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To solve this compounding interest problem, we will use the formula for compound interest, which is:

\[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
\]

Where:
- $A$ is the amount of money in the account after interest (final value).
- $P$ is the principal amount (initial investment, which we are solving for).
- $r$ is the annual interest rate (expressed as a decimal).
- $n$ is the number of times the interest is compounded per year.
- $t$ is the number of years.
- $A = 138,000$ (the future value of the account),
- $r = 5.3\% = 0.053$,
- $n = 365$ (since it's compounded daily),
- $t = 12$ years.

We need to find $P$, the principal. Rearranging the formula to solve for $P$:

\[
P = \frac{A}{\left( 1 + \frac{r}{n} \right)^{nt}}
\]

Now, plugging in the known values:

\[
P = \frac{138,000}{\left( 1 + \frac{0.053}{365} \right)^{365 \times 12}}
\]

Let me calculate this for you.

\[
P = \frac{138,000}{\left( 1 + 0.000145205 \right)^{4380}}
\]

\[
P = \frac{138,000}{(1.000145205)^{4380}}
\]

Now, calculate the exponentiation:

\[
P = \frac{138,000}{1.922495}
\]

Finally:

\[
P \approx \frac{138,000}{1.922495} \approx 71,799.51
\]

Rounding to the nearest dollar, Kiran needs to invest **$71,800**.

Would you like more details on this calculation or have any questions?

### Related Questions:
1. How does changing the compounding frequency (e.g., monthly instead of daily) affect the required initial investment?
2. What would be the required principal if the interest rate was 6% instead of 5.3%?
3. How does the length of the investment period affect the principal required?
4. What happens if Kiran can only invest $60,000? What would the account value be after 12 years?
5. How would the investment differ if the interest were compounded continuously?

**Tip:** Always check if the interest rate is compounded annually, semi-annually, or daily, as this impacts the final amount significantly.

Kiran is going to invest in an account paying an interest rate of 5.3% compounded daily. How much would Kiran need to invest, to the nearest dollar, for the value of the account to reach $138,000 in 12 years?

Determine the initial investment needed for an account with 5.3% interest compounded daily to reach $138,000 in 12 years. This compound interest problem covers exponential growth and financial mathematics. Suitable for high school students.

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