Answer :
Alright, let's break this problem down step-by-step to find the solution.
### Part 1: Compound Amount Formula
The formula to find the compound amount when interest is compounded periodically is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the compound amount.
- [tex]\( P \)[/tex] is the principal amount.
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year.
- [tex]\( t \)[/tex] is the time in years.
Given:
- Principal [tex]\( P = 12000 \)[/tex]
- Annual interest rate [tex]\( r = 10\% = 0.10 \)[/tex]
- Time [tex]\( t = 1 \)[/tex] year
- Compounded half-yearly [tex]\( n = 2 \)[/tex]
### Part 2: Calculate Compound Amount and Compound Interest
First, plug the values into the compound amount formula:
[tex]\[ A = 12000 \left(1 + \frac{0.10}{2}\right)^{2 \times 1} \][/tex]
Evaluate the terms inside the parentheses and the exponent:
[tex]\[ A = 12000 \left(1 + 0.05\right)^{2} \][/tex]
[tex]\[ A = 12000 \left(1.05\right)^{2} \][/tex]
[tex]\[ A = 12000 \times 1.1025 \][/tex]
[tex]\[ A = 13230.0 \][/tex]
The compound amount after 1 year is [tex]\( A = 13230.0 \)[/tex].
The compound interest is the difference between the compound amount and the principal:
[tex]\[ \text{Compound Interest} = A - P \][/tex]
[tex]\[ \text{Compound Interest} = 13230.0 - 12000 \][/tex]
[tex]\[ \text{Compound Interest} = 1230.0 \][/tex]
### Part 3: Calculate Number of Copies
To find out how many copies of books costing Rs. 41 each can be purchased with the compound interest, we use the following calculation:
[tex]\[ \text{Number of Copies} = \frac{\text{Compound Interest}}{\text{Cost per Copy}} \][/tex]
[tex]\[ \text{Number of Copies} = \frac{1230.0}{41} \][/tex]
[tex]\[ \text{Number of Copies} \approx 30.0 \][/tex]
Therefore, with the compound interest of Rs. 1230.0, you can purchase approximately 30 copies of books costing Rs. 41 each.
### Summary:
1. Compound Amount: [tex]\( A = 13230.0 \)[/tex]
2. Compound Interest: [tex]\( \text{Compound Interest} = 1230.0 \)[/tex]
3. Number of Copies: [tex]\( \text{Number of Copies} = 30 \)[/tex]
This completes the detailed step-by-step solution.
### Part 1: Compound Amount Formula
The formula to find the compound amount when interest is compounded periodically is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the compound amount.
- [tex]\( P \)[/tex] is the principal amount.
- [tex]\( r \)[/tex] is the annual interest rate (as a decimal).
- [tex]\( n \)[/tex] is the number of times the interest is compounded per year.
- [tex]\( t \)[/tex] is the time in years.
Given:
- Principal [tex]\( P = 12000 \)[/tex]
- Annual interest rate [tex]\( r = 10\% = 0.10 \)[/tex]
- Time [tex]\( t = 1 \)[/tex] year
- Compounded half-yearly [tex]\( n = 2 \)[/tex]
### Part 2: Calculate Compound Amount and Compound Interest
First, plug the values into the compound amount formula:
[tex]\[ A = 12000 \left(1 + \frac{0.10}{2}\right)^{2 \times 1} \][/tex]
Evaluate the terms inside the parentheses and the exponent:
[tex]\[ A = 12000 \left(1 + 0.05\right)^{2} \][/tex]
[tex]\[ A = 12000 \left(1.05\right)^{2} \][/tex]
[tex]\[ A = 12000 \times 1.1025 \][/tex]
[tex]\[ A = 13230.0 \][/tex]
The compound amount after 1 year is [tex]\( A = 13230.0 \)[/tex].
The compound interest is the difference between the compound amount and the principal:
[tex]\[ \text{Compound Interest} = A - P \][/tex]
[tex]\[ \text{Compound Interest} = 13230.0 - 12000 \][/tex]
[tex]\[ \text{Compound Interest} = 1230.0 \][/tex]
### Part 3: Calculate Number of Copies
To find out how many copies of books costing Rs. 41 each can be purchased with the compound interest, we use the following calculation:
[tex]\[ \text{Number of Copies} = \frac{\text{Compound Interest}}{\text{Cost per Copy}} \][/tex]
[tex]\[ \text{Number of Copies} = \frac{1230.0}{41} \][/tex]
[tex]\[ \text{Number of Copies} \approx 30.0 \][/tex]
Therefore, with the compound interest of Rs. 1230.0, you can purchase approximately 30 copies of books costing Rs. 41 each.
### Summary:
1. Compound Amount: [tex]\( A = 13230.0 \)[/tex]
2. Compound Interest: [tex]\( \text{Compound Interest} = 1230.0 \)[/tex]
3. Number of Copies: [tex]\( \text{Number of Copies} = 30 \)[/tex]
This completes the detailed step-by-step solution.
