Suppose the price of a certain item increases by 3.5% a total of 4 times, and then decreases by 1.4% a total of 3 times. By what overall percent did the price increase? 

Round your answer to the nearest tenth of a percent. 



To determine the overall percent change in price after multiple increases and decreases, we need to apply the percent changes sequentially. 

Let’s denote the initial price of the item as $ P $. 

1. **Four 3.5% increases**:  
   Each time the price increases by 3.5%, the new price is $ P \times (1 + 0.035) $. After four such increases, the price becomes:
   \[
   P \times (1.035)^4
   \]

2. **Three 1.4% decreases**:  
   Each time the price decreases by 1.4%, the new price is $ P \times (1 - 0.014) $. After three such decreases, the price becomes:
   \[
   P \times (1.035)^4 \times (0.986)^3
   \]

Now, we calculate the overall percent change by comparing the final price to the initial price $ P $. The formula for overall percent change is:
\[
\text{Percent change} = \left( \frac{\text{Final Price}}{\text{Initial Price}} - 1 \right) \times 100
\]

Substitute the expression for the final price:
\[
\text{Percent change} = \left( (1.035)^4 \times (0.986)^3 - 1 \right) \times 100
\]

Let’s compute this value.The overall percent increase in the price is 10.0%, rounded to the nearest tenth of a percent.

Would you like any additional details or have any questions? Here are 5 related questions to expand on this topic:

1. How would the overall percentage change if the number of increases and decreases were different?
2. What would the final price be if the initial price was $100?
3. How does compounding affect price changes when there are multiple increases and decreases?
4. How does rounding the percentage after each change affect the overall result?
5. Can you calculate the exact percentage change if the number of increases or decreases were continuous (e.g., a smooth change over time)?

**Tip:** When dealing with multiple percentage changes, always apply them sequentially, as the impact compounds over time.

Suppose the price of a certain item increases by 3.5% a total of 4 times, and then decreases by 1.4% a total of 3 times. By what overall percent did the price increase? Round your answer to the nearest tenth of a percent.

Calculate the overall percentage change in price after four 3.5% increases and three 1.4% decreases. Learn how to apply percentage increases and decreases sequentially to determine the final result. Suitable for high school students (Grades 9-12).

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