use percent errors and percent differences A supermarket has a promotion where they give away gift cards to customers who guess closest to the number of balls in a large bin. Jenny guesses there are 152 balls. The manager of the store announces that there are 177 balls in the bin.

To analyze Jenny's guess using percent errors and percent differences, we can calculate as follows:

### 1. Percent Error
The percent error measures how far Jenny's guess was from the actual number of balls.

The formula for percent error is:

\[
\text{Percent Error} = \left|\frac{\text{Actual Value} - \text{Estimated Value}}{\text{Actual Value}}\right| \times 100\%
\]

Plugging in the values:

\[
\text{Percent Error} = \left|\frac{177 - 152}{177}\right| \times 100\%
\]

\[
\text{Percent Error} = \left|\frac{25}{177}\right| \times 100\% \approx 14.12\%
\]

So, Jenny's percent error is approximately **14.12%**.

### 2. Percent Difference
The percent difference is used when comparing two values to see how much they differ relative to their average value.

The formula for percent difference is:

\[
\text{Percent Difference} = \left|\frac{\text{Value 1} - \text{Value 2}}{\frac{\text{Value 1} + \text{Value 2}}{2}}\right| \times 100\%
\]

Plugging in Jenny's guess and the actual number of balls:

\[
\text{Percent Difference} = \left|\frac{177 - 152}{\frac{177 + 152}{2}}\right| \times 100\%
\]

\[
\text{Percent Difference} = \left|\frac{25}{164.5}\right| \times 100\% \approx 15.20\%
\]

So, the percent difference between Jenny's guess and the actual number of balls is approximately **15.20%**.

### Summary:
- **Percent Error**: 14.12%
- **Percent Difference**: 15.20%

Would you like further details or explanations?

Here are some related questions you might find useful:
1. How is percent error different from percent difference?
2. What is the significance of a lower percent error in real-world applications?
3. How can we use percent difference in comparing experimental and theoretical values?
4. What are some practical examples where percent error is used?
5. What factors can cause a high percent error in an experiment?

**Tip**: Always ensure to express percent errors and differences clearly, as they provide valuable insights into the accuracy and precision of estimates or measurements.

Explore how to calculate percent error and percent difference using a real-world example. Learn the formulas and understand their significance in accuracy estimation. Suitable for high school students and those interested in mathematical applications.

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