Westonci.ca is the premier destination for reliable answers to your questions, provided by a community of experts. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
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.
We appreciate your visit. Hopefully, the answers you found were beneficial. Don't hesitate to come back for more information. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.
