Step 1: Identify given parameters.
Let the principal amount be $P$.
The amount $A$ after 10 years will be double the principal, so $A=2P$.
Time period, $T=10$ years.

Step 2: Calculate the Simple Interest (SI).
$$SI=A-P=2P-P=P$$

Step 3: Apply the Simple Interest formula to find the rate ($R$).
$$SI=\frac{P\times R\times T}{100}$$ $$P=\frac{P\times R\times 10}{100}$$ Dividing both sides by $P$ and simplifying: $$1=\frac{R\times 10}{100}$$ $$1=\frac{R}{10}$$ $$R=10\%$$

Step 1: Identify given parameters for compound interest.
Principal $P=12000$
Annual rate $r=10\%$, compounded semi-annually.
Interest rate per period $i=\frac{10\%}{2}=5\%=0.05$
Number of periods $n=1\text{year}\times 2=2$ periods.

Step 2: Apply the Compound Amount formula.
$$A=P{(1+i)}^n$$ $$A=12000{(1+0.05)}^2$$ $$A=12000{\left(1.05\right)}^2$$ $$A=12000\times 1.1025$$ $$A=13230$$

Step 1: Calculate Simple Interest (SI).
$P=50000$, $R=4\%$, $T=2$
$$SI=\frac{P\times R\times T}{100}=\frac{50000\times 4\times 2}{100}=4000$$

Step 2: Calculate Compound Interest (CI).
$i=0.04$, $n=2$
$$A=P{(1+i)}^n=50000{\left(1.04\right)}^2$$ $$A=50000\times 1.0816=54080$$ $$CI=A-P=54080-50000=4080$$

Step 3: Find the difference.
$$Difference=CI-SI=4080-4000=80$$ Alternatively, use the formula for the difference for 2 years: $D=P{\left(\frac{R}{100}\right)}^2=50000\times {\left(0.04\right)}^2=50000\times 0.0016=80$.

Step 1: Identify the variables.
Nominal Rate $r=12\%$ per annum.
Number of compounding periods per year $n=4$ (since it is compounded quarterly).
Interest rate per period $i=\frac{12\%}{4}=3\%=0.03$.

Step 2: Apply the Effective Rate of Interest formula.
$$R_{eff}={\left(1+i\right)}^n-1$$ $$R_{eff}={(1+0.03)}^4-1$$ $$R_{eff}={\left(1.03\right)}^4-1$$ $$R_{eff}\approx 1.1255-1=0.1255$$

Step 1: Calculate the effective rate for the first investment.
Nominal rate = 10% compounded semi-annually.
$i=\frac{0.10}{2}=0.05$, $n=2$.
$$R_{eff1}={(1+0.05)}^2-1={\left(1.05\right)}^2-1$$ $$R_{eff1}=1.1025-1=0.1025=10.25\%$$

Step 2: Calculate the effective rate for the second investment.
Nominal rate = 9.8% compounded quarterly.
$i=\frac{0.098}{4}=0.0245$, $n=4$.
$$R_{eff2}={(1+0.0245)}^4-1$$ Using the given value ${\left(1.0245\right)}^4=1.1017$:
$$R_{eff2}=1.1017-1=0.1017=10.17\%$$

Step 3: Compare both effective rates.
Since $10.25\%>10.17\%$, the first investment yields a higher real return.

Step 1: Identify the type of annuity and parameters.
Since the payment is made at the end of every year, this is an Ordinary Annuity.
Periodic Payment $R=10000$
Interest rate $i=10\%=0.10$
Number of periods $n=3$

Step 2: Apply the Future Value (FV) formula for an Ordinary Annuity.
$$FV=R\times \left[\frac{{(1+i)}^n-1}{i}\right]$$ $$FV=10000\times \left[\frac{{(1+0.10)}^3-1}{0.10}\right]$$ $$FV=10000\times \left[\frac{{\left(1.1\right)}^3-1}{0.10}\right]$$ $$FV=10000\times \left[\frac{1.331-1}{0.10}\right]$$ $$FV=10000\times \frac{0.331}{0.10}=10000\times 3.31=33100$$

Step 1: Identify the type of annuity and parameters.
Since deposits are made at the beginning of every period, this is an Annuity Due.
Periodic Payment $R=5000$
Interest rate $i=\frac{8\%}{2}=4\%=0.04$
Number of periods $n=2$

Step 2: Apply the Future Value (FV) formula for an Annuity Due.
$$FV=R\times \left[\frac{{(1+i)}^n-1}{i}\right]\times (1+i)$$ $$FV=5000\times \left[\frac{{(1+0.04)}^2-1}{0.04}\right]\times (1+0.04)$$ $$FV=5000\times \left[\frac{{\left(1.04\right)}^2-1}{0.04}\right]\times 1.04$$ $$FV=5000\times \left[\frac{1.0816-1}{0.04}\right]\times 1.04$$ $$FV=5000\times \left[\frac{0.0816}{0.04}\right]\times 1.04$$ $$FV=5000\times 2.04\times 1.04=10608$$

