040 \\[6pt] \text{C.I.} &= 27040 - 25000 \\ & = \color{green} \text{₹ }2040 \end{aligned} \]\[ \begin{aligned} \text{Difference in C.I.} &= 2040-2000 \\ &=\text{₹ }40 \end{aligned} \]

Case I: Half-yearly compounding

\[ \begin{aligned} & \color{brown} \boxed{\textbf{Method - } 2} \\[6pt] P &= \text{₹ }25000 \\[6pt] R &= 16 \% \\[6pt] n &= 6 \text{ months } = \frac{6}{12} \implies \frac{1}{2} \text{ years} \\[6pt] & \color{magenta} \textbf{CI - half yearly} \\[6pt] A &= P\left(1 + \frac{R}{200}\right)^{2n} \\[6pt] &= 25000\left(1 + \frac{16}{200}\right)^{2 \times \frac{1}{2}} \\[6pt] &= 25000\left(1 + \frac{2}{25}\right)^1 \\[6pt] &= 25000\left(\frac{25 +2}{25}\right) \\[6pt] &= 25000 \times \frac{27}{25} \\[4pt] &= 1000 \times 27 \\[4pt] &= 27000 \\[4pt] \text{C.I.} &= A - P \\ &= 27000 - 25000 \\ &= \text{₹ }2000 \end{aligned} \]

Case II: Quarterly compounding

\[ \begin{aligned} P &= \text{₹ }25000 \\ R &= 16 \% \\ n &= 6 \text{ months } = \frac{6}{12} \implies \frac{1}{2} \text{ years} \\[6pt] & \color{magenta} \textbf{CI - quarterly} \\[6pt] A &= P\left(1 + \frac{R}{400}\right)^{4n} \\[6pt] &= 25000\left(1 + \frac{16}{400}\right)^{4 \times \frac{1}{2}} \\[4pt] &= 25000\left(1 + \frac{4}{100}\right)^2 \\[4pt] &= 25000\left(1 + \frac{1}{25}\right)^2 \\[4pt] &= 25000\left(\frac{26}{25}\right)^2 \\[6pt] &= 25000 \times \frac{26 \times 26}{25 \times 25} \\[4pt] &= \frac{\cancel{25000}^{1000} \times 676}{\cancel{25} \times 25} \\[4pt] &= \frac{\cancel{1000}^{\ 40} \times 676}{\cancel{25}} \\[4pt] &= 40 \times 676 \\[4pt] &= 27040 \\[4pt] \text{C.I.} &= A - P \\ &= 27040 - 25000 \\ &= \text{₹ }2040 \end{aligned} \]\[ \begin{aligned} \text{Difference in C.I.} &= 2040 - 2000 \\ &= \text{₹ }40 \end{aligned} \]

Answer Difference in \( C.I = \color{red}{\text{₹}40}. \) Quarterly compounding is better as it gives more interest.

Solution

\[ \begin{aligned} & \color{magenta}\textbf{Given}: \\[6pt] P &= \text{₹ }25000 \\[6pt] R &= \frac{20\%}{2} \implies 10\% \\[6pt] T &= 1\dfrac{1}{2}\text{ years } \implies 3 \text{ (half-years)} \end{aligned} \] \[ \begin{aligned} & \color{magenta}\textbf{First Half-year} \\[6pt] \text{S.I.} &= \frac{25000 \times 10 \times 1}{100} \\[6pt] & = 250 \times 10 \\ & =\text{₹ }2500 \\[6pt] \text{Amount} &= 25000 + 2500 \\ & = \text{₹ }27500 \\ \\ & \color{magenta}\textbf{Second Half-year} \\[6pt] \text{S.I.} &= \frac{27500\times10}{100} \\[6pt] &= 275 \times 10 \\ &=\text{₹ }2750 \\[6pt] \text{Amount} &= 27500 + 2750 \\ & = \text{₹ }30250 \\ \\ & \color{magenta}\textbf{Third Half-year} \\[6pt] \text{S.I.} &= \frac{30250\times10}{100} \\[6pt] & =\text{₹ }3025 \\[6pt] \text{Amount}& = 30250 + 3025 \\ & = \text{₹ }33275 \\ \\ \end{aligned} \]\[ \begin{aligned} & \color{brown} \boxed{\textbf{Method - } 2} \\[6pt] P &= \text{₹ }25000 \\[6pt] R &= 20 \% \\[6pt] n &= 1\dfrac{1}{2} \text{ years } \implies \frac{3}{2} \text{ years } \\[6pt] & \color{magenta} \textbf{CI - half yearly} \\[6pt] A &= P\left(1 + \frac{R}{200}\right)^{2n} \\[6pt] &= 25000\left(1 + \frac{20}{200}\right)^{2 \times \frac{3}{2}} \\[6pt] &= 25000\left(1 + \frac{1}{10}\right)^{3} \\[6pt] &= 25000\left(\frac{10 + 1}{10}\right)^{3} \\[6pt] &= 25000\left(\frac{11}{10}\right)^3 \\[6pt] &= 25000 \times \frac{1331}{1000} \\[6pt] &= 25 \times 1331 \\[4pt] &= 33275 \end{aligned} \]

Answer Amount to clear the debt \(= \color{red}{\text{₹ }33{,}275}\)