Simple and Compound Interest: Practice Questions & Notes

What is Interest?

Interest is the extra money paid by a borrower to a lender for using their money over a period of time. It is calculated on the principal amount (initial sum borrowed or invested).

There are two primary types of interest:

Simple Interest (SI): Interest calculated on the original principal throughout the period.
Compound Interest (CI): Interest calculated on the principal plus the accumulated interest of previous periods.

Core Concepts

Principal (P)

The original sum of money invested or borrowed.

Time (T)

The duration for which the money is lent or borrowed (typically in years).

Rate of Interest (R)

The annual percentage of interest charged or earned.

Simple Interest (SI)

Formula

\text{SI} = \frac{P \times R \times T}{100}

Total Amount (A)

A = P + \text{SI} = P \left(1 + \frac{R \times T}{100}\right)

Compound Interest (CI)

Formula (Compounded Annually)

A = P \left(1 + \frac{R}{100}\right)^T

\text{CI} = A - P = P \left( \left(1 + \frac{R}{100} \right)^T - 1 \right)

Compounded Half-Yearly

A = P \left(1 + \frac{R}{2 \times 100}\right)^{2T}

Compounded Quarterly

A = P \left(1 + \frac{R}{4 \times 100}\right)^{4T}

Key Differences: SI vs. CI

Feature	Simple Interest (SI)	Compound Interest (CI)
Interest On	Only Principal	Principal + Accumulated Interest
Growth	Linear	Exponential
Formula	$\frac{P \times R \times T}{100}$	$P \left(1 + \frac{R}{100} \right)^T$
Use Cases	Short-term loans, education loans	Bank savings, investments, EMIs

Advanced Formulas

CI for Two Years (Shortcut)

If $R = r\%$ , $T = 2$ years:

\text{CI} = P \left( \frac{2R}{100} + \frac{R^2}{100^2} \right)

Difference Between CI and SI for 2 Years

\text{CI} - \text{SI} = P \times \left( \frac{R}{100} \right)^2

Conceptual Tips & Tricks

When rate and time are the same, CI will always be greater than SI.
For 1 year, SI = CI (because no interest is yet compounded).
Use compounding frequency adjustments only when explicitly stated.
When interest is compounded more frequently, the effective rate increases.

Effective Annual Rate (EAR)

To compare two compound interests with different compounding frequencies:

\text{EAR} = \left(1 + \frac{R}{n \times 100} \right)^{n} - 1

Multiply EAR by 100 to convert to percentage.

Visual Intuition

Imagine stacking blocks:

In SI: Same size blocks stacked one over another (interest constant).
In CI: Each new block is slightly bigger (interest grows on itself).

Common Mistakes to Avoid

Mistake	Correction
Assuming SI = CI always	Only true for 1 year
Using wrong compounding formula	Use frequency-specific version
Forgetting to subtract P when calculating CI	CI = A − P
Not converting months/quarters to years	Time (T) must always be in years

Examples

Example 1: Simple Interest

Q: What is the SI on ₹10,000 at 8% per annum for 3 years?

Solution:

\text{SI} = \frac{10000 \times 8 \times 3}{100} = \text{Rs.~}2400

Example 2: Compound Interest (Annually)

Q: Find CI on ₹8000 at 10% for 2 years.

Solution:

A = 8000 \left(1 + \frac{10}{100}\right)^2 = 8000 \times 1.21 = \text{Rs.~}9680 \\ \text{CI} = 9680 - 8000 = \text{Rs.~}1680

Example 3: CI Compounded Half-Yearly

Q: Find CI on ₹5000 at 8% p.a. for 1 year, compounded half-yearly.

Solution:

A = 5000 \left(1 + \frac{8}{2 \times 100}\right)^2 = 5000 \left(1.04\right)^2 = 5000 \times 1.0816 = \text{Rs.~}5408 \\ \text{CI} = 5408 - 5000 = \text{Rs.~}408

Example 4: Difference Between SI and CI

Q: Find the difference between CI and SI on ₹4000 at 10% for 2 years.

Solution:

\text{Difference} = 4000 \left( \frac{10}{100} \right)^2 = 4000 \times \frac{1}{100} = \text{Rs.~}40

Example 5: Finding Rate

Q: CI on ₹6250 for 2 years is ₹650. Find the rate.

Solution:

Let $R = r$ , and use the difference formula:

\text{CI} - \text{SI} = 6250 \left( \frac{r}{100} \right)^2 = 650 - \frac{6250 \times r \times 2}{100} \Rightarrow \text{Solve the quadratic}