Step 1: Identify the type of annuity and parameters.
The payments are made at the end of the period, so it is an Ordinary Annuity.
The original loan amount is the Present Value (PV) of this annuity.
Periodic Payment $R=6000$
Interest rate $i=\frac{10\%}{2}=5\%=0.05$
Number of periods $n=1.5\text{years}\times 2=3$

Step 2: Apply the Present Value (PV) formula.
$$PV=R\times \left[\frac{1-{(1+i)}^{-n}}{i}\right]$$ $$PV=6000\times \left[\frac{1-{(1+0.05)}^{-3}}{0.05}\right]$$ $$PV=6000\times \left[\frac{1-{\left(1.05\right)}^{-3}}{0.05}\right]$$ $$PV=6000\times \left[\frac{1-0.8638376}{0.05}\right]$$ $$PV=6000\times \left[\frac{0.1361624}{0.05}\right]$$ $$PV=6000\times 2.723248\approx 16339.49$$ (Note: Depending on rounding during intermediate steps, the value evaluates to approximately $16339.20$)

Step 1: Evaluate the cost of the Cash Option.
The cash down payment is directly given as $₹25,000$.

Step 2: Evaluate the Present Value of the Instalment Option.
The instalments form an Ordinary Annuity since they are paid at the end of each year.
Periodic Payment $R=9000$
Interest rate $i=6\%=0.06$
Number of periods $n=3$
$$PV=R\times \left[\frac{1-{(1+i)}^{-n}}{i}\right]$$ $$PV=9000\times \left[\frac{1-{\left(1.06\right)}^{-3}}{0.06}\right]$$ Using ${\left(1.06\right)}^{-3}\approx 0.839619$ : $$PV=9000\times \left[\frac{1-0.839619}{0.06}\right]$$ $$PV=9000\times \left[\frac{0.160381}{0.06}\right]$$ $$PV=9000\times 2.6730=24057.15$$ (Depending on rounding, the Present Value is approximately $₹24,057$)

Step 3: Compare both options.
The Present Value of the instalment option ($₹24,057$) is less than the cash price ($₹25,000$).
Difference = $25000-24057=943$

Step 1: Identify the type of annuity and parameters.
Since the money is needed at the start of each year, this is an Annuity Due.
The amount to be invested today represents the Present Value (PV) of the annuity.
Periodic Payment $R=50000$
Interest rate $i=5\%=0.05$
Number of periods $n=2$

Step 2: Apply the Present Value (PV) formula for an Annuity Due.
$$PV=R\times \left[\frac{1-{(1+i)}^{-n}}{i}\right]\times (1+i)$$ $$PV=50000\times \left[\frac{1-{\left(1.05\right)}^{-2}}{0.05}\right]\times 1.05$$ First, find ${\left(1.05\right)}^{-2}\approx 0.907029$: $$PV=50000\times \left[\frac{1-0.907029}{0.05}\right]\times 1.05$$ $$PV=50000\times \left[\frac{0.092971}{0.05}\right]\times 1.05$$ $$PV=50000\times 1.85942\times 1.05=97619.55$$ (Alternative direct calculation: Invest $50000$ today for the first year + $\frac{50000}{1.05}=47619.05$ for the second year. Total = $50000+47619.05=97619.05$. Rounding gives approximately $₹97,618.50$)

Let the principal be $P$. The amount $A$ after 8 years is $2P$.
Simple Interest, $SI=A-P=2P-P=P$.
Using the SI formula: $SI=\frac{P\times R\times T}{100}$
$$P=\frac{P\times R\times 8}{100}$$ $$1=\frac{8R}{100}\Rightarrow R=\frac{100}{8}=12.5\%$$

The effective interest rate formula is $R_{eff}={\left(1+\frac{r}{n}\right)}^n-1$.
As the compounding frequency $n$ increases, the value of ${\left(1+\frac{r}{n}\right)}^n$ mathematically approaches $e^r$, which means the total yield (effective rate) increases.

By definition, an Annuity Due consists of periodic payments that are made at the beginning of each payment period.

$P=1000$, Nominal rate $r=10\%$.
Compounded half-yearly, so $i=\frac{10\%}{2}=5\%=0.05$ and $n=1\times 2=2$.
$$A=P{(1+i)}^n=1000{\left(1.05\right)}^2=1000\times 1.1025=1102.50$$ $$CI=A-P=1102.50-1000=102.50$$

Nominal rate $r=6\%$. Compounded semi-annually, so $i=\frac{6\%}{2}=3\%=0.03$ and $n=2$.
$$R_{eff}={(1+i)}^n-1$$ $$R_{eff}={\left(1.03\right)}^2-1=1.0609-1=0.0609=6.09\%$$

Using the Simple Interest formula: $SI=\frac{P\times R\times T}{100}$
$$SI=\frac{5000\times 6\times 3}{100}=50\times 18=900$$

The effective rate is greater than the nominal rate when compounding happens more than once a year. When compounding happens exactly once a year (Annually), the nominal rate equals the effective rate.

