Answer :
Sure! Let's work through the problem step by step to find the value of [tex]\( n \)[/tex] when we know the principal amount [tex]\( p \)[/tex], the interest rate [tex]\( i \)[/tex], and the total amount after interest has been added.
1. Understand the Variables:
- [tex]\( p \)[/tex] is the principal amount, which is given as [tex]\( 20,000 \)[/tex].
- [tex]\( i \)[/tex] is the interest rate, given as [tex]\( 4.5\% \)[/tex], which we convert to a decimal by dividing by 100: [tex]\( i = 0.045 \)[/tex].
- [tex]\( n \)[/tex] is the unknown number of years.
- The total amount after interest is added is [tex]\( 24,500 \)[/tex].
2. Calculate Interest Earned:
- The interest earned is the difference between the total amount and the principal:
[tex]\[
\text{Interest Earned} = \text{Total Amount} - p = 24,500 - 20,000 = 4500
\][/tex]
3. Set Up the Interest Formula:
- The simple interest formula is given by [tex]\( \text{Interest Earned} = p \times i \times n \)[/tex].
- Substitute the known values:
[tex]\[
4500 = 20,000 \times 0.045 \times n
\][/tex]
4. Solve for [tex]\( n \)[/tex]:
- Rearrange the equation to solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{4500}{20,000 \times 0.045}
\][/tex]
- Calculate [tex]\( n \)[/tex]:
[tex]\[
n = 5
\][/tex]
5. Verify the Calculations:
- Calculate the interest for [tex]\( n = 5 \)[/tex] to ensure it gives the correct interest earned:
[tex]\[
\text{Interest} = 20,000 \times 0.045 \times 5 = 4,500
\][/tex]
- Add the interest to the principal to check if it results in the total amount:
[tex]\[
20,000 + 4,500 = 24,500
\][/tex]
Thus, the number of years [tex]\( n \)[/tex] is [tex]\( 5 \)[/tex].
1. Understand the Variables:
- [tex]\( p \)[/tex] is the principal amount, which is given as [tex]\( 20,000 \)[/tex].
- [tex]\( i \)[/tex] is the interest rate, given as [tex]\( 4.5\% \)[/tex], which we convert to a decimal by dividing by 100: [tex]\( i = 0.045 \)[/tex].
- [tex]\( n \)[/tex] is the unknown number of years.
- The total amount after interest is added is [tex]\( 24,500 \)[/tex].
2. Calculate Interest Earned:
- The interest earned is the difference between the total amount and the principal:
[tex]\[
\text{Interest Earned} = \text{Total Amount} - p = 24,500 - 20,000 = 4500
\][/tex]
3. Set Up the Interest Formula:
- The simple interest formula is given by [tex]\( \text{Interest Earned} = p \times i \times n \)[/tex].
- Substitute the known values:
[tex]\[
4500 = 20,000 \times 0.045 \times n
\][/tex]
4. Solve for [tex]\( n \)[/tex]:
- Rearrange the equation to solve for [tex]\( n \)[/tex]:
[tex]\[
n = \frac{4500}{20,000 \times 0.045}
\][/tex]
- Calculate [tex]\( n \)[/tex]:
[tex]\[
n = 5
\][/tex]
5. Verify the Calculations:
- Calculate the interest for [tex]\( n = 5 \)[/tex] to ensure it gives the correct interest earned:
[tex]\[
\text{Interest} = 20,000 \times 0.045 \times 5 = 4,500
\][/tex]
- Add the interest to the principal to check if it results in the total amount:
[tex]\[
20,000 + 4,500 = 24,500
\][/tex]
Thus, the number of years [tex]\( n \)[/tex] is [tex]\( 5 \)[/tex].
