Simple and Compound Interest

Overview

Simple and Compound Interest form a crucial quantitative topic in MP TET, appearing regularly in the Mathematics section. These concepts test your ability to calculate the cost of borrowing money or the returns on savings over time. Understanding the distinction between SI and CI is essential—SI grows linearly while CI grows exponentially due to "interest on interest."

For the exam, you must be comfortable with direct formula application, conversion between SI and CI problems, and word problems involving loans, deposits, and instalments. Questions typically involve 2-3 year periods and may require you to find principal, rate, time, or the difference between SI and CI. Mastery of this topic also builds foundation for profit-loss and percentage problems.

Key Concepts

• **Principal (P)**: The original sum of money borrowed or invested before any interest is added.

• **Rate of Interest (R)**: The percentage charged or earned per unit time, usually expressed as "per annum" (per year).

• **Time (T or n)**: The duration for which money is borrowed or invested, typically in years.

• **Simple Interest (SI)**: Interest calculated only on the original principal throughout the entire period. The interest amount remains constant each year.

• **Compound Interest (CI)**: Interest calculated on principal plus accumulated interest from previous periods. Each year's interest is added to principal, creating a snowball effect.

• **Amount (A)**: The total money at the end of the period = Principal + Interest earned.

• **Compounding Frequency**: CI can be compounded annually, half-yearly (twice a year), quarterly (four times a year), or monthly. More frequent compounding yields higher returns.

• **CI always exceeds SI** for the same principal, rate, and time (when time > 1 year), because CI earns interest on previously earned interest.

Formulas / Key Facts

**Simple Interest:**

• SI = (P × R × T) / 100
• Amount = P + SI = P(1 + RT/100)

**Compound Interest:**

• Amount = P(1 + R/100)ⁿ where n = number of years
• CI = Amount − P = P[(1 + R/100)ⁿ − 1]

**Half-yearly Compounding:**

• Rate becomes R/2, Time becomes 2n
• Amount = P(1 + R/200)²ⁿ

**Quarterly Compounding:**

• Rate becomes R/4, Time becomes 4n
• Amount = P(1 + R/400)⁴ⁿ

**Difference between CI and SI (2 years):**

• CI − SI = P(R/100)² = SI × R / (2 × 100)

**Difference between CI and SI (3 years):**

• CI − SI = P(R/100)² × (3 + R/100)

**When Amount doubles:**

• For SI: T = 100/R years
• For CI: Use (1 + R/100)ⁿ = 2

**Effective Rate for CI:**

• For 2 years: Effective rate = 2R + R²/100

Worked Examples

**Example 1: Basic SI Calculation**

*Find the simple interest on Rs 8000 at 12% per annum for 3 years.*

Solution:

• P = 8000, R = 12%, T = 3 years
• SI = (P × R × T) / 100
• SI = (8000 × 12 × 3) / 100
• SI = 288000 / 100 = Rs 2880
• Amount = 8000 + 2880 = Rs 10880

**Example 2: CI Calculation**

*Find the compound interest on Rs 5000 at 10% per annum for 2 years, compounded annually.*

Solution:

• P = 5000, R = 10%, n = 2
• Amount = P(1 + R/100)ⁿ
• Amount = 5000(1 + 10/100)²
• Amount = 5000 × (1.1)²
• Amount = 5000 × 1.21 = Rs 6050
• CI = 6050 − 5000 = Rs 1050

**Example 3: Difference between CI and SI**

*The difference between CI and SI on a certain sum at 5% per annum for 2 years is Rs 15. Find the sum.*

Solution:

• For 2 years: CI − SI = P(R/100)²
• 15 = P × (5/100)²
• 15 = P × (1/20)²
• 15 = P × 1/400
• P = 15 × 400 = Rs 6000

**Example 4: Half-yearly Compounding**

*Find CI on Rs 10000 at 8% per annum for 1 year, compounded half-yearly.*

Solution:

• P = 10000, R = 8%, Time = 1 year
• For half-yearly: Rate = 8/2 = 4%, n = 2 half-years
• Amount = 10000(1 + 4/100)²
• Amount = 10000 × (1.04)²
• Amount = 10000 × 1.0816 = Rs 10816
• CI = 10816 − 10000 = Rs 816

Common Mistakes

• **Confusing SI and CI formulas**: Students apply the CI formula when SI is asked or vice versa. → Always identify whether the problem states "simple" or "compound" interest explicitly.

• **Forgetting to adjust rate and time for non-annual compounding**: When interest is compounded half-yearly or quarterly, students use annual rate directly. → Divide rate by compounding frequency and multiply time by the same frequency.

• **Calculating CI instead of Amount**: Some questions ask for total amount, but students give only CI (or vice versa). → Read the question carefully—Amount = P + CI, not just CI.

• **Using the 2-year shortcut formula for 3 years**: The formula CI − SI = P(R/100)² works only for 2 years. → For 3 years, use CI − SI = P(R/100)²(3 + R/100).

• **Percentage to decimal conversion errors**: Writing 5% as 5 instead of 5/100 in formulas. → Always divide the rate by 100 when substituting in formulas.

Quick Reference

• SI = PRT/100; CI = P[(1 + R/100)ⁿ − 1]

• For 2 years: CI − SI = P(R/100)² — this is a direct exam shortcut

• Half-yearly compounding: Halve the rate, double the time periods

• CI > SI always when n > 1 year (same P, R, T)

• To find time when sum doubles at SI: T = 100/R years

• Amount in CI = P(1 + R/100)ⁿ — memorise for n = 2 and 3