Answer :
Answer:
Step-by-step explanation:
Let's solve for the time it takes for the investment to double using both the future value formula and the rule of 72.
**Using the Future Value Formula:**
The future value formula for compound interest is:
\[ A = P \left( 1 + \frac{r}{m} \right)^{mt} \]
Where:
- \( A \) is the future value (double the principal in this case),
- \( P \) is the principal amount ($1800),
- \( r \) is the annual nominal interest rate (5.1% or 0.051 as a decimal),
- \( m \) is the number of times interest is compounded per year (annual compounding, so \( m = 1 \)),
- \( t \) is the time in years.
We need to solve for \( t \):
\[ 2P = P \left( 1 + \frac{r}{m} \right)^{mt} \]
\[ 2 = \left( 1 + \frac{0.051}{1} \right)^{1 \cdot t} \]
\[ 2 = \left( 1.051 \right)^t \]
Taking the natural logarithm (ln) of both sides to solve for \( t \):
\[ \ln(2) = t \cdot \ln(1.051) \]
\[ t = \frac{\ln(2)}{\ln(1.051)} \]
Calculating the values:
\[ t \approx \frac{0.693147}{0.050961} \]
\[ t \approx 13.60 \]
So, using the future value formula, it will take approximately \( \boxed{13.60} \) years for her investment to double.
**Using the Rule of 72:**
The rule of 72 states that you can approximate the time it takes for an investment to double by dividing 72 by the annual growth rate (in percent):
\[ \text{Years to double} = \frac{72}{r} \]
Where \( r \) is the annual interest rate.
Given \( r = 5.1\% = 5.1 \),
\[ \text{Years to double} = \frac{72}{5.1} \]
\[ \text{Years to double} \approx 14.12 \]
Therefore, using the rule of 72, it will take approximately \( \boxed{14.12} \) years for her investment to double.
Answer:
To calculate the balance of the $200,000 mortgage after one month, we can use the following formula:
Balance = Loan Amount - (Monthly Payment - (Loan Amount * Interest Rate / 12))
Given information:
- Loan Amount: $200,000
- Interest Rate: 4% per year
- Loan Term: 20 years
- Monthly Payment: $1,211.96
Step 1: Calculate the interest for the first month.
Interest for the first month = Loan Amount * Interest Rate / 12
Interest for the first month = $200,000 * 0.04 / 12 = $666.67
Step 2: Calculate the balance after the first month.
Balance = Loan Amount - (Monthly Payment - Interest for the first month)
Balance = $200,000 - ($1,211.96 - $666.67)
Balance = $200,000 - $545.29
Balance = $199,454.71
Rounding the answer to the nearest dollar, the balance of the loan after one month is $199,455.
To calculate the time it takes for the woman's investment to double using the future value formula:
A) The formula is A = P(1 + r/m)^n, where A is the future value, P is the principal, r is the nominal rate, m is the number of times the investment is compounded in a year, and n is the number of compounding periods.
Given:
- Principal (P) = $1800
- Nominal rate (r) = 5.1%
- Compounding frequency (m) = 1 (annually)
A = $1800(1 + 0.051/1)^n
A = $1800(1.051)^n
To find the number of years (n) that it takes for the investment to double:
$3600 = $1800(1.051)^n
$3600 / $1800 = (1.051)^n
2 = (1.051)^n
ln(2) = ln((1.051)^n)
ln(2) = n * ln(1.051)
n = ln(2) / ln(1.051)
n ≈ 13.43 years
So, it will take approximately 13.43 years for her investment to double using the future value formula.
To calculate the time it takes for the woman's investment to double using the rule of 72:
B) The rule of 72 formula is years to double = 72 / growth rate.
Given:
- Growth rate = 5.1%
Years to double = 72 / 5.1
Years to double ≈ 14.12 years
So, it will take approximately 14.12 years for her investment to double using the rule of 72.
