RRB Group D Maths Compound Interest By Manoj Sir

Understanding Compound Interest is crucial for competitive exams like RRB Group D. This concept involves interest earning interest, leading to significant growth over time. This lecture demystifies Compound Interest (CI) by explaining its core principles and clearly differentiating it from Simple Interest (SI), equipping you with essential problem-solving techniques.

Simple Interest vs. Compound Interest

The fundamental difference between Simple Interest and Compound Interest lies in how the principal is treated over time.

• Simple Interest (SI): In SI, the principal remains constant throughout the entire period. Interest is calculated only on the original principal amount for each time period.

• Compound Interest (CI): In CI, the principal does not remain constant. The interest earned in one period is added to the principal of the previous period to form the new principal for the next period. This is the concept of interest on interest. (Memory Tip: As the instructor says, "बाप ना भैया और सबसे बड़ा रुपैया" – "Neither father nor brother, money is the biggest of all," emphasizing how money itself earns more money relentlessly.)

Also Read: RRB Group D Maths Syllabus

Traditional Formula for Amount

While other efficient methods exist, understanding the traditional formula for calculating the final amount in compound interest is essential.

The formula is:

A = P (1 + R/100)ⁿ

Where:

• A = Amount (मिश्रधन): The total sum after adding interest (Principal + Interest).

• P = Principal (मूलधन): The initial sum of money.

• R = Rate (दर): The rate of interest per annum.

• n or T = Time (समय): The duration in years.

The Compound Interest (CI) can then be found by subtracting the principal from the amount:

CI = Amount – Principal

Example 1: Multiple Calculation Methods

Let's find the Compound Interest on a Principal of ₹1000 at a Rate of 10% per annum for a Time of 2 years.

Method 1: Using the Formula

1. Formula: A = P(1 + R/100)ⁿ

2. Substitute values: A = 1000 * (1 + 10/100)²

3. Simplify: A = 1000 * (11/10)² = 1000 * (11/10) * (11/10)

4. Calculate Amount: A = 10 * 121 = ₹1210.

5. Calculate CI: CI = Amount - Principal = 1210 - 1000 = ₹210.

Method 2: The Tree Method (or "Jhumka Method")

This method visually breaks down the interest calculation year by year. It depends on the Principal.

1. Principal: ₹1000. Rate: 10% (Fractional value = 1/10).

2. Year 1 Interest: ₹1000 * (1/10) = ₹100.

3. Year 2 Interest:

• Interest on Principal: ₹1000 * (1/10) = ₹100.

• Interest on Year 1's Interest: ₹100 * (1/10) = ₹10.

4. Total CI: Sum of all interest components = 100 (Year 1) + 100 (Year 2 on P) + 10 (Year 2 on I) = ₹210.

Method 3: The Ratio/Fraction Method

This method works with the rate of interest.

1. Rate: 10% = 1/10.

2. This fraction means for a Principal of ₹10, there is ₹1 of Interest, making the Amount ₹11.

3. Ratio (Principal : Amount): 10 : 11

4. For 2 years, we square the ratio: 10² : 11² → 100 : 121.

5. This implies a Principal of ₹100 yields an Amount of ₹121, so the CI is ₹21 (121 - 100).

6. Unitary Method:

• Our actual Principal is ₹1000.

• If 100 units = ₹1000, then 1 unit = ₹10.

• CI = 21 units = 21 * 10 = ₹210.

Method 4: The Shortcut for 2 Years (2:1 Rule)

This is a direct trick for calculating 2-year CI.

1. Step 1: Calculate interest on the principal: ₹1000 * (1/10) = ₹100.

2. Step 2: Calculate interest on the result from Step 1: ₹100 * (1/10) = ₹10.

3. Apply the 2:1 Rule: (2 * Step 1 Result) + (1 * Step 2 Result)

• (2 * 100) + (1 * 10) = 200 + 10 = ₹210.

