To solve these problems, we need to use formulas related to compound interest and the future value of an annuity. Let's break down each part of the problem:

Part (a): Time to Reach $150,000 with Quarterly Contributions of $4,000

Given:

• Quarterly contribution ($R$) = $4,000
• Annual interest rate ($r$) = 10%
• Compounded quarterly, so the quarterly interest rate = $\frac{10\%}{4} = 2.5\% = 0.025$
• Future value ($FV$) = $150,000

We use the future value formula for an ordinary annuity (contributions at the end of each period):

$FV = R \times \frac{(1 + i)^n - 1}{i}$

Where:

• $FV$ is the future value
• $R$ is the contribution per period
• $i$ is the interest rate per period
• $n$ is the number of periods

Plugging in the known values:

$150,000 = 4,000 \times \frac{(1 + 0.025)^n - 1}{0.025}$

Simplifying further:

$150,000 = 4,000 \times \frac{(1.025)^n - 1}{0.025}$

$\frac{150,000}{4,000} = \frac{(1.025)^n - 1}{0.025}$

$37.5 = \frac{(1.025)^n - 1}{0.025}$

$37.5 \times 0.025 = (1.025)^n - 1$

$0.9375 = (1.025)^n - 1$

$1.9375 = (1.025)^n$

To find $n$, we take the natural logarithm of both sides:

$n = \frac{\ln(1.9375)}{\ln(1.025)}$

Let's calculate $n$.The account will be worth at least $150,000 after 27 quarters.

Part (b): Future Value After 15 More Years with Increased Contributions

Now, when the account reaches $150,000, the contributions increase to $8,000 at the end of each quarter, and this continues for another 15 years. We need to find the total value of the account after these 15 years.

Given:

• New quarterly contribution ($R$) = $8,000
• Quarterly interest rate ($i$) = 2.5% = 0.025
• Number of periods ($n$) = 15 years $\times 4$ quarters per year = 60 quarters
• Initial amount ($P$) = $150,000 (the amount from part a)

The future value of the account after 15 years can be found using the combined future value formula for the initial amount compounded over time and the new annuity contributions:

$FV_{\text{total}} = P \times (1 + i)^n + R \times \frac{(1 + i)^n - 1}{i}$

Let's plug in the values to find the total value.The total value of the account after 15 more years, with increased contributions of $8,000 at the end of each quarter, will be $1,747,901.

Would you like further details on the calculations, or do you have any questions?

Here are some related questions that might interest you:

1. How would the time change if the contribution amount or interest rate was different?
2. What if the contributions were made at the beginning of each quarter instead?
3. How would inflation affect the value of the retirement account in real terms?
4. How can changes in compounding frequency (e.g., monthly or annually) impact the final amount?
5. What is the impact of starting with an initial deposit rather than beginning with no initial funds?

Tip: Regularly reviewing your investment strategy, including contribution amounts and investment options, can significantly impact the growth of your retirement savings over time.