Interest (Simple and Compound) | Mathematics Class 8 ICSE PDF Download

Introduction

Interest is the additional money paid or earned when borrowing or lending money. This chapter introduces the concepts of simple and compound interest, explaining key terms like principal, rate, time, and amount. It covers how to calculate interest using formulas, understand time periods, and differentiate between simple and compound interest, including calculations for half-yearly compounding. The notes provide clear, step-by-step explanations to help understand these financial concepts.

Review (Simple and Compound)

• Principal (P): The original amount of money borrowed or lent.
• Interest (I): The extra money paid by the borrower to the lender for using the borrowed money. Simple interest is referred to as S.I.
• Rate (R): The percentage of interest charged or earned per ₹100.
  • If the rate is 12% per year, it means ₹12 interest on ₹100 for one year.
  • If the rate is 2% per month, it means ₹2 interest on ₹100 for one month.
• Time (T): The duration for which money is borrowed or lent.
  • If the rate is per year (e.g., 8% per year), time must be in years.
  • If the rate is per month (e.g., 1.5% per month), time must be in months.
• Amount (A): The total of the principal and the interest earned.
  • Formula: A = P + I
• Factors affecting Interest: Interest depends on:
  • Principal (P)
  • Rate percent (R)
  • Time (T)
• Simple Interest Formula:
  • I = (P × R × T) / 100
  • Steps to calculate:
    1. Identify the principal (P), rate (R), and time (T).
    2. Multiply P, R, and T, then divide by 100 to find the interest (I).
    3. To find the amount, add interest to the principal: A = P + I.
• Amount Formula:
  • A = P + I 
  • A = P + (P × R × T) / 100
  • Simplified: A = P(1 + (RT / 100))
• Example: Calculate simple interest on ₹1300 at 7.5% per year for 146 days.
  • Step 1: P = ₹1300, R = 7.5% = 15/2, T = 146/365 years = 2/5 years.
  • Step 2: I = (1300 × 15/2 × 2/5) / 100 = ₹39.
  • Step 3: Amount = P + I = 1300 + 39 = ₹1339.
• Calculating Time (T):
  • Exclude the borrowing date but include the repayment date.
  • Example: From December 23 to May 18:
    • December: 31 − 23 = 8 days
    • January: 31 days
    • February: 28 days
    • March: 31 days
    • April: 30 days
    • May: 18 days
    • Total: 146 days = 146/365 years.

To Find the Principal (P), Rate Percent (R), and Time (T)

Formulas Derived from Simple Interest

• Principal: P = (100 × I) / (R × T)
• Rate: R = (100 × I) / (P × T)
• Time: T = (100 × I) / (P × R)

Steps to Find Principal:

• Identify the interest (I), rate (R), and time (T).
• Use P = (100 × I) / (R × T) to calculate the principal.
• Example: Find the principal if interest is ₹1480 in 2 years at 10% per year.
  • Step 1: I = ₹1480, T = 2 years, R = 10%.
  • Step 2: P = (100 × 1480) / (10 × 2) = ₹7400.

Steps to Find Rate:

• Identify the interest (I), principal (P), and time (T).
• Use R = (100 × I) / (P × T) to calculate the rate.
• Example: Find the rate if interest is ₹6 on ₹100 in 8 months.
  • Step 1: I = ₹6, P = ₹100, T = 8/12 = 2/3 years.
  • Step 2: R = (6 × 100) / (100 × 2/3) = 9% per year.

Steps to Find Time:

• Identify the interest (I), principal (P), and rate (R).
• Use T = (100 × I) / (P × R) to calculate the time.
• Example: Find the time if ₹950 produces ₹399 interest at 7% per year.
  • Step 1: I = ₹399, P = ₹950, R = 7%.
  • Step 2: T = (100 × 399) / (950 × 7) = 6 years.

Compound Interest (By Simple Interest Method)

• Definition: In compound interest, the interest earned each period is added to the principal, and the new principal earns interest in the next period.
• Formula: Compound Interest (C.I.) = Final Amount − Original Principal (C.I. = A − P).