Step 1: Identify the parameters.
Since it is an ordinary annuity, payments are at the end of each year.
$R=5000$, $i=10\%=0.10$, $n=3$.

Step 2: Apply the FV formula for an ordinary annuity.
$$FV=R\times \left[\frac{{(1+i)}^n-1}{i}\right]$$ $$FV=5000\times \left[\frac{{(1+0.10)}^3-1}{0.10}\right]$$ $$FV=5000\times \left[\frac{1.331-1}{0.10}\right]$$ $$FV=5000\times \frac{0.331}{0.10}=5000\times 3.31=16550$$

Step 1: Use the difference formula for 2 years.
The difference $D$ between CI and SI for 2 years is given by the formula:
$$D=P{\left(\frac{R}{100}\right)}^2$$ Where $D=100$ and $R=10\%$.

Step 2: Substitute and solve for $P$.
$$100=P{\left(\frac{10}{100}\right)}^2$$ $$100=P{\left(0.1\right)}^2$$ $$100=P\times 0.01$$ $$P=\frac{100}{0.01}=10000$$

Step 1: Identify the type of annuity and parameters.
It is an Annuity Due, so payments are at the beginning of each year.
$R=8000$, $i=5\%=0.05$, $n=4$.
Given value: ${\left(1.05\right)}^{-4}=0.8227$.

Step 2: Apply the PV formula for an annuity due.
$$PV=R\times \left[\frac{1-{(1+i)}^{-n}}{i}\right]\times (1+i)$$ $$PV=8000\times \left[\frac{1-{\left(1.05\right)}^{-4}}{0.05}\right]\times 1.05$$ $$PV=8000\times \left[\frac{1-0.8227}{0.05}\right]\times 1.05$$ $$PV=8000\times \left[\frac{0.1773}{0.05}\right]\times 1.05$$ $$PV=8000\times 3.546\times 1.05=29786.40$$ (Note: Using exact value of ${1.05}^{-4}$ yields approximately $₹29,785.56$)

Step 1: Calculate the repayment amount for Bank P (Simple Interest).
$P=50000$, $R=10\%$, $T=2$ years.
$$SI=\frac{P\times R\times T}{100}=\frac{50000\times 10\times 2}{100}=10000$$ Total repayment to Bank P = $50000+10000=60000$.
Extra charge = $₹10,000$.

Step 2: Calculate the repayment amount for Bank Q (Compound Interest).
$P=50000$, $i=10\%=0.10$, $n=2$ years.
$$A=P{(1+i)}^n=50000{(1+0.10)}^2$$ $$A=50000{\left(1.1\right)}^2=50000\times 1.21=60500$$ Total repayment to Bank Q = $60500$.
Extra charge = $60500-50000=10500$.

Step 3: Compare and conclude.
Bank P requires a total repayment of $₹60,000$ and charges $₹10,000$ extra.
Bank Q requires a total repayment of $₹60,500$ and charges $₹10,500$ extra.
Therefore, Bank P is cheaper.

Step 1: Calculate Effective Rate for Bank A.
Nominal rate $r=10\%$ compounded semi-annually. $i=\frac{10\%}{2}=5\%=0.05$, $n=2$.
$$R_{effA}={(1+0.05)}^2-1={\left(1.05\right)}^2-1$$ $$R_{effA}=1.1025-1=0.1025=10.25\%$$

Step 2: Calculate Effective Rate for Bank B.
Nominal rate $r=10.25\%$ compounded annually. $i=10.25\%=0.1025$, $n=1$.
$$R_{effB}={(1+0.1025)}^1-1=0.1025=10.25\%$$

Step 3: Compare the rates.
Both banks provide exactly the same effective rate of $10.25\%$. Difference = $0$.

Step 1: Identify the parameters.
Principal $P=200000$.
Nominal rate $r=18\%$ compounded monthly. $i=\frac{18\%}{12}=1.5\%=0.015$.
Total periods $n=2\times 12=24$.

