Monthly payment of the future value of P50,000 for 1 year with an interest rate of 10% compounded monthly

To calculate the monthly payment required for a future value (FV) of 50,000 over 1 year with an interest rate of 10% compounded monthly, we can use the **future value of an annuity formula**:

\[
FV = P \cdot \frac{(1 + r)^n - 1}{r}
\]

where:
- \(FV\) is the future value, which is 50,000.
- \(P\) is the monthly payment (what we're solving for).
- \(r\) is the monthly interest rate (annual rate divided by 12 months).
- \(n\) is the total number of payments (12 for 1 year).

### Step 1: Define the Variables
1. **Future Value (\(FV\))**: 50,000
2. **Interest Rate (monthly, \(r\))**: \(10\% \text{ per year} = \frac{10\%}{12} = \frac{0.10}{12} \approx 0.008333\)
3. **Number of Payments (\(n\))**: 12

### Step 2: Substitute into the Formula and Solve for \(P\)

Rearranging the formula to solve for \(P\):
\[
P = \frac{FV \cdot r}{(1 + r)^n - 1}
\]

Substituting in the values:
\[
P = \frac{50000 \times 0.008333}{(1 + 0.008333)^{12} - 1}
\]

### Step 3: Calculate the Monthly Payment

Let's calculate this step-by-step.

1. Calculate \(1 + r = 1 + 0.008333 = 1.008333\).
2. Raise this to the power of 12:
   \[
   (1.008333)^{12} \approx 1.104713
   \]
3. Subtract 1 from the result:
   \[
   1.104713 - 1 = 0.104713
   \]
4. Multiply \(50000 \times 0.008333\):
   \[
   50000 \times 0.008333 = 416.65
   \]
5. Finally, divide by the denominator:
   \[
   P = \frac{416.65}{0.104713} \approx 3977.17
   \]

### Answer
The monthly payment required is approximately **3,977.17**.

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

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Here are some related questions to explore:
1. How would the payment change if the compounding frequency were quarterly instead of monthly?
2. What if the term were extended to 2 years with the same future value and rate?
3. How would the calculation differ if using continuous compounding?
4. What impact does a higher or lower interest rate have on monthly payments?
5. What’s the difference between present value and future value in these types of problems?

**Tip:** Always ensure that the interest rate aligns with the payment frequency (monthly, quarterly, etc.) when calculating financial values.

Learn how to calculate the monthly payment for a future value of 50,000 over 1 year with a 10% interest rate compounded monthly. This example covers annuity concepts and compound interest formulas, ideal for high school students in Grades 11-12.

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