Steps to Calculate:

• For each year, calculate interest using I = (P × R × T) / 100, with T = 1 year.
• Add interest to the principal to get the amount: A = P + I.
• Use the amount as the new principal for the next year.
• Repeat for the given number of years.
• Subtract the original principal from the final amount to find C.I.
• Example: Calculate compound interest on ₹6000 for 2 years at 10% per year.
  • Year 1: P = ₹6000, R = 10%, T = 1 year.
  • Interest = (6000 × 10 × 1) / 100 = ₹600.
  • Amount = 6000 + 600 = ₹6600.
  • Year 2: P = ₹6600, R = 10%, T = 1 year.
  • Interest = (6600 × 10 × 1) / 100 = ₹660.
  • Amount = 6600 + 660 = ₹7260.
  • C.I. = 7260 − 6000 = ₹1260.

Difference from Simple Interest:

• In simple interest, the principal stays the same each year.
• In compound interest, the principal increases each year as interest is added.
• This leads to higher interest in compound interest compared to simple interest.
• Example: Compare simple and compound interest on ₹8000 at 5% for 3 years.
  • Simple Interest: I = (8000 × 5 × 3) / 100 = ₹1200, Amount = 8000 + 1200 = ₹9200.
  • Compound Interest:
    • Year 1: I = (8000 × 5 × 1) / 100 = ₹400, A = 8000 + 400 = ₹8400.
    • Year 2: I = (8400 × 5 × 1) / 100 = ₹420, A = 8400 + 420 = ₹8820.
    • Year 3: I = (8820 × 5 × 1) / 100 = ₹441, A = 8820 + 441 = ₹9261.
    • C.I. = 9261 − 8000 = ₹1261.
  • Difference: ₹1261 − ₹1200 = ₹61.

Interest Compounded Half-Yearly

Concept: Interest is calculated every six months, with the interest added to the principal each half-year.
Steps to Calculate:

• Divide the annual rate by 2 for the half-yearly rate.
• Calculate interest for 6 months using I = (P × R/2 × 1/2) / 100.
• Add interest to the principal to get the new amount.
• Repeat for each half-year period.
• Subtract the original principal from the final amount to find C.I.

Example: Calculate compound interest on ₹8000 for 1 year at 10% per year, compounded half-yearly.

• First half-year: P = ₹8000, R = 10/2 = 5%, T = 1/2 year.
  • Interest = (8000 × 10 × 1) / (100 × 2) = ₹400.
  • Amount = 8000 + 400 = ₹8400.
• Second half-year: P = ₹8400, R = 5%, T = 1/2 year.
  • Interest = (8400 × 10 × 1) / (100 × 2) = ₹420.
  • Amount = 8400 + 420 = ₹8820.
• C.I. = 8820 − 8000 = ₹820.

Using Formulae

Compound Interest Formula (Yearly)

• A = P(1 + r/100)ⁿ, where A is the amount, P is the principal, r is the rate per year, and n is the number of years.
• C.I. = A − P.
• Example: Find the amount and C.I. on ₹16000 for 3 years at 10% per year.
  • Step 1: P = ₹16000, r = 10%, n = 3.
  • Step 2: A = 16000 × (1 + 10/100)³ = 16000 × (11/10)³ = ₹21296.
  • Step 3: C.I. = 21296 − 16000 = ₹5296.

Compound Interest Formula (Half-Yearly)

• A = P(1 + (r/2)/100)^{(n × 2)}.
• C.I. = A − P.
• Example: Find the amount and C.I. on ₹8000 for 1.5 years at 10% per year, compounded half-yearly.
  • Step 1: P = ₹8000, r = 10%, n = 1.5, so n × 2 = 3 periods.
  • Step 2: A = 8000 × (1 + 10/(2 × 100))³ = 8000 × (21/20)³ = ₹9261.
  • Step 3: C.I. = 9261 − 8000 = ₹1261.

Different Rates for Successive Years:

• A = P(1 + R₁/100)(1 + R₂/100)(1 + R₃/100)..., where R₁, R₂, R₃ are rates for each year.
• C.I. = A − P.
• Example: Find the amount and C.I. on ₹12000 for 3 years at 10%, 12%, and 15% for successive years.
  • Step 1: P = ₹12000, R₁ = 10%, R₂ = 12%, R₃ = 15%.
  • Step 2: A = 12000 × (1 + 10/100) × (1 + 12/100) × (1 + 15/100) = 12000 × (11/10) × (28/25) × (23/20) = ₹17001.60.
  • Step 3: C.I. = 17001.60 − 12000 = ₹5001.60.