Amount = $P(1 + \frac{R}{100})^n \Rightarrow 4410 = 4000(1 + \frac{5}{100})^n \Rightarrow \frac{4410}{4000} = (1.05)^n \Rightarrow 1.1025 = (1.05)^n \Rightarrow (1.05)^2 = (1.05)^n \Rightarrow n=2$ years.

Explanation - Let the principal be P and rate be R%.
$P(1 + \frac{R}{100})^3 = 900$
$P(1 + \frac{R}{100})^4 = 945$
Dividing the two equations:
$1 + \frac{R}{100} = \frac{945}{900} = 1.05$
$\frac{R}{100} = 0.05 \Rightarrow R = 5\%$.

$1331 = 1000(1 + \frac{10}{100})^t \Rightarrow 1.331 = (1.1)^t \Rightarrow (1.1)^3 = (1.1)^t \Rightarrow t = 3$ years.

$A = P(1 + R/100)^T 24200 = 20000(1 + R/100)^2 1.21 = (1 + R/100)^2 1.1 = 1 + R/100 R/100 = 0.1 R = 10\%$

Difference for 2 years $(D) = P(\frac{r}{100})^2 \Rightarrow 75 = P(\frac{5}{100})^2 \Rightarrow 75 = P(\frac{1}{20})^2 \Rightarrow 75 = P(\frac{1}{400}) \Rightarrow P = 30000$ Rupees.

Formula: (n1-1)/t1 = (n2-1)/t2 => (2-1)/10 = (3-1)/T => 1/10 = 2/T => T = 20 years.

Principal = 125000, Rate = 8% per annum = 2% quarterly, Time = 9 months = 3 quarters. Amount = $125000(1 + \frac{2}{100})^3 = 125000 \times (1.02)^3 = 125000 \times 1.061208 = 132651$. Interest = 132651 - 125000 = 7651 Rs.

Let total money be $P$.
$P/3 \times 7\% + P/4 \times 8\% + (P - P/3 - P/4) \times 10\% = 1683$
$P[ 7/300 + 8/400 + (5/12) \times 10/100 ] = 1683 P [ 7/300 + 2/100 + 5/120 ] = 1683 P [ 28 + 24 + 50 / 1200 ] = 1683 P [ 102/1200 ] = 1683 P = (1683 \times 1200) / 102 = 19800$ rupees

Let amount at 8% be x, then at 6% be (2500-x). Interest: $\frac{x \times 8 \times 1}{100} + \frac{(2500-x) \times 6 \times 1}{100} = 180 \Rightarrow 8x + 15000 - 6x = 18000 \Rightarrow 2x = 3000 \Rightarrow x = 1500$ Rs.

Difference = $P \times (\frac{R}{100})^2 = 8690 \times (\frac{8}{100})^2 = 8690 \times \frac{64}{10000} = 8690 \times 0.0064 = 55.616$ Rs. Approx 55.61 Rs.

3 years interest = 776 - 660 = 116. 1 year interest = 116/3. 5 years interest = 193.33. Principal = 660 - 193.33 = 466.67. Rate = $8.28\% \approx 8 \frac{2}{7}\%$.

Rate = 5% semi-annually. Formula: $92610 = 80000(1 + 5/100)^T$. $(21/20)^3 = (21/20)^T$. T = 3 half-years = 1.5 years.

Doubles in 15 yrs. 8 times = $2^3$. So $3 \times 15 = 45$ years.

Let sum be $100x$. Amount after first year $110x$, amount after second year $121x$. Interest for second year $121x - 110x = 11x$. Given $11x = 132 \Rightarrow x = 12$. So, principal = $100 \times 12 = 1200$ Rs.

Let the money deposited in bank A and B be $P_1$ and $P_2$.
Ratio of interest $(R) = 5 : 4 Simple Interest$= \frac{Principal \times Time \times Rate}{100}
\frac{P_1 \times 5 \times \frac{1}{2}}{100} = \frac{P_2 \times 4 \times \frac{1}{2}}{100}
P_1 \times 5 \times \frac{1}{2} = P_2 \times 4 \times \frac{1}{2}
\frac{P_1}{P_2} = \frac{4}{5}
P_1 : P_2 = 4 : 5$

SI = $\frac{3}{5}P$. $SI = \frac{P \times R \times T}{100} \Rightarrow \frac{3}{5}P = \frac{P \times R \times 5}{100} \Rightarrow 0.6 = \frac{5R}{100} \Rightarrow R = \frac{0.6 \times 100}{5} = 12\%$.

Let principal be $x$. Interest = $\frac{x \times 4 \times 8}{100} = 0.32x$. Condition: $x - 0.32x = 340 \Rightarrow 0.68x = 340 \Rightarrow x = \frac{340}{0.68} = 500$ Rs.

Time = 2 years + 4 months + 20 days = 2 + 4/12 + 20/360 = 2 + 1/3 + 1/18 = (36+6+1)/18 = 43/18 years
S.I. = (1800 × 9 × 43) / (100 × 18) = 387 Rupees

Explanation - Amount on 48000 rupees for 3 years at 5% rate = $48000 \times (1 + \frac{5}{100})^3$
(i) Half-yearly: $T = 3$ years, $R = 5\%$. Calculation gives amount = $48000 \times (\frac{105}{100})^3$.
(ii) Calculated every 4 months: $T = 3$ years, $R = 5\%$. Calculation gives amount = $48000 \times (\frac{105}{100})^3$.
(iii) Quarterly: $T = 3$ years, $R = 5\%$. Calculation gives amount = $48000 \times (\frac{105}{100})^3$.
(iv) In option (4), the calculation results differently, hence it is not equal.

Total Interest = 2200. Interest = (P1RT1)/100 + (P2RT2)/100 => 2200 = (5000R2)/100 + (3000R4)/100 => 2200 = 100R + 120R => 2200 = 220R => R = 10%.