(i) Effective Annual Rate of Interest:
$$R_{eff}={(1+i)}^{12}-1={\left(1.015\right)}^{12}-1$$ $$R_{eff}=1.1956-1=0.1956=19.56\%$$

(ii) Total amount repays at the end of 2 years:
$$A=P{(1+i)}^n=200000\times {\left(1.015\right)}^{24}$$ $$A=200000\times 1.4258=285160$$

(iii) Total interest paid:
$$CI=A-P=285160-200000=85160$$

Part 1: Interest earned after 3 years.
It is an ordinary annuity with $R=50000$, $i=10\%=0.10$, $n=3$.
$$FV=R\times \left[\frac{{(1+i)}^n-1}{i}\right]$$ $$FV=50000\times \left[\frac{{\left(1.1\right)}^3-1}{0.10}\right]$$ $$FV=50000\times \left[\frac{1.331-1}{0.10}\right]=50000\times 3.31=165500$$ Total principal invested in 3 years = $50000\times 3=150000$.
Interest earned = $FV-\text{Total Invested}=165500-150000=15500$.

Part 2: Annual investment for a target of $₹50,00,000$ by age 50.
Target $FV=5000000$.
Number of years $n=50-30=20$.
We need to find the annual payment $R$.
$$5000000=R\times \left[\frac{{\left(1.1\right)}^{20}-1}{0.10}\right]$$ $$5000000=R\times \left[\frac{6.7275-1}{0.10}\right]$$ $$5000000=R\times 57.275$$ $$R=\frac{5000000}{57.275}\approx 87301.61$$

(i) Calculate the amount under PPF (Option A) after 15 years.
$P=500000$, $i=7.1\%=0.071$, $n=15$.
$$A=P{(1+i)}^n=500000{\left(1.071\right)}^{15}$$ $$A=500000\times 2.8059\approx 1402950$$

(ii) Calculate the amount under FD (Option B) after 15 years.
$P=500000$, $r=7\%$, compounded quarterly.
$i=\frac{7\%}{4}=1.75\%=0.0175$, $n=15\times 4=60$.
$$A=P{(1+i)}^n=500000{\left(1.0175\right)}^{60}$$ $$A=500000\times 2.8327\approx 1416350$$

(iii) Which option gives higher returns and by how much?
Option B yields $₹14,16,350$ while Option A yields $₹14,02,950$.
Difference = $1416350-1402950=13400$.
Therefore, Option B (SBI FD) gives higher returns by $₹13,400$.

(iv) Calculate the Effective Annual Rate for SBI FD with quarterly compounding.
$$R_{eff}={\left(1+0.0175\right)}^4-1$$ $$R_{eff}={\left(1.0175\right)}^4-1=1.07186-1=0.07186=7.19\%$$

(i) Nominal interest rate for Option A:
The stated annual rate for Option A is exactly the nominal rate: $12\%$.

(ii) Compounding frequency in Option B:
Compounded "semi-annually" means it happens twice a year. So, $2$ times.

(iii) Effective Annual Rate for Option A:
Nominal rate $r=12\%$ compounded quarterly. $i=\frac{12\%}{4}=3\%=0.03$, $n=4$.
$$R_{effA}={(1+0.03)}^4-1=1.1255-1=0.1255=12.55\%$$

(iv) Effective Annual Rate for Option B:
Nominal rate $r=12.2\%$ compounded semi-annually. $i=\frac{12.2\%}{2}=6.1\%=0.061$, $n=2$.
$$R_{effB}={(1+0.061)}^2-1=1.125721-1=0.1257=12.57\%$$ Option A has a lower effective rate, thus it is cheaper.

(i) Total interest to Lender A:
$P=1000000$, $R=10\%$, $T=4$ years.
$$SI=\frac{1000000\times 10\times 4}{100}=400000$$

(ii) Values for Lender B:
$P=1000000$, $n=4$, $i=9\%=0.09$.

(iii) Total Amount for Lender B:
$$A=P{(1+i)}^n=1000000{(1+0.09)}^4$$ $$A=1000000\times 1.4116=1411600$$

(iv) Compare the two options:
Interest paid to Lender B = $1411600-1000000=411600$.
Interest paid to Lender A = $400000$.
Lender A is cheaper. The difference is $411600-400000=11600$.

Because interest accumulates periodically inside the year (e.g., quarterly, monthly), you earn interest on interest. Thus, the effective rate—reflecting the actual yearly growth—will always equal (if compounded annually) or exceed the nominal rate. Reason (R) directly explains why Assertion (A) is correct.

Assertion (A) is true because for the first year, no previous interest has been accumulated to generate "interest on interest". Hence, SI and CI are equal for year 1.
Reason (R) is also true as a general statement about SI. However, (R) does not explain (A). The correct explanation for (A) is that in the first year under CI, there is no accumulated interest yet to apply the compounding effect.

Assertion (A) is false. Doubling the time period more than doubles the Compound Interest because CI grows exponentially, not linearly (as SI does).
Reason (R) is true; Compound Interest is indeed an exponential function.

Assertion (A) is true because an annuity aggregates recurring payments. Increasing the periods adds more payments to the account.
Reason (R) perfectly explains this phenomenon: a greater number of periods means larger total principal deposits as well as extra time for compound interest to accumulate.