Simple & Compound Interest

Interest वह extra money है जो borrower lender को principal amount (मूलधन) के उपयोग के बदले में देता है।

• Principal (P) — मूलधन: वह मूल राशि जो उधार ली/जमा की जाती है।
• Rate (R) — दर: प्रति वर्ष प्रतिशत ब्याज दर।
• Time (T) — समय: वह अवधि जिसके लिए पैसा उधार लिया/जमा किया जाता है।
• Amount (A) — मिश्रधन: Principal + Interest (A = P + I)

Two Types of Interest:

• Simple Interest (SI) — साधारण ब्याज: Interest केवल Principal पर calculate होता है।
• Compound Interest (CI) — चक्रवृद्धि ब्याज: Interest Principal + accrued interest पर calculate होता है।

Simple Interest वह ब्याज है जो केवल Original Principal पर calculate होता है, न कि पहले accrued interest पर।

Key Points:

• Interest हर साल same रहता है (constant) — जब तक P, R, T same हैं।
• Principal constant रहता है।
• Amount = P + SI

Example:

• Principal = ₹1000, Rate = 10% p.a., Time = 3 years
• SI = (1000 × 10 × 3) / 100 = ₹300
• Amount = 1000 + 300 = ₹1300

• To find P: P = (SI × 100) / (R × T)
• To find R: R = (SI × 100) / (P × T)
• To find T: T = (SI × 100) / (P × R)
• Amount: A = P + SI = P × (1 + RT/100)

Example 1: SI = ₹300, R = 10%, T = 3 years, find P.

• P = (300 × 100) / (10 × 3) = 30000/30 = ₹1000

Example 2: P = ₹2000, R = 8%, T = 2 years, find SI and Amount.

• SI = (2000 × 8 × 2) / 100 = ₹320
• A = 2000 + 320 = ₹2320

Compound Interest वह ब्याज है जो Principal + previously earned interest पर calculate होता है। इसे "interest on interest" भी कहा जाता है।

Key Points:

• Interest हर साल increasing होता है।
• Principal changes each year (P + interest).
• CI हमेशा same time/rate पर SI से greater होता है (for T > 1 year).

Example:

• Principal = ₹1000, Rate = 10% p.a., Time = 3 years
• A = 1000 × (1 + 10/100)³ = 1000 × (1.1)³ = 1000 × 1.331 = ₹1331
• CI = 1331 − 1000 = ₹331
• SI = ₹300 (comparison: CI > SI)

• To find P: P = A / (1 + R/100)ⁿ
• To find R: R = 100 × [(A/P)¹⁄ⁿ − 1]
• To find n: n = log(A/P) / log(1 + R/100)

Example: A = ₹1331, R = 10%, n = 3 years, find P.

• P = 1331 / (1.1)³ = 1331 / 1.331 = ₹1000

Important: For Compound Interest, always calculate Amount first, then CI = A − P.

When interest is compounded more than once in a year:

• Half-Yearly Compounding: Rate = R/2, Time = 2n (n = number of years)
• Quarterly Compounding: Rate = R/4, Time = 4n
• Monthly Compounding: Rate = R/12, Time = 12n

Where m = number of compounding periods per year.

Example 1 (Half-Yearly): P = ₹1000, R = 10% p.a., n = 2 years, compounded half-yearly.

• Rate = 10/2 = 5% per half-year, Time = 2×2 = 4 half-years
• A = 1000 × (1 + 5/100)⁴ = 1000 × (1.05)⁴ = 1000 × 1.2155 = ₹1215.50

Example 2 (Quarterly): P = ₹1000, R = 12% p.a., n = 1 year, compounded quarterly.

