The annual income of a person decreases by Rs. 64 if the rate of interest decreases from 4% to 3.75%. What is his original annual income?
Rs. 25600
Understanding the Problem: Annual Income and Interest Rate Changes
The question asks us to find the original annual income of a person. We are told that if the interest rate decreases, the person's annual income also decreases by a specific amount. This decrease in income is directly related to the decrease in the interest rate applied to their original annual income (which acts like a principal amount earning interest).
Identifying the Key Information
- Original Interest Rate: 4%
- New Interest Rate: 3.75%
- Decrease in Annual Income: Rs. 64
The decrease in annual income is caused by the difference between the interest earned at 4% and the interest earned at 3.75% on the original annual income.
Calculating the Decrease in Interest Rate
The difference between the two interest rates is:
\( \text{Decrease in Rate} = \text{Original Rate} - \text{New Rate} \)
\( \text{Decrease in Rate} = 4\% - 3.75\% \)
\( \text{Decrease in Rate} = 0.25\% \)
So, the interest rate decreases by 0.25 percentage points.
Setting up the Equation
Let the original annual income be represented by \( P \). The annual income earned from interest is calculated as a percentage of this original annual income.
The decrease in annual income (Rs. 64) is the result of earning 0.25% less interest on the original annual income \( P \).
Therefore, we can write the equation:
\( 0.25\% \text{ of } P = \text{Decrease in Annual Income} \)
\( \frac{0.25}{100} \times P = 64 \)
Solving for the Original Annual Income
Now, we need to solve the equation for \( P \):
\( \frac{0.25}{100} \times P = 64 \)
To make the calculation easier, we can write 0.25 as \(\frac{1}{4}\):
\( \frac{\frac{1}{4}}{100} \times P = 64 \)
\( \frac{1}{400} \times P = 64 \)
Multiply both sides of the equation by 400 to isolate \( P \):
\( P = 64 \times 400 \)
\( P = 25600 \)
So, the original annual income is Rs. 25600.
Verifying the Solution
Let's check if the decrease in income is indeed Rs. 64 with an original income of Rs. 25600.
- Annual income at 4% interest: \( 25600 \times \frac{4}{100} = 256 \times 4 = 1024 \)
- Annual income at 3.75% interest: \( 25600 \times \frac{3.75}{100} = 256 \times 3.75 \)
Calculation for \( 256 \times 3.75 \):
\( 256 \times 3.75 = 256 \times (3 + 0.75) = 256 \times 3 + 256 \times 0.75 \)
\( 256 \times 3 = 768 \)
\( 256 \times 0.75 = 256 \times \frac{3}{4} = \frac{256}{4} \times 3 = 64 \times 3 = 192 \)
\( 768 + 192 = 960 \)
So, annual income at 3.75% is Rs. 960.
- Decrease in income: \( 1024 - 960 = 64 \)
The calculated decrease matches the given decrease of Rs. 64. This confirms our solution is correct.
Final Answer
The original annual income of the person is Rs. 25600.
Revision Table: Annual Income Calculation
| Concept | Explanation | Formula/Relation |
|---|---|---|
| Annual Income (from Interest) | The amount earned on the principal amount based on the interest rate over one year. | Principal \( \times \) Rate \( \times \) Time (Here Time = 1 year) |
| Interest Rate Decrease | The difference between the initial and final interest rates. | Initial Rate - Final Rate |
| Income Decrease | The reduction in annual income corresponding to the decrease in interest rate. This represents the interest on the principal at the decreased rate. | Principal \( \times \) (Interest Rate Decrease) |
Additional Information: Simple Interest Basics
This problem is based on the concept of simple interest. Simple interest is calculated only on the principal amount.
The formula for simple interest (SI) is:
\( SI = \frac{P \times R \times T}{100} \)
Where:
- \( P \) = Principal amount (the original annual income in this case)
- \( R \) = Rate of interest per annum (yearly rate)
- \( T \) = Time period in years (1 year in this case)
In this problem, the difference in income is the difference in simple interest earned over one year due to the change in rate, with the original annual income acting as the principal.
- Interest at 4% = \( \frac{P \times 4 \times 1}{100} \)
- Interest at 3.75% = \( \frac{P \times 3.75 \times 1}{100} \)
- Difference = \( \frac{P \times 4}{100} - \frac{P \times 3.75}{100} = \frac{P \times (4 - 3.75)}{100} = \frac{P \times 0.25}{100} \)
This difference is given as Rs. 64, leading back to the equation \( \frac{P \times 0.25}{100} = 64 \).
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