Diana wants to earn ₱200 on the ₱12000 that she lead for 2½ months. What simple interest rate must she give in order to earn the said amount? show me solution 

To determine the simple interest rate that Diana must offer to earn ₱200 on a loan of ₱12,000 over 2½ months, we can use the simple interest formula:

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

Where:
- \( I \) = Interest earned (₱200 in this case)
- \( P \) = Principal amount (₱12,000)
- \( r \) = Interest rate (this is what we're trying to find)
- \( t \) = Time period (in years)

### Step 1: Convert Time to Years
Given time is 2½ months. First, we convert this to years:
\[
t = \frac{2.5}{12} \approx 0.2083 \text{ years}
\]

### Step 2: Rearrange the Formula to Solve for \( r \)
\[
r = \frac{I}{P \times t}
\]
Substitute the known values:
\[
r = \frac{200}{12000 \times 0.2083}
\]

### Step 3: Perform the Calculation
\[
r = \frac{200}{2500} = 0.08
\]

### Step 4: Convert the Decimal to a Percentage
\[
r = 0.08 \times 100 = 8\%
\]

### Conclusion:
Diana must give an interest rate of **8%** to earn ₱200 on the ₱12,000 loan over 2½ months.

Would you like more details or have any questions?

#### 5 Related Questions:
1. How would the required interest rate change if Diana wanted to earn ₱300 instead?
2. What would be the interest earned if the rate was reduced to 5% for the same period?
3. How does the time period affect the required interest rate to achieve a specific earning?
4. How would you calculate the interest rate if the principal was ₱15,000 instead of ₱12,000?
5. What is the difference between simple and compound interest when calculating earnings?

**Tip:** When dealing with simple interest, always ensure the time period aligns with the interest rate's basis (usually annual).

Learn how to calculate the simple interest rate Diana needs to offer to earn ₱200 on a ₱12,000 loan over 2½ months. Includes step-by-step solution and formula explanation.

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