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Sagot :
Let's carefully solve this problem step by step. We're given an initial amount (principal amount) and the amount after three years, and we need to determine the annual compound interest rate.
### Given:
1. Principal amount (P): $55125
2. Amount after 3 years (A): $5788.25
3. Time period (t): 3 years
We need to find the annual compound interest rate (r).
### Step-by-Step Solution:
1. Understanding the Formula:
The formula for compound interest when compounded annually is:
[tex]\[ A = P(1 + r)^t \][/tex]
Where:
- \(A\) is the amount after time \(t\)
- \(P\) is the principal amount
- \(r\) is the annual interest rate
- \(t\) is the time period in years
2. Rearranging the Formula:
To solve for the interest rate (\(r\)), we first rearrange the formula:
[tex]\[ (1 + r)^t = \frac{A}{P} \][/tex]
Taking the \(t\)-th root of both sides:
[tex]\[ 1 + r = \left(\frac{A}{P}\right)^{\frac{1}{t}} \][/tex]
Subtracting 1 from both sides to solve for \(r\):
[tex]\[ r = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1 \][/tex]
3. Substituting the Values:
Let's substitute the values \(A = 5788.25\), \(P = 55125\), and \(t = 3\) into the formula:
[tex]\[ r = \left(\frac{5788.25}{55125}\right)^{\frac{1}{3}} - 1 \][/tex]
4. Calculating the Values:
We first compute the fraction:
[tex]\[ \frac{5788.25}{55125} = 0.105 \][/tex]
Next, we take the cube root of 0.105:
[tex]\[ (0.105)^{\frac{1}{3}} \approx 0.473 \][/tex]
Finally, we subtract 1:
[tex]\[ r = 0.473 - 1 = -0.527 \][/tex]
Therefore, the annual compound interest rate is approximately:
[tex]\[ -0.5282 \][/tex]
### Conclusion:
The annual interest rate is approximately -0.5282 or -52.82%. This negative value indicates that the value of the principal amount has decreased over the period of three years, suggesting a decrease in value rather than growth due to compound interest.
### Given:
1. Principal amount (P): $55125
2. Amount after 3 years (A): $5788.25
3. Time period (t): 3 years
We need to find the annual compound interest rate (r).
### Step-by-Step Solution:
1. Understanding the Formula:
The formula for compound interest when compounded annually is:
[tex]\[ A = P(1 + r)^t \][/tex]
Where:
- \(A\) is the amount after time \(t\)
- \(P\) is the principal amount
- \(r\) is the annual interest rate
- \(t\) is the time period in years
2. Rearranging the Formula:
To solve for the interest rate (\(r\)), we first rearrange the formula:
[tex]\[ (1 + r)^t = \frac{A}{P} \][/tex]
Taking the \(t\)-th root of both sides:
[tex]\[ 1 + r = \left(\frac{A}{P}\right)^{\frac{1}{t}} \][/tex]
Subtracting 1 from both sides to solve for \(r\):
[tex]\[ r = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1 \][/tex]
3. Substituting the Values:
Let's substitute the values \(A = 5788.25\), \(P = 55125\), and \(t = 3\) into the formula:
[tex]\[ r = \left(\frac{5788.25}{55125}\right)^{\frac{1}{3}} - 1 \][/tex]
4. Calculating the Values:
We first compute the fraction:
[tex]\[ \frac{5788.25}{55125} = 0.105 \][/tex]
Next, we take the cube root of 0.105:
[tex]\[ (0.105)^{\frac{1}{3}} \approx 0.473 \][/tex]
Finally, we subtract 1:
[tex]\[ r = 0.473 - 1 = -0.527 \][/tex]
Therefore, the annual compound interest rate is approximately:
[tex]\[ -0.5282 \][/tex]
### Conclusion:
The annual interest rate is approximately -0.5282 or -52.82%. This negative value indicates that the value of the principal amount has decreased over the period of three years, suggesting a decrease in value rather than growth due to compound interest.
Let's denote the principal amount by \( P \) and the rate of interest by \( r \) (expressed as a decimal).
Given:
- The amount after 2 years is \( A_2 \).
- The amount after 3 years is \( A_3 \).