• Rate = 12/4 = 3% per quarter, Time = 1×4 = 4 quarters
• A = 1000 × (1 + 3/100)⁴ = 1000 × (1.03)⁴ = ₹1125.51

For 2 years:

For 3 years:

Example: P = ₹1000, R = 10%, T = 2 years

• SI = (1000 × 10 × 2)/100 = ₹200
• CI = 1000 × [(1.1)² − 1] = 1000 × 0.21 = ₹210
• Difference = ₹10
• Formula: 1000 × (10/100)² = 1000 × 0.01 = ₹10 ✓

• Simple Interest: SI = (P × R × T) / 100
• SI Amount: A = P + SI = P × (1 + RT/100)
• Compound Interest (Annual): A = P × (1 + R/100)ⁿ
• CI: CI = A − P = P × [(1 + R/100)ⁿ − 1]
• Half-Yearly: A = P × (1 + R/200)²ⁿ
• Quarterly: A = P × (1 + R/400)⁴ⁿ
• CI − SI (2 years): P × (R/100)²
• CI − SI (3 years): P × (R/100)² × (300 + R)/100

• Trap 1: For T=1 year, SI = CI → True. For T>1, CI > SI.
• Trap 2: Half-yearly compounding: don't forget to adjust both Rate (R/2) and Time (2n).
• Trap 3: CI formula: A = P(1 + R/100)ⁿ — always calculate Amount first, then CI = A − P.
• Trap 4: Difference formula for 2 years: P × (R/100)² — not P × R²/100.
• Trap 5: Simple Interest: Principal remains constant — don't add interest to principal.
• Trap 6: Rate और Time की units match होनी चाहिए — if time is in months, convert to years.

Find the simple interest on ₹5000 at 8% p.a. for 3 years.

• A. ₹1000
• B. ₹1200
• C. ₹1500
• D. ₹1800

Explanation: SI = (5000 × 8 × 3)/100 = ₹1200.

Find the compound interest on ₹5000 at 10% p.a. for 2 years (compounded annually).

• A. ₹1000
• B. ₹1025
• C. ₹1050
• D. ₹1100

Explanation: A = 5000 × (1.1)² = 5000 × 1.21 = ₹6050; CI = 6050 − 5000 = ₹1050.

Find the amount on ₹2000 at 10% p.a. for 1 year, compounded half-yearly.

• A. ₹2100
• B. ₹2150
• C. ₹2200
• D. ₹2205

Explanation: Rate = 10/2 = 5%, Time = 2 half-years; A = 2000 × (1.05)² = 2000 × 1.1025 = ₹2205.

The difference between CI and SI on ₹10000 for 2 years at 10% p.a. is:

• A. ₹50
• B. ₹100
• C. ₹150
• D. ₹200

Explanation: Difference = P × (R/100)² = 10000 × (10/100)² = 10000 × 0.01 = ₹100.

At 10% p.a., the amount after 2 years (compounded annually) is ₹12100. Find the principal.

• A. ₹9000
• B. ₹10000
• C. ₹11000
• D. ₹12000

Explanation: P = A / (1.1)² = 12100 / 1.21 = ₹10000.

Find the amount on ₹10000 at 12% p.a. for 1 year, compounded quarterly.

• A. ₹11200
• B. ₹11250
• C. ₹11255
• D. ₹11300

Explanation: Rate = 12/4 = 3%, Time = 4 quarters; A = 10000 × (1.03)⁴ = 10000 × 1.1255 = ₹11255.

Key Formulae:

• SI = (P×R×T)/100
• A (SI) = P + SI = P(1 + RT/100)
• A (CI) = P(1 + R/100)ⁿ
• CI = A − P

Compounding Periods:

• Half-Yearly: R/2, 2n
• Quarterly: R/4, 4n
• Monthly: R/12, 12n

SI vs CI:

• T=1: SI = CI
• T>1: CI > SI
• T<1: SI > CI

Difference Formulae:

• 2 years: P × (R/100)²
• 3 years: P × (R/100)² × (300+R)/100

Exam Formula:

UGC NET में Simple & Compound Interest के questions formula-based और application-based होते हैं। Compounding periods और SI-CI difference पर विशेष ध्यान दें।