Work out difference between simple interest and compound interest for one year at the rate 8% annual interest on ₹ 5,000.
Model Answer
Step-by-Step Financial Arithmetic Solution:
- Given Parameters: Principal (P) = ₹ 5,000; Annual Interest Rate (R) = 8%; Time period (T) = 1 Year.
- 1. Simple Interest (SI):
SI = (P × R × T) / 100 = (5000 × 8 × 1) / 100 = ₹ 400 - 2. Compound Interest (Compounded Semi-Annually):
- Semi-annual interest rate (r) = 8% / 2 = 4% per half-year.
- Number of conversion periods (n) = 1 × 2 = 2 half-years.
- Maturity Amount (A) = P × (1 + r/100)ⁿ = 5000 × (1.04)² = 5000 × 1.0816 = ₹ 5,408.
- CI = A - P = 5408 - 5000 = ₹ 408.
- Difference between CI and SI:
Difference = CI - SI = ₹ 408 - ₹ 400 = ₹ 8
(Direct Formula: Difference = P × (r/100)² = 5000 × (4/100)² = ₹ 8)
Final Answer: The net difference is ₹ 8.
हिंदी में प्रश्न एवं आदर्श उत्तर
रुपये पाँच हजार पर 8% वार्षिक ब्याज की दर से एक वर्ष के लिए संयोजित होने वाले चक्रवृद्धि ब्याज तथा साधारण ब्याज का अन्तर बताइये।
हल:
- दिए गए मान: मूलधन (P) = ₹ 5,000; वार्षिक दर (R) = 8%; समय (T) = 1 वर्ष।
- 1. साधारण ब्याज (Simple Interest - SI):
SI = (P × R × T) / 100
SI = (5000 × 8 × 1) / 100 = ₹ 400 - 2. चक्रवृद्धि ब्याज (Compound Interest - CI - जब ब्याज अर्धवार्षिक देय हो):
- अर्धवार्षिक दर (r) = 8% / 2 = 4% प्रति छमाही
- अवधियों की संख्या (n) = 1 × 2 = 2 छमाही
- मिश्रधन (A) = P × (1 + r/100)ⁿ
A = 5000 × (1 + 4/100)² = 5000 × (1.04)² = 5000 × 1.0816 = ₹ 5408 - चक्रवृद्धि ब्याज CI = A - P = 5408 - 5000 = ₹ 408
- साधारण ब्याज तथा चक्रवृद्धि ब्याज में अंतर:
अंतर = CI - SI = 408 - 400 = ₹ 8
(वैकल्पिक सूत्र: अंतर = P × (r/100)² = 5000 × (4/100)² = 5000 × 0.0016 = ₹ 8)
उत्तर: साधारण ब्याज और चक्रवृद्धि ब्याज के बीच का अभीष्ट अंतर ₹ 8 है।