A company's sales increased by 2.0% in the first quarter of the year and decreased by 1.5% in the next quarter compared to the first quarter. What is the overall percentage change in the company's sales for the first half of the year?

To find the overall percentage change in the company's sales over the first half of the year, we need to calculate the compounded percentage change over the two quarters.

1. Let the initial sales amount be \( S_0 \).

2. **After the first quarter**, the sales increased by 2.0%. This can be represented as:
   \[
   S_1 = S_0 \times (1 + 0.02) = S_0 \times 1.02
   \]
   where \( S_1 \) is the sales after the first quarter.

3. **After the second quarter**, the sales decreased by 1.5% compared to the first quarter. So, the new sales after the second quarter are:
   \[
   S_2 = S_1 \times (1 - 0.015) = S_0 \times 1.02 \times 0.985
   \]
   where \( S_2 \) is the sales after the second quarter.

4. Now, we calculate the overall percentage change in sales from the beginning to the end of the second quarter:
   \[
   \text{Overall change} = \left( \frac{S_2}{S_0} - 1 \right) \times 100
   \]
   Substituting the values:
   \[
   \frac{S_2}{S_0} = 1.02 \times 0.985 = 1.0047
   \]
   Therefore, the overall percentage change is:
   \[
   \text{Overall change} = (1.0047 - 1) \times 100 = 0.47\%
   \]

Thus, the overall percentage change in the company's sales for the first half of the year is approximately **0.47%**.

Would you like more details on any step?

Here are some related questions to think about:
1. How would the result change if the sales decrease was 2% instead of 1.5%?
2. What is the effect of consecutive percentage increases instead of a mix of increase and decrease?
3. How can we apply this method to calculate percentage changes over multiple periods?
4. How does compounding percentage changes differ from adding simple percentage changes?
5. What happens if the sales decrease in both quarters?

**Tip:** When dealing with percentage changes over multiple periods, always multiply the individual factors rather than simply adding or subtracting percentages.

Learn how to calculate the overall percentage change in sales when sales increase by 2.0% in the first quarter and decrease by 1.5% in the second quarter. This solution uses key percentage change formulas and shows the impact of compounding percentage changes. Suitable for students in Grades 9-11.

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