The Tree Method or the Ratio/Fraction Method are highly recommended for solving problems quickly and accurately.

Example 2: Using Fractional Rates

Problem: P = ₹3600, R = 16 ⅔ %, T = 2 years. Find the CI.

1. Convert Rate to Fraction: 16 ⅔ % = (50/3)% = 50 / (3 * 100) = 1/6.

Solution (Tree Method)

1. Principal: ₹3600.

2. Year 1 Interest: ₹3600 * (1/6) = ₹600.

3. Year 2 Interest:

• On Principal: ₹3600 * (1/6) = ₹600.

• On Year 1's Interest: ₹600 * (1/6) = ₹100.

4. Total CI: 600 + 600 + 100 = ₹1300.

Solution (Ratio Method)

1. Ratio (P:A): For a rate of 1/6, the ratio is 6 : 7.

2. For 2 years: 6² : 7² → 36 : 49.

3. CI in units: 49 - 36 = 13 units.

4. Unitary Method:

• Principal is 36 units = ₹3600.

• 1 unit = ₹100.

• CI = 13 units = 13 * 100 = ₹1300.

Further Examples with Different Rates

The same methods apply consistently for various rates.

Calculating CI for Fractional Time Periods

When the time period is not a whole number (e.g., 1 year 6 months), calculate the interest for the next full integer year using the Tree Method, and then take the required fraction of the interest for that final year.

Problem 1: P = ₹5000, R = 20% (1/5), T = 1 year 6 months.

1. Calculate CI for 2 full years using the Tree Method:

• Year 1 Interest: ₹5000 * (1/5) = ₹1000.

• Year 2 Interest:

• On Principal: ₹5000 * (1/5) = ₹1000.

• On Year 1's Interest: ₹1000 * (1/5) = ₹200.

• Total interest accumulated during Year 2 = 1000 + 200 = ₹1200.

2. Calculate CI for the required period:

• CI for 1st year: ₹1000 (complete).

• CI for the next 6 months: We need half of the second year's interest.

• (6 months / 12 months) * ₹1200 = ½ * ₹1200 = ₹600.

3. Total CI: 1000 (from Year 1) + 600 (from Year 2's fraction) = ₹1600.

Problem 2: P=₹4800, R=12.5% (1/8), T=1 year 4 months

1. Calculate interest for Year 1: ₹4800 * (1/8) = ₹600.

2. Calculate total interest for Year 2 (if it were full):

• On Principal: ₹4800 * (1/8) = ₹600.

• On Year 1's Interest: ₹600 * (1/8) = ₹75.

• Total interest for Year 2 = 600 + 75 = ₹675.

3. Calculate CI for the required period:

• CI for 1st Year: ₹600.

• CI for next 4 months: (4/12) * ₹675 = (1/3) * ₹675 = ₹225.

4. Total CI: 600 + 225 = ₹825.

Calculating the Difference Between CI and SI

The Tree Method simplifies finding the difference between Compound and Simple Interest for 2 years.

Problem: P = ₹8100, R = 11 ¹/₉ % (1/9), T = 2 years. Find CI - SI.

1. Construct the Tree:

• Year 1 Interest: ₹8100 * (1/9) = ₹900.

• Year 2 Interest:

• On Principal: ₹8100 * (1/9) = ₹900.

• On Year 1's Interest: ₹900 * (1/9) = ₹100.

2. Identify SI and CI:

• Simple Interest (SI) is the sum of interest calculated only on the principal: 900 + 900 = ₹1800.

• Compound Interest (CI) is the sum of all interest components: 900 + 900 + 100 = ₹1900.

3. Find the Difference:

• CI - SI = 1900 - 1800 = ₹100.

Key Insight: In the Tree Method, the first line of interest components (all from principal) represents the Simple Interest. All subsequent interest components (the "interest on interest," which are the branches off the initial interest amounts) directly represent the difference between CI and SI. In this case, the ₹100 calculated on the first year's interest is the exact difference.