Regular semi-annual payments of 400 are deposited into an account paying 6.15% interest, compounded semi-annually. If the final value of the account is 46000 how long was the money invested? Select one: a. 24.17 years b. 24.96 years c. 26.84 years d. 25.33 years

## Step 1: Identify given information<br />### We are given semi-annual payments $PMT = \$400$, an interest rate of $r = 6.15\% = 0.0615$ compounded semi-annually, and a future value $FV = \$46000$. We need to find the time $t$ in years.<br /><br />## Step 2: Determine the semi-annual interest rate and number of periods<br />### Since the interest is compounded semi-annually, the semi-annual interest rate is $i = \frac{0.0615}{2} = 0.03075$. Let $n$ be the number of semi-annual periods. We are looking for $t$ in years, so $t = \frac{n}{2}$.<br /><br />## Step 3: Apply the future value of an ordinary annuity formula<br />### The future value of an ordinary annuity is given by the formula: $FV = PMT \cdot \frac{(1+i)^n - 1}{i}$. Substituting the given values, we have: $46000 = 400 \cdot \frac{(1+0.03075)^n - 1}{0.03075}$.<br /><br />## Step 4: Solve for n<br />### Simplify the equation: $115 = \frac{(1.03075)^n - 1}{0.03075}$. Multiply both sides by 0.03075: $3.53625 = (1.03075)^n - 1$. Add 1 to both sides: $4.53625 = (1.03075)^n$. Take the logarithm of both sides: $\ln(4.53625) = n \ln(1.03075)$. Solve for $n$: $n = \frac{\ln(4.53625)}{\ln(1.03075)} \approx 50.65$.<br /><br />## Step 5: Calculate the time in years<br />### Since $n$ represents the number of semi-annual periods, the time in years is $t = \frac{n}{2} = \frac{50.65}{2} \approx 25.33$ years.