A certain sum amounts to Rs. 15,500 in 2 years at 12% p.a. simple interest. If the same sum is compounded half-yearly at 10% per annum for \(1 \frac{1}{2}\) years, what will be the amount received?
Rs. 14,470
Understanding the Problem: Simple and Compound Interest
The question involves two parts related to interest calculations. First, we need to find the original principal sum using information about simple interest. Second, we need to calculate the final amount for the same principal sum but under compound interest, specifically compounded half-yearly.
Step 1: Finding the Principal Sum Using Simple Interest
We are given the amount received after 2 years at 12% per annum simple interest. The formula for the amount (A) in simple interest is:
\(A = P + \text{SI}\)
where P is the Principal and SI is the Simple Interest.
The formula for Simple Interest is:
\(\text{SI} = \frac{P \times R \times T}{100}\)
where R is the rate of interest per annum and T is the time in years.
Substituting the SI formula into the Amount formula:
\(A = P + \frac{P \times R \times T}{100}\)
\(A = P \left(1 + \frac{R \times T}{100}\right)\)
We are given:
- Amount (A) = Rs. 15,500
- Rate (R) = 12% per annum
- Time (T) = 2 years
Now, let's plug these values into the formula to find the Principal (P):
\(15500 = P \left(1 + \frac{12 \times 2}{100}\right)\)
\(15500 = P \left(1 + \frac{24}{100}\right)\)
\(15500 = P \left(1 + 0.24\right)\)
\(15500 = P \times 1.24\)
To find P, we rearrange the equation:
\(P = \frac{15500}{1.24}\)
\(P = 12500\)
So, the principal sum is Rs. 12,500.
Step 2: Calculating the Amount with Compound Interest (Half-Yearly)
Now we need to calculate the amount for the principal sum P = Rs. 12,500 under compound interest. The conditions are:
- Principal (P) = Rs. 12,500
- Annual Rate = 10% per annum
- Time = \(1 \frac{1}{2}\) years
- Compounding frequency: Half-yearly
When interest is compounded half-yearly, we need to adjust the rate and the time period.
- Rate per compounding period (R') = Annual Rate / 2 = 10% / 2 = 5% per half-year.
- Number of compounding periods (n) = Time in years \(\times\) Number of compounding periods per year = \(1.5 \text{ years} \times 2 \text{ periods/year} = 3\) half-years.
The formula for the amount (A) with compound interest is:
\(A = P \left(1 + \frac{R'}{100}\right)^n\)
Plugging in the values:
\(A = 12500 \left(1 + \frac{5}{100}\right)^3\)
\(A = 12500 \left(1 + 0.05\right)^3\)
\(A = 12500 \left(1.05\right)^3\)
Calculate \((1.05)^3\):
\((1.05)^1 = 1.05\)
\((1.05)^2 = 1.05 \times 1.05 = 1.1025\)
\((1.05)^3 = 1.1025 \times 1.05 = 1.157625\)
Now, substitute this back into the amount formula:
\(A = 12500 \times 1.157625\)
\(A = 14470.3125\)
Rounding the amount to the nearest rupee, we get Rs. 14,470.
Therefore, the amount received after \(1 \frac{1}{2}\) years compounded half-yearly at 10% per annum will be approximately Rs. 14,470.
Revision Table: Key Formulas
| Concept | Formula | Notes |
|---|---|---|
| Simple Interest (SI) | \(SI = \frac{P \times R \times T}{100}\) | R = annual rate, T = years |
| Simple Interest Amount | \(A = P \left(1 + \frac{R \times T}{100}\right)\) | A = Amount, P = Principal |
| Compound Interest Amount (Annual) | \(A = P \left(1 + \frac{R}{100}\right)^T\) | R = annual rate, T = years |
| Compound Interest Amount (Half-yearly) | \(A = P \left(1 + \frac{R/2}{100}\right)^{2T}\) | R = annual rate, T = years |
| Compound Interest Amount (General) | \(A = P \left(1 + \frac{R'}{100}\right)^n\) | R' = Rate per period, n = Number of periods |
Additional Information: Simple vs. Compound Interest and Compounding Frequency
Simple Interest: In simple interest, the interest is calculated only on the initial principal amount for the entire duration. The interest earned does not get added back to the principal for calculating future interest.
Compound Interest: In compound interest, the interest earned in each period is added to the principal for calculating the interest in the next period. This leads to earning interest on interest, which results in a higher total amount compared to simple interest over the same period and rate (assuming rate > 0).
Compounding Frequency: The frequency at which interest is compounded affects the final amount. Common frequencies include:
- Annually: Interest is compounded once a year.
- Half-yearly: Interest is compounded twice a year (rate is halved, time is doubled in terms of periods).
- Quarterly: Interest is compounded four times a year (rate is divided by 4, time is multiplied by 4 in terms of periods).
- Monthly: Interest is compounded twelve times a year (rate is divided by 12, time is multiplied by 12 in terms of periods).
Higher compounding frequency generally leads to a slightly higher amount for the same annual rate and time period, due to the effect of earning interest on interest more frequently.
Important Questions from Simple and Compound Both
What is the compound interest (in Rs.) at the rate of 10%, compounded annually, for 3 years on the principal which in 8 years at the rate of 12% per annum gives Rs. 4,800 as simple interest?
The difference in compound interest on a certain sum at 10% p.a. for one year, when the interest is compounded half-yearly and yearly, is Rs. 88.80. What is the simple interest on the same sum for \(1\frac{2}{3}\) years at the same rate?
A sum of Rs. 7500 amounts to Rs. 9075 at 10% p.a, interest being compounded yearly in a certain time. The simple interest (in Rs.) on the same sum for the same time and the same rate is:
A certain sum amounts to Rs.291600 in 2 years and to Rs.314928 in 3 years on compound interest compounded annually. How much will be the simple interest (in Rs.) on Rs.40000 at the same rate for 2 years?
The difference between compound interest and simple interest on x at 15% per annum for 2 years is 9. What is the value of x?