The formula for compound interest compounded annually is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
For 2 years:
\[ A_2 = P \left(1 + \frac{r}{100}\right)^2 \]
For 3 years:
\[ A_3 = P \left(1 + \frac{r}{100}\right)^3 \]
To find the rate of interest \( r \), we can use the relationship between \( A_2 \) and \( A_3 \):
\[ A_3 = A_2 \left(1 + \frac{r}{100}\right) \]
This follows from the fact that the amount after 3 years can be considered as the amount after 2 years compounded for one more year.
So, we have:
\[ A_3 = P \left(1 + \frac{r}{100}\right)^3 = A_2 \left(1 + \frac{r}{100}\right) \]
Divide both sides by \( A_2 \):
\[ \frac{A_3}{A_2} = 1 + \frac{r}{100} \]
Subtract 1 from both sides to isolate \( \frac{r}{100} \):
\[ \frac{A_3}{A_2} - 1 = \frac{r}{100} \]
Multiply both sides by 100 to find \( r \):
\[ r = 100 \left( \frac{A_3}{A_2} - 1 \right) \]
Once \( r \) is known, we can substitute it back into the equation for \( A_2 \) to find the principal amount \( P \):
\[ A_2 = P \left(1 + \frac{r}{100}\right)^2 \]
Solving for \( P \):
\[ P = \frac{A_2}{\left(1 + \frac{r}{100}\right)^2} \]
Let's plug in hypothetical values for \( A_2 \) and \( A_3 \) to illustrate the calculation. Suppose:
- \( A_2 = 1210 \)
- \( A_3 = 1331 \)
Using these values:
\[ \frac{A_3}{A_2} = \frac{1331}{1210} \approx 1.1 \]
\[ r = 100 \left( 1.1 - 1 \right) = 100 \times 0.1 = 10 \% \]
Now, finding \( P \):
\[ P = \frac{1210}{(1 + 0.1)^2} = \frac{1210}{1.21} \approx 1000 \]
Therefore, with these hypothetical values:
- The rate of interest is \( 10\% \).
- The principal amount is \( 1000 \).
To find the exact rate and principal, substitute the actual values of \( A_2 \) and \( A_3 \) provided.
Given:
- The amount after 2 years is \( A_2 \).
- The amount after 3 years is \( A_3 \).
The formula for compound interest compounded annually is:
\[ A = P \left(1 + \frac{r}{100}\right)^n \]
For 2 years:
\[ A_2 = P \left(1 + \frac{r}{100}\right)^2 \]
For 3 years:
\[ A_3 = P \left(1 + \frac{r}{100}\right)^3 \]
To find the rate of interest \( r \), we can use the relationship between \( A_2 \) and \( A_3 \):
\[ A_3 = A_2 \left(1 + \frac{r}{100}\right) \]
This follows from the fact that the amount after 3 years can be considered as the amount after 2 years compounded for one more year.
So, we have:
\[ A_3 = P \left(1 + \frac{r}{100}\right)^3 = A_2 \left(1 + \frac{r}{100}\right) \]
Divide both sides by \( A_2 \):
\[ \frac{A_3}{A_2} = 1 + \frac{r}{100} \]
Subtract 1 from both sides to isolate \( \frac{r}{100} \):
\[ \frac{A_3}{A_2} - 1 = \frac{r}{100} \]
Multiply both sides by 100 to find \( r \):
\[ r = 100 \left( \frac{A_3}{A_2} - 1 \right) \]
Once \( r \) is known, we can substitute it back into the equation for \( A_2 \) to find the principal amount \( P \):
\[ A_2 = P \left(1 + \frac{r}{100}\right)^2 \]
Solving for \( P \):
\[ P = \frac{A_2}{\left(1 + \frac{r}{100}\right)^2} \]
Let's plug in hypothetical values for \( A_2 \) and \( A_3 \) to illustrate the calculation. Suppose:
- \( A_2 = 1210 \)
- \( A_3 = 1331 \)
Using these values:
\[ \frac{A_3}{A_2} = \frac{1331}{1210} \approx 1.1 \]
\[ r = 100 \left( 1.1 - 1 \right) = 100 \times 0.1 = 10 \% \]
Now, finding \( P \):
\[ P = \frac{1210}{(1 + 0.1)^2} = \frac{1210}{1.21} \approx 1000 \]
Therefore, with these hypothetical values:
- The rate of interest is \( 10\% \).
- The principal amount is \( 1000 \).
To find the exact rate and principal, substitute the actual values of \( A_2 \) and \( A_3 \) provided.
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