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?
Rs. 5120
Calculating Original Capital with Annual Growth
This problem involves finding the initial amount of money (original capital) a merchant invested, given the final amount after a certain number of years and a fixed annual growth rate. This type of problem can be solved using the formula for compound growth, similar to compound interest.
Let's define the terms:
- Original Capital (Principal): Let this be \(P\). This is the amount we need to find.
- Annual Growth Rate: The problem states the merchant gains annually at the rate of 25%. So, \(R = 25\%\).
- Time Period: The growth occurs over 3 years. So, \(T = 3\) years.
- Final Amount: At the end of 3 years, the merchant has Rs. 10,000. This is the final amount, let's call it \(A\). So, \(A = 10000\).
The formula for compound growth is given by:
\[A = P \times \left(1 + \frac{R}{100}\right)^T\]We need to find the value of \(P\). We can plug in the given values into the formula:
\[10000 = P \times \left(1 + \frac{25}{100}\right)^3\]Now, let's simplify the expression inside the parenthesis:
\[1 + \frac{25}{100} = 1 + \frac{1}{4} = \frac{4+1}{4} = \frac{5}{4}\]So, the equation becomes:
\[10000 = P \times \left(\frac{5}{4}\right)^3\]Let's calculate the value of \(\left(\frac{5}{4}\right)^3\):
\[\left(\frac{5}{4}\right)^3 = \frac{5^3}{4^3} = \frac{5 \times 5 \times 5}{4 \times 4 \times 4} = \frac{125}{64}\]Substitute this back into the equation:
\[10000 = P \times \frac{125}{64}\]To find \(P\), we need to isolate \(P\) by multiplying both sides of the equation by \(\frac{64}{125}\):
\[P = 10000 \times \frac{64}{125}\]Now, we can perform the calculation. We can simplify this by dividing 10000 by 125:
We know that \(1000 / 125 = 8\). Therefore, \(10000 / 125 = 80\).
So, the calculation becomes:
\[P = 80 \times 64\]Let's calculate \(80 \times 64\):
\[80 \times 64 = 8 \times 10 \times 64 = 8 \times 640\] \[8 \times 640 = 8 \times (600 + 40) = (8 \times 600) + (8 \times 40) = 4800 + 320 = 5120\]So, the original amount that the merchant invested (the original capital) was Rs. 5120.
Let's verify the options:
- Option 1: Rs. 5120
- Option 2: Rs. 5210
- Option 3: Rs. 5350
- Option 4: Rs. 5500
Our calculated value is Rs. 5120, which matches Option 1.
Summary of Calculation Steps
- Identify the given values: Final Amount (A), Rate (R), Time (T).
- Identify the unknown: Original Capital (P).
- Use the compound growth formula: \(A = P \times (1 + \frac{R}{100})^T\).
- Substitute the known values into the formula.
- Solve the equation for P.
- Perform the arithmetic calculation to find the value of P.
| Parameter | Value | Formula Symbol |
|---|---|---|
| Final Amount | Rs. 10,000 | \(A\) |
| Annual Growth Rate | 25% | \(R\) |
| Time Period | 3 years | \(T\) |
| Original Capital (To find) | ? | \(P\) |
The calculation shows that an original capital of Rs. 5120 growing at 25% annually for 3 years will result in Rs. 10,000.
\[ \text{Year 1 End:} \quad 5120 \times \left(1 + \frac{25}{100}\right) = 5120 \times \frac{5}{4} = 1280 \times 5 = 6400 \] \[ \text{Year 2 End:} \quad 6400 \times \frac{5}{4} = 1600 \times 5 = 8000 \] \[ \text{Year 3 End:} \quad 8000 \times \frac{5}{4} = 2000 \times 5 = 10000 \]This confirms our result.
Revision Table: Financial Formulas
| Concept | Formula | Variables |
|---|---|---|
| Simple Interest (I) | \(I = \frac{P \times R \times T}{100}\) | P=Principal, R=Rate, T=Time |
| Simple Amount (A) | \(A = P + I = P \left(1 + \frac{R \times T}{100}\right)\) | P=Principal, R=Rate, T=Time, I=Simple Interest |
| Compound Amount (A) | \(A = P \left(1 + \frac{R}{100}\right)^T\) | P=Principal, R=Rate per period, T=Number of periods |
| Compound Interest (CI) | \(CI = A - P = P \left[\left(1 + \frac{R}{100}\right)^T - 1\right]\) | P=Principal, R=Rate per period, T=Number of periods, A=Compound Amount |
Additional Information: Compound Growth Concepts
The problem describes a merchant's capital gaining annually at a certain rate. This is a classic example of compound growth, where the gain each year is calculated on the accumulated amount from the previous year, including the initial capital and any previous gains. This is different from simple growth (or simple interest), where the gain would only be calculated on the original principal amount.
- Compound Growth: Growth is added to the principal for the next calculation period. The base for calculation increases over time, leading to faster growth. This is often used for investments, population growth, etc.
- Simple Growth: Growth is calculated only on the initial principal amount. The base for calculation remains constant. This is less common for long-term financial scenarios but is easier to calculate.
Understanding the difference between simple and compound calculations is crucial in financial mathematics and quantitative aptitude problems.
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