Certainly! Let's break down the calculations for Minakshi's investment of Rs 85,000 at an 8% annual interest rate over 1 year with different compounding periods.
### (i) Annual Compounding
When interest is compounded annually, it is added to the principal once at the end of the year. Here’s how we calculate it:
Formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n years, including interest.
- [tex]\( P \)[/tex] is the principal amount (Rs 85,000).
- [tex]\( r \)[/tex] is the annual interest rate (8% or 0.08).
- [tex]\( n \)[/tex] is the number of times interest is compounded per year (1 for annual).
- [tex]\( t \)[/tex] is the number of years the money is invested for (1 year).
Plugging in the values:
[tex]\[ A = 85000 \left(1 + \frac{0.08}{1}\right)^{1 \times 1} \][/tex]
[tex]\[ A = 85000 \left(1 + 0.08\right) \][/tex]
[tex]\[ A = 85000 \times 1.08 \][/tex]
[tex]\[ A = 91800 \][/tex]
The interest earned is:
[tex]\[ \text{Interest} = A - P \][/tex]
[tex]\[ \text{Interest} = 91800 - 85000 \][/tex]
[tex]\[ \text{Interest} = 6800 \][/tex]
So, Minakshi will receive Rs 6,800 as interest if it is compounded annually.
### (ii) Semi-Annual Compounding (Every 6 Months)
When interest is compounded semi-annually, it is added to the principal twice a year. Here’s how we calculate it:
Formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( n \)[/tex] is the number of times interest is compounded per year (2 for semi-annual).
Plugging in the values:
[tex]\[ A = 85000 \left(1 + \frac{0.08}{2}\right)^{2 \times 1} \][/tex]
[tex]\[ A = 85000 \left(1 + 0.04\right)^{2} \][/tex]
[tex]\[ A = 85000 \times 1.04^{2} \][/tex]
[tex]\[ A = 85000 \times 1.0816 \][/tex]
[tex]\[ A = 91936 \][/tex]
The interest earned is:
[tex]\[ \text{Interest} = A - P \][/tex]
[tex]\[ \text{Interest} = 91936 - 85000 \][/tex]
[tex]\[ \text{Interest} = 6936 \][/tex]
So, Minakshi will receive Rs 6,936 as interest if it is compounded semi-annually.
### (iii) Quarterly Compounding (Every 3 Months)
When interest is compounded quarterly, it is added to the principal four times a year. Here’s how we calculate it:
Formula:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
Where:
- [tex]\( n \)[/tex] is the number of times interest is compounded per year (4 for quarterly).
Plugging in the values:
[tex]\[ A = 85000 \left(1 + \frac{0.08}{4}\right)^{4 \times 1} \][/tex]
[tex]\[ A = 85000 \left(1 + 0.02\right)^{4} \][/tex]
[tex]\[ A = 85000 \times 1.02^{4} \][/tex]
[tex]\[ A = 85000 \times 1.08243216 \][/tex]
[tex]\[ A = 92006.7336 \][/tex]
The interest earned is:
[tex]\[ \text{Interest} = A - P \][/tex]
[tex]\[ \text{Interest} = 92006.7336 - 85000 \][/tex]
[tex]\[ \text{Interest} = 7006.7336 \][/tex]
So, Minakshi will receive Rs 7,006.73 (approximately) as interest if it is compounded quarterly.
