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Sagot :
Sure, let's work through this problem step-by-step.
### Step 1: Understand the given values
We have the following information:
- Interest accrued: [tex]\( \text{Rs.} 524.95 \)[/tex]
- Time period: 2 years
- Annual interest rate: 12% per year
- Compounding frequency: semi-annually (2 times a year)
### Step 2: Calculate the effective interest rate per compounding period
The annual rate is 12%, and since it is compounded semi-annually, the effective interest rate per compounding period is:
[tex]\[ \text{Effective rate per period} = \frac{12\%}{2} = 6\% = 0.06 \][/tex]
### Step 3: Determine the total number of compounding periods
Since it is compounded semi-annually over 2 years:
[tex]\[ \text{Total compounding periods} = 2 \text{ years} \times 2 \text{ times/year} = 4 \text{ periods} \][/tex]
### Step 4: Use the compound interest formula
The compound interest formula is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n periods, including interest.
- [tex]\( P \)[/tex] is the principal amount (which we need to find).
- [tex]\( r \)[/tex] is the annual interest rate (12% or 0.12).
- [tex]\( n \)[/tex] is the number of times interest is compounded per year.
- [tex]\( t \)[/tex] is the number of years.
Since the interest accrued is given (not the final accumulated amount), we need to rearrange the formula to solve for [tex]\( P \)[/tex].
### Step 5: Re-arrange the formula to find [tex]\( P \)[/tex]
We know the interest [tex]\( I = A - P \)[/tex], which rearranges to:
[tex]\[ P \left( (1 + \frac{r}{n})^{nt} - 1 \right) = I \][/tex]
Given values:
- [tex]\( I = \text{Rs.} 524.95 \)[/tex]
- [tex]\( r = 0.12 \)[/tex]
- [tex]\( n = 2 \)[/tex]
- [tex]\( t = 2 \)[/tex]
So:
[tex]\[ \overset{\sim}{(\text{Value obtained previously})\text{ }} = (1 + 0.06)^{4} \][/tex]
### Step 6: Calculate the compound factor
[tex]\[ \text{Accumulated factor} = (1.06)^4 \approx 1.26248 \][/tex]
It's worth noting that this leads to:
[tex]\[ (1.06)^4 \approx 1.26248 \][/tex]
### Step 7: Solve for [tex]\( P \)[/tex]
[tex]\[ P = \frac{524.95}{1.26248 - 1} = \frac{524.95}{0.26248} = 1999.985 \][/tex]
### Conclusion
The principal sum of money that would produce an interest of Rs. 524.95 in 2 years at a 12% annual interest rate compounded semi-annually is approximately Rs. 1999.99.
### Step 1: Understand the given values
We have the following information:
- Interest accrued: [tex]\( \text{Rs.} 524.95 \)[/tex]
- Time period: 2 years
- Annual interest rate: 12% per year
- Compounding frequency: semi-annually (2 times a year)
### Step 2: Calculate the effective interest rate per compounding period
The annual rate is 12%, and since it is compounded semi-annually, the effective interest rate per compounding period is:
[tex]\[ \text{Effective rate per period} = \frac{12\%}{2} = 6\% = 0.06 \][/tex]
### Step 3: Determine the total number of compounding periods
Since it is compounded semi-annually over 2 years:
[tex]\[ \text{Total compounding periods} = 2 \text{ years} \times 2 \text{ times/year} = 4 \text{ periods} \][/tex]
### Step 4: Use the compound interest formula
The compound interest formula is:
[tex]\[ A = P \left(1 + \frac{r}{n}\right)^{nt} \][/tex]
where:
- [tex]\( A \)[/tex] is the amount of money accumulated after n periods, including interest.
- [tex]\( P \)[/tex] is the principal amount (which we need to find).
- [tex]\( r \)[/tex] is the annual interest rate (12% or 0.12).
- [tex]\( n \)[/tex] is the number of times interest is compounded per year.
- [tex]\( t \)[/tex] is the number of years.
Since the interest accrued is given (not the final accumulated amount), we need to rearrange the formula to solve for [tex]\( P \)[/tex].
### Step 5: Re-arrange the formula to find [tex]\( P \)[/tex]
We know the interest [tex]\( I = A - P \)[/tex], which rearranges to:
[tex]\[ P \left( (1 + \frac{r}{n})^{nt} - 1 \right) = I \][/tex]
Given values:
- [tex]\( I = \text{Rs.} 524.95 \)[/tex]
- [tex]\( r = 0.12 \)[/tex]
- [tex]\( n = 2 \)[/tex]
- [tex]\( t = 2 \)[/tex]
So:
[tex]\[ \overset{\sim}{(\text{Value obtained previously})\text{ }} = (1 + 0.06)^{4} \][/tex]
### Step 6: Calculate the compound factor
[tex]\[ \text{Accumulated factor} = (1.06)^4 \approx 1.26248 \][/tex]
It's worth noting that this leads to:
[tex]\[ (1.06)^4 \approx 1.26248 \][/tex]
### Step 7: Solve for [tex]\( P \)[/tex]
[tex]\[ P = \frac{524.95}{1.26248 - 1} = \frac{524.95}{0.26248} = 1999.985 \][/tex]
### Conclusion
The principal sum of money that would produce an interest of Rs. 524.95 in 2 years at a 12% annual interest rate compounded semi-annually is approximately Rs. 1999.99.
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