What is the principal amount which earns Rs. 210 as compound interest for the second year at 5% per annum?
Rs. 4000
Finding the Principal Amount from Second Year Compound Interest
This question asks us to find the initial principal amount that generates a specific amount of compound interest specifically in the second year at a given annual interest rate. Let's break down how compound interest works and how to use the information provided to find the principal.
Understanding Compound Interest for the Second Year
Compound interest means that the interest earned in the first year is added to the principal, and then in the second year, interest is calculated on this new, larger amount. The interest earned in the second year is the interest on the original principal plus the interest on the interest earned in the first year.
- In the first year, the interest earned is simple interest on the original principal.
- At the end of the first year, the amount is Principal + Interest of the first year.
- In the second year, the interest is calculated on the amount at the end of the first year.
- The compound interest for the second year is the difference between the amount at the end of the second year and the amount at the end of the first year. Alternatively, it is the simple interest calculated on the amount at the end of the first year for one year.
Step-by-Step Calculation
Let the principal amount be \( P \).
The annual rate of interest is \( R = 5\% \).
Step 1: Calculate Interest for the First Year (I1)
The interest for the first year is simple interest on the principal \( P \) at \( 5\% \) for 1 year.
\( I_1 = P \times \frac{R}{100} \times 1 \)
\( I_1 = P \times \frac{5}{100} \times 1 = 0.05P \)
Step 2: Calculate the Amount at the end of the First Year
This is the principal plus the interest earned in the first year.
Amount after 1 year \( = P + I_1 = P + 0.05P = 1.05P \)
Step 3: Calculate Interest for the Second Year (I2)
The interest for the second year is the simple interest on the amount at the end of the first year (\( 1.05P \)) at \( 5\% \) for 1 year.
\( I_2 = (1.05P) \times \frac{R}{100} \times 1 \)
\( I_2 = (1.05P) \times \frac{5}{100} \times 1 = 1.05P \times 0.05 \)
\( I_2 = 0.0525P \)
Step 4: Use the Given Information to Find P
We are given that the compound interest for the second year is Rs. 210.
So, \( I_2 = 210 \).
We have the equation: \( 0.0525P = 210 \)
Step 5: Solve for P
To find \( P \), divide 210 by 0.0525.
\( P = \frac{210}{0.0525} \)
To make the division easier, we can multiply the numerator and denominator by 10000 to remove the decimal:
\( P = \frac{210 \times 10000}{0.0525 \times 10000} = \frac{2100000}{525} \)
Now, simplify the fraction. We can divide both numerator and denominator by common factors. For example, divide by 25:
\( 2100000 \div 25 = 84000 \)
\( 525 \div 25 = 21 \)
So, \( P = \frac{84000}{21} \)
Now, divide 84000 by 21:
\( P = \frac{84}{21} \times 1000 = 4 \times 1000 = 4000 \)
The principal amount is Rs. 4000.
Let's verify this:
| Year | Starting Amount | Interest (5%) | Ending Amount |
|---|---|---|---|
| 1 | Rs. 4000 | \( 4000 \times \frac{5}{100} = \) Rs. 200 | \( 4000 + 200 = \) Rs. 4200 |
| 2 | Rs. 4200 | \( 4200 \times \frac{5}{100} = \) Rs. 210 | \( 4200 + 210 = \) Rs. 4410 |
The interest earned in the second year is indeed Rs. 210, which matches the information given in the question. Therefore, the principal amount is Rs. 4000.
Revision Table: Key Concepts in Compound Interest
| Concept | Explanation | Formula (P=Principal, R=Rate, n=Time) |
|---|---|---|
| Simple Interest (SI) | Interest calculated only on the principal amount. | \( SI = \frac{P \times R \times n}{100} \) |
| Compound Interest (CI) | Interest calculated on the principal and accumulated interest from previous periods. | \( \text{Amount (A)} = P \left(1 + \frac{R}{100}\right)^n \) \( CI = A - P \) |
| CI for a Specific Year (k) | Interest earned only during the k-th year. Calculated as Amount after k years - Amount after (k-1) years. | \( CI_{\text{year k}} = P \left(1 + \frac{R}{100}\right)^k - P \left(1 + \frac{R}{100}\right)^{k-1} \) |
Additional Information: Compound vs. Simple Interest
It's important to understand the difference between simple and compound interest. Simple interest is a fixed amount calculated yearly on the original principal. Compound interest grows faster because the interest earned is reinvested, earning interest itself. In this problem, the interest for the second year is higher than the interest for the first year (Rs. 210 vs Rs. 200, if P=4000) because it includes interest on the first year's interest. This compounding effect is key to financial growth over time.
Similar Questions
A sum of money at 20% rate of compound interest per annum becomes more than 100 times in n years. What is the least value of n? (Use log10 2 = 0.301, log10 3 = 0.477)
- If C is the compound interest on Rs. 10,000 for one year at 4% per annum when compounded quarterly, then which one of the following is correct ?
The rate of interest on two different schemes is the same and it is 20%. But in one of the schemes, the interest is compounded half-yearly and in the other, the interest is compounded annually. Equal amounts are invested in the schemes. If the difference of the returns after 2 years is Rs. 482, then what is the principal amount in each scheme?
What is the least number of complete years in which a sum of money put out at 40% annual compound interest will be more than tripled?
A merchant commences with a certain capital and gains annually at the rate of 25%. At the end of 3 years he has Rs. 10,000. What is the original amount that the merchant invested?
A sum of money compounded annually doubles itself in 5 years. In how many years will it become four times of itself ?
A person borrowed Rs. 10,000 at 12% rate of interest per annum compounded quarterly for a period of 9 months. What is the interest paid by him to settle his account after 9 months?
Important Questions from Compound Interest
A person borrowed Rs. 10000 on compound interest at the rate of 40 percent per annum. If the interest is compounded half yearly, then what will be the amount to be paid after 1.5 years?
The difference between the compound interest (compounding annually) and the simple interest on a sum of money at the rate of 40 per cent per annum for 2 years is Rs. 2400. What is the amount?
In how many years will a sum of Rs.1875 amount to Rs.2187 at 8 percent p.a. compound interest?
A sum of money has increased by 45% in 9 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?
At a certain rate of compound interest a certain sum amounts to Rs. 64800 in 4 years and Rs. 93312 in 6 years. What is the compound interest earned in fifth year?
