In how many years will a sum of Rs. 1,25,000 become Rs. 1,57,464 at 16% per annum interest, if the interest is compounded half-yearly?
Compound Interest Calculation: Finding the Time Period
This problem asks us to determine the time it takes for a principal sum to grow to a specific amount under compound interest, compounded half-yearly. Understanding the formula for compound interest and how to apply it for half-yearly compounding is crucial here.
Understanding the Key Terms
- Principal Amount (P): This is the initial sum of money deposited or invested. In this question, the principal amount is Rs. 1,25,000.
- Amount (A): This is the total sum after the interest has been added to the principal. Here, the final amount is Rs. 1,57,464.
- Interest Rate (R): This is the rate at which interest is charged or earned annually. The given annual interest rate is 16% per annum.
- Compounding Frequency: This indicates how often the interest is calculated and added to the principal. In this problem, the interest is compounded half-yearly, meaning twice a year.
- Time (t): This is the duration for which the money is invested or borrowed, which we need to find in years.
Formula for Half-Yearly Compounding
When interest is compounded half-yearly, the annual interest rate is halved, and the number of compounding periods is doubled. The formula used is:
\[A = P \left(1 + \frac{R/2}{100}\right)^{2t}\]
Where:
- \(A\) = Final Amount
- \(P\) = Principal Amount
- \(R\) = Annual Interest Rate (in %)
- \(t\) = Time (in years)
- \(R/2\) = Half-yearly interest rate
- \(2t\) = Total number of half-years (compounding periods)
Step-by-Step Solution
Let's substitute the given values into the compound interest formula:
Given:
- Principal (P) = Rs. 1,25,000
- Amount (A) = Rs. 1,57,464
- Annual Rate (R) = 16%
Since the interest is compounded half-yearly:
- Half-yearly Rate = \(R/2 = 16\% / 2 = 8\%\)
Now, apply the formula:
\[157464 = 125000 \left(1 + \frac{8}{100}\right)^{2t}\]
Simplify the term inside the parenthesis:
\[157464 = 125000 \left(1 + 0.08\right)^{2t}\]
\[157464 = 125000 \left(1.08\right)^{2t}\]
To isolate the term with the exponent, divide both sides by the principal amount (1,25,000):
\[\frac{157464}{125000} = (1.08)^{2t}\]
Perform the division:
\[1.259712 = (1.08)^{2t}\]
Now, we need to find what power of 1.08 equals 1.259712. We can do this by testing powers of 1.08:
- \(1.08^1 = 1.08\)
- \(1.08^2 = 1.08 \times 1.08 = 1.1664\)
- \(1.08^3 = 1.1664 \times 1.08 = 1.259712\)
From the above calculation, we see that \( (1.08)^3 = 1.259712 \). Therefore, we can equate the exponents:
\[2t = 3\]
Finally, solve for \(t\):
\[t = \frac{3}{2}\]
\[t = 1.5 \text{ years}\]
Summary of Values
| Parameter | Value |
|---|---|
| Principal (P) | Rs. 1,25,000 |
| Amount (A) | Rs. 1,57,464 |
| Annual Rate (R) | 16% |
| Half-Yearly Rate | 8% |
| Time (t) | 1.5 years |
Thus, it will take 1.5 years for a sum of Rs. 1,25,000 to become Rs. 1,57,464 at 16% per annum interest, compounded half-yearly.
Important Questions from Miscellaneous
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