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To solve this problem, let's break it down step by step.

1. **Price of the pony**: The pony costs \( P \) dollars.
2. **Pony tax**: Jack also has to pay a 25% tax on the price of the pony, which is equivalent to \( 0.25P \).

So, the total amount Jack pays is the sum of the price of the pony and the tax:

\[
\text{Total cost} = P + 0.25P
\]

This can be simplified as:

\[
\text{Total cost} = P(1 + 0.25) = 1.25P = \frac{5}{4}P
\]

### Now let's analyze the options:
- **Option A**: \( P + \frac{1}{4} \) — This is incorrect because the tax should be a percentage of \( P \), not a flat fraction added to \( P \).
- **Option B**: \( P + 0.25P \) — This is correct and represents \( P + 25\% \) of \( P \), which is exactly how we calculate the total.
- **Option C**: \( \left(P + \frac{1}{4}\right)P \) — This is incorrect because it incorrectly multiplies \( P \) by \( P + \frac{1}{4} \).
- **Option D**: \( \frac{5}{4}P \) — This is also correct as it simplifies \( P + 0.25P \) to \( \frac{5}{4}P \).
- **Option E**: \( 0.25P \) — This is incorrect because it only represents the tax amount, not the total amount Jack pays.

### Correct Answers: **B** and **D**.

Would you like further explanation or any more details?

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Here are some related questions:
1. How do you convert a percentage to a decimal?
2. Why is the option \( \frac{5}{4}P \) equivalent to \( P + 0.25P \)?
3. Can you simplify the expression \( P + 0.25P \) differently?
4. What happens to the total cost if the tax rate increases to 30%?
5. How would the expression change if the tax was a fixed dollar amount instead?

**Tip**: When adding a percentage, always convert the percentage to a decimal and multiply it by the base value.

Handsome Jack is buying a pony made of diamonds. The price of the pony is P dollars, and Jack also has to pay a 25% diamond pony tax. Which of the following expressions could represent how much Jack pays in total for the pony?

Learn how to calculate the total cost when purchasing an item with a 25% tax. This example problem involves adding percentages to a base price and covers key algebra concepts for students in Grades 6-8.

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