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The difference between the compound interest and the simple interest accrued on an amount of Rs. 18,000 in 2 years was Rs. 405. What was the rate of interest p.c.p.a. ?
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The Problem:
The problem states that the difference between the compound interest and the simple interest accrued on an amount of Rs. 18,000 in 2 years was Rs. 405. We are asked to find the rate of interest per annum.
Understanding the Concepts:
Before we proceed, let's understand the concepts of simple interest and compound interest.
Simple Interest (SI):
Simple interest is calculated based on the initial principal amount and the rate of interest for a specified period of time.
Compound Interest (CI):
Compound interest is calculated based on the initial principal amount and the rate of interest for a specified period of time, but with the addition of interest earned in previous periods.
Approach:
To solve this problem, we can use the following formulae:
- Simple Interest (SI) = (Principal * Rate * Time) / 100
- Compound Interest (CI) = Principal * (1 + Rate/100) ^ Time - Principal
We are given the difference between compound interest and simple interest, which is Rs. 405. This can be expressed as CI - SI = 405.
Substituting the formulae, we get:
Principal * (1 + Rate/100) ^ Time - Principal - Principal * Rate * Time / 100 = 405.
Solving the Equation:
Let's solve the equation step by step to find the rate of interest.
Step 1: Substitute the given values in the equation.
18000 * (1 + Rate/100) ^ 2 - 18000 - 18000 * Rate * 2 / 100 = 405.
Step 2: Simplify the equation.
18000 * (1 + Rate/100) ^ 2 - 18000 - 360 * Rate = 405.
Step 3: Expand and rearrange the equation.
18000 * (1 + Rate/100) ^ 2 - 18000 = 405 + 360 * Rate.
Step 4: Simplify further.
(1 + Rate/100) ^ 2 - 1 = (405 + 360 * Rate) / 18000.
Step 5: Expand and simplify the equation again.
(1 + 2 * Rate/100 + Rate^2/10000) - 1 = (405 + 360 * Rate) / 18000.
Step 6: Simplify and rearrange the equation.
Rate^2/10000 + 2 * Rate/100 = (405 + 360 * Rate) / 18000.
Step 7: Multiply both sides of the equation by 10000.
Rate^2 + 200 * Rate = (405 + 360 * Rate) * 100.
Step 8: Simplify and rearrange the equation.
Rate^2 + 200 * Rate - 36000 * Rate - 40500 = 0.
Step 9: Combine like terms.
Rate^2 - 35800 * Rate - 40500 = 0.
Step 10: Solve the quadratic equation.
Using the quadratic formula, we can find the values of Rate.
Conclusion:
By solving the quadratic equation, we can find the rate of interest per annum.
The problem states that the difference between the compound interest and the simple interest accrued on an amount of Rs. 18,000 in 2 years was Rs. 405. We are asked to find the rate of interest per annum.
Understanding the Concepts:
Before we proceed, let's understand the concepts of simple interest and compound interest.
Simple Interest (SI):
Simple interest is calculated based on the initial principal amount and the rate of interest for a specified period of time.
Compound Interest (CI):
Compound interest is calculated based on the initial principal amount and the rate of interest for a specified period of time, but with the addition of interest earned in previous periods.
Approach:
To solve this problem, we can use the following formulae:
- Simple Interest (SI) = (Principal * Rate * Time) / 100
- Compound Interest (CI) = Principal * (1 + Rate/100) ^ Time - Principal
We are given the difference between compound interest and simple interest, which is Rs. 405. This can be expressed as CI - SI = 405.
Substituting the formulae, we get:
Principal * (1 + Rate/100) ^ Time - Principal - Principal * Rate * Time / 100 = 405.
Solving the Equation:
Let's solve the equation step by step to find the rate of interest.
Step 1: Substitute the given values in the equation.
18000 * (1 + Rate/100) ^ 2 - 18000 - 18000 * Rate * 2 / 100 = 405.
Step 2: Simplify the equation.
18000 * (1 + Rate/100) ^ 2 - 18000 - 360 * Rate = 405.
Step 3: Expand and rearrange the equation.
18000 * (1 + Rate/100) ^ 2 - 18000 = 405 + 360 * Rate.
Step 4: Simplify further.
(1 + Rate/100) ^ 2 - 1 = (405 + 360 * Rate) / 18000.
Step 5: Expand and simplify the equation again.
(1 + 2 * Rate/100 + Rate^2/10000) - 1 = (405 + 360 * Rate) / 18000.
Step 6: Simplify and rearrange the equation.
Rate^2/10000 + 2 * Rate/100 = (405 + 360 * Rate) / 18000.
Step 7: Multiply both sides of the equation by 10000.
Rate^2 + 200 * Rate = (405 + 360 * Rate) * 100.
Step 8: Simplify and rearrange the equation.
Rate^2 + 200 * Rate - 36000 * Rate - 40500 = 0.
Step 9: Combine like terms.
Rate^2 - 35800 * Rate - 40500 = 0.
Step 10: Solve the quadratic equation.
Using the quadratic formula, we can find the values of Rate.
Conclusion:
By solving the quadratic equation, we can find the rate of interest per annum.
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The difference between the compound interest and the simple interest accrued on an amount of Rs. 18,000 in 2 years was Rs. 405. What was the rate of interest p.c.p.a. ?
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The difference between the compound interest and the simple interest accrued on an amount of Rs. 18,000 in 2 years was Rs. 405. What was the rate of interest p.c.p.a. ? for Class 8 2024 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about The difference between the compound interest and the simple interest accrued on an amount of Rs. 18,000 in 2 years was Rs. 405. What was the rate of interest p.c.p.a. ? covers all topics & solutions for Class 8 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The difference between the compound interest and the simple interest accrued on an amount of Rs. 18,000 in 2 years was Rs. 405. What was the rate of interest p.c.p.a. ?.
The difference between the compound interest and the simple interest accrued on an amount of Rs. 18,000 in 2 years was Rs. 405. What was the rate of interest p.c.p.a. ? for Class 8 2024 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about The difference between the compound interest and the simple interest accrued on an amount of Rs. 18,000 in 2 years was Rs. 405. What was the rate of interest p.c.p.a. ? covers all topics & solutions for Class 8 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The difference between the compound interest and the simple interest accrued on an amount of Rs. 18,000 in 2 years was Rs. 405. What was the rate of interest p.c.p.a. ?.
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