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?
4
Understanding Compound Interest and Investment Growth
This question asks about the least number of complete years required for a sum of money to grow to more than three times its original amount when invested at a 40% annual compound interest rate.
Let's break down the problem using the principles of compound interest.
Compound Interest Formula
The formula for compound interest is:
\(A = P(1 + r)^n\)
Where:
- \(A\) is the final amount after \(n\) years.
- \(P\) is the principal amount (initial investment).
- \(r\) is the annual interest rate (as a decimal).
- \(n\) is the number of years.
Setting up the Problem
We are given that the annual interest rate \(r\) is 40%, which as a decimal is \(r = 40/100 = 0.4\).
The condition is that the sum of money will be "more than tripled". This means the final amount \(A\) must be strictly greater than three times the principal amount \(P\).
So, we need to find the smallest integer \(n\) such that:
\(A > 3P\)
Solving the Inequality
Substitute the compound interest formula into the inequality:
\(P(1 + r)^n > 3P\)
Assuming the principal amount \(P\) is positive (which it must be for an investment), we can divide both sides of the inequality by \(P\) without changing the direction of the inequality sign:
\((1 + r)^n > 3\)
Now, substitute the value of \(r = 0.4\):
\((1 + 0.4)^n > 3\)
\((1.4)^n > 3\)
Calculating for Different Years (n)
We need to find the smallest whole number \(n\) for which \((1.4)^n\) is greater than 3. Let's test values for \(n\) starting from 1:
| Number of Years (\(n\)) | Calculation of \((1.4)^n\) | Result (\((1.4)^n\)) | Is Result > 3? |
|---|---|---|---|
| 1 | \((1.4)^1\) | 1.4 | No (1.4 is not > 3) |
| 2 | \((1.4)^2 = 1.4 \times 1.4\) | 1.96 | No (1.96 is not > 3) |
| 3 | \((1.4)^3 = 1.96 \times 1.4\) | 2.744 | No (2.744 is not > 3) |
| 4 | \((1.4)^4 = 2.744 \times 1.4\) | 3.8416 | Yes (3.8416 is > 3) |
From the calculations, we can see that after 1, 2, or 3 years, the amount is not yet more than triple the principal. However, after 4 years, the amount is more than 3.84 times the principal, which is indeed more than triple.
Since we are looking for the least number of complete years, the smallest integer \(n\) that satisfies \((1.4)^n > 3\) is 4.
Conclusion
The least number of complete years in which a sum of money put out at 40% annual compound interest will be more than tripled is 4 years.
Revision Table: Compound Interest Growth
| Concept | Formula/Rule | Application |
|---|---|---|
| Compound Interest Formula | \(A = P(1 + r)^n\) | Used to calculate future value |
| Condition for Tripling | \(A > 3P\) | Sets up the required growth |
| Solving for Time (\(n\)) | Test values or use logarithms | We tested integer years to find the smallest \(n\) |
Additional Information on Investment Growth
Difference from Simple Interest: In simple interest, only the principal earns interest each year. In compound interest, the interest earned in previous years is added to the principal, and the next year's interest is calculated on this new, larger amount. This leads to exponential growth, which is why compound interest can cause money to grow much faster over time compared to simple interest, especially at high rates like 40%.
Impact of Rate and Time: The higher the interest rate (\(r\)) and the longer the time period (\(n\)), the faster the investment grows. A 40% annual rate is very high, leading to rapid growth.
Using Logarithms: For more complex problems or rates that don't yield nice integer solutions quickly, you can solve the inequality \((1 + r)^n > 3\) using logarithms:
\(n \log(1 + r) > \log(3)\)
\(n > \frac{\log(3)}{\log(1 + r)}\)
In this case, \(n > \frac{\log(3)}{\log(1.4)}\). Using a calculator, \(\log(3) \approx 0.4771\) and \(\log(1.4) \approx 0.1461\). So, \(n > \frac{0.4771}{0.1461} \approx 3.265\). Since \(n\) must be a complete year and must be greater than 3.265, the least complete year is 4.
Similar Questions
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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?
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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?
