Math Problem Statement
The difference between simple interest and compound interest on a principal for 2 years is 76.8. The simple interest on the same principal for 4 years is 3,840. Find the principal and the rate of interest
Solution
Let's solve the problem step-by-step to find the principal (P) and the rate of interest (R).
Step 1: Understanding the Problem
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Simple Interest (SI) for 4 years is given as .
Simple Interest formula: Where:
- is the principal
- is the rate of interest
- is the time period in years
For 4 years, we have:
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Difference between Compound Interest (CI) and Simple Interest (SI) for 2 years is given as .
The formula for Compound Interest compounded annually is:
For 2 years:
The difference between CI and SI for 2 years is: The Simple Interest for 2 years is: So:
Step 2: Solving for Principal (P) and Rate of Interest (R)
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From the Simple Interest equation for 4 years: Simplifying for :
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From the Compound Interest and Simple Interest difference equation:
Let's substitute into the equations and solve for and .The solutions for the principal (P) and the rate of interest (R) are:
- Principal (P) = 12,000
- Rate of Interest (R) = 8%
Verification
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Simple Interest for 4 years: This matches the given simple interest.
-
Difference between Compound Interest and Simple Interest for 2 years: This confirms the given difference.
Thus, the answers are correct.
Would you like more details, or do you have any questions? Here are some related questions for further exploration:
- How would the solution change if the interest was compounded semi-annually?
- What is the formula for compound interest if compounded quarterly?
- How does the difference between simple and compound interest vary with different time periods?
- What are the implications of different compounding frequencies on the effective interest rate?
- How do you calculate continuous compound interest?
Tip: For longer investment durations, compound interest grows significantly compared to simple interest due to compounding effects.
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Math Problem Analysis
Mathematical Concepts
Simple Interest
Compound Interest
Equations
Formulas
Simple Interest formula
Compound Interest formula
Theorems
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Suitable Grade Level
High School
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