Saving For A House: A $100/Month Plan

 Problem:

Solution:

We need to determine how long it will take for your bank account balance to grow from $20,000 to $30,000, given that your account earns 0.5% interest per month and you add $100 each month.

The account balance grows in two ways:

1. The initial $20,000 earns interest.
2. You make monthly deposits of $100 that also earn interest.

The formula for the future value of an account with an initial principal $P$ that earns interest at rate $r$ per period for $t$ periods, with a deposit $PMT$ at the end of each period, is:

$$FV = P(1 + r)^t + PMT \times \frac{(1 + r)^t - 1}{r}$$

Here:

• $P = 20{,}000$
  
• $r = 0.005$ (0.5% per month)
• $PMT = 100$
  
• We want $FV = 30{,}000$
  

So, our equation is:

$$30{,}000 = 20{,}000(1.005)^t + 100 \times \frac{(1.005)^t - 1}{0.005}$$

Notice that $\frac{100}{0.005} = 20{,}000$. Thus, the equation simplifies to:

$$30{,}000 = 20{,}000(1.005)^t + 20{,}000\left[(1.005)^t - 1\right]$$

Combine like terms:

$$30{,}000 = 20{,}000(1.005)^t + 20{,}000(1.005)^t - 20{,}000$$
$$30{,}000 = 40{,}000(1.005)^t - 20{,}000$$

Add 20,000 to both sides:

$$50{,}000 = 40{,}000(1.005)^t$$

Divide both sides by 40,000:

$$(1.005)^t = \frac{50{,}000}{40{,}000} = 1.25$$

Now, solve for $t$ by taking the natural logarithm of both sides:

$$t \ln(1.005) = \ln(1.25)$$
$$t = \frac{\ln(1.25)}{\ln(1.005)}$$

Using approximate values:

• $\ln(1.25) \approx 0.223144$
  
• $\ln(1.005) \approx 0.0049875$
  

Thus,

$$t \approx \frac{0.223144}{0.0049875} \approx 44.77 \text{ months}$$

Rounding, it will take about 45 months (or approximately 3 years and 9 months) for your account to reach $30,000.