If $16,000 is invested in an account for 15 years. Calculate the total interest earned at the end of 15 years if the interest is: 

(a) 7% simple interest: $ Correct 

(b) 7% compounded annually: $ Incorrect 

(c) 7% compounded quarterly: $ Incorrect 

(d) 7% compounded monthly: $ 

Let's break down the calculations for each case of interest.

### Given:
- Principal ($ P $) = $16,000
- Interest rate ($ r $) = 7% = 0.07
- Time ($ t $) = 15 years

### (a) Simple Interest (Correct)

The formula for simple interest is:

\[
I = P \cdot r \cdot t
\]

Substituting the values:

\[
I = 16,000 \cdot 0.07 \cdot 15 = 16,800
\]

The total interest earned is **$16,800**.

### (b) Compounded Annually (Incorrect)

The formula for compound interest is:

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

Where:
- $ P $ is the principal ($16,000),
- $ r $ is the interest rate (0.07),
- $ n $ is the number of times interest is compounded per year (for annually, $ n = 1 $),
- $ t $ is the time (15 years).

Substitute the values:

\[
A = 16,000 \left( 1 + \frac{0.07}{1} \right)^{1 \cdot 15} = 16,000 \left( 1.07 \right)^{15}
\]

Now calculate:

\[
A = 16,000 \cdot 2.759031 \approx 44,144.50
\]

The total amount after 15 years is **$44,144.50**. The total interest earned is:

\[
I = A - P = 44,144.50 - 16,000 = 28,144.50
\]

So the interest earned is **$28,144.50**.

### (c) Compounded Quarterly (Incorrect)

For quarterly compounding, $ n = 4 $. Use the same compound interest formula:

\[
A = 16,000 \left( 1 + \frac{0.07}{4} \right)^{4 \cdot 15}
\]

Substitute the values:

\[
A = 16,000 \left( 1 + 0.0175 \right)^{60} = 16,000 \left( 1.0175 \right)^{60}
\]

Now calculate:

\[
A = 16,000 \cdot 2.848571 \approx 45,577.14
\]

The total amount after 15 years is **$45,577.14**. The total interest earned is:

\[
I = A - P = 45,577.14 - 16,000 = 29,577.14
\]

So the interest earned is **$29,577.14**.

### (d) Compounded Monthly (Final Calculation)

For monthly compounding, $ n = 12 $. Again, use the compound interest formula:

\[
A = 16,000 \left( 1 + \frac{0.07}{12} \right)^{12 \cdot 15}
\]

Substitute the values:

\[
A = 16,000 \left( 1 + 0.0058333 \right)^{180} = 16,000 \left( 1.0058333 \right)^{180}
\]

Now calculate:

\[
A = 16,000 \cdot 2.85941 \approx 45,750.56
\]

The total amount after 15 years is **$45,750.56**. The total interest earned is:

\[
I = A - P = 45,750.56 - 16,000 = 29,750.56
\]

So the interest earned is **$29,750.56**.

---

### Summary:

- **(a)** Simple interest: **$16,800** (Correct)
- **(b)** Compounded annually: **$28,144.50** (Incorrect previously)
- **(c)** Compounded quarterly: **$29,577.14** (Incorrect previously)
- **(d)** Compounded monthly: **$29,750.56**

Would you like any further details on these calculations?

---

### Related Questions:
1. What would the interest be if the rate was 6% compounded quarterly?
2. How would the total interest change if the time period was 10 years instead of 15?
3. What would the interest be if the principal was $20,000, compounded monthly at 7%?
4. How does continuous compounding compare to quarterly compounding in this scenario?
5. What would be the interest if compounded daily at 7% for 15 years?

**Tip:** The more frequently interest is compounded, the greater the total amount accrued over time!

If $16,000 is invested in an account for 15 years. Calculate the total interest earned at the end of 15 years if the interest is: (a) 7% simple interest (b) 7% compounded annually (c) 7% compounded quarterly (d) 7% compounded monthly.

Calculate the total interest for a $16,000 investment over 15 years at 7% interest. Explore simple interest and compound interest (annually, quarterly, monthly) with clear, step-by-step explanations. Suitable for Grades 10-12 students learning about finance and interest calculations.

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