In January, the consumer price index increased by 6%. In February, the index increased compared to January
7% based on the price level, and in March it decreased by 10% based on the February price level. What was the average relative increase in the consumer price index in one month?

Let's break down the problem to calculate the average relative increase in the consumer price index (CPI) over the three months.

### Step 1: Define Variables

Let the initial CPI value in January be \( P_0 = 100 \) (since we are dealing with percentage changes, this is a common starting point).

- January: The CPI increased by 6%. So, the CPI at the end of January is:
  \[
  P_1 = P_0 \times (1 + 0.06) = 100 \times 1.06 = 106
  \]

- February: The CPI increased by 7% relative to January. So, the CPI at the end of February is:
  \[
  P_2 = P_1 \times (1 + 0.07) = 106 \times 1.07 = 113.42
  \]

- March: The CPI decreased by 10% relative to February. So, the CPI at the end of March is:
  \[
  P_3 = P_2 \times (1 - 0.10) = 113.42 \times 0.90 = 102.078
  \]

### Step 2: Calculate the Total Change Over Three Months

Now, we calculate the overall relative change from January to March:

\[
\text{Total change} = \frac{P_3 - P_0}{P_0} = \frac{102.078 - 100}{100} = 0.02078 = 2.078\%
\]

### Step 3: Calculate the Average Relative Increase Per Month

To find the average relative increase per month, we take the total change over three months and divide it by 3:

\[
\text{Average monthly increase} = \frac{2.078\%}{3} = 0.6927\%
\]

Thus, the **average relative increase** in the consumer price index per month over the three-month period is approximately **0.693%**.

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Would you like further details or have any questions?

Here are 5 related questions for deeper understanding:

1. What would the average increase be if the changes in each month were the same percentage?
2. How does a negative percentage change affect the overall calculation of the index?
3. How would the average increase change if the index increased by 5% in March instead of decreasing by 10%?
4. What would happen to the average monthly increase if the decrease in March was larger, say 20%?
5. Can we generalize this method to calculate average monthly changes over any time period?

**Tip:** When calculating average changes over time, small variations in monthly percentages can significantly impact the overall trend.

In January, the consumer price index increased by 6%. In February, the index increased compared to January 7% based on the price level, and in March it decreased by 10% based on the February price level. What was the average relative increase in the consumer price index in one month?

Learn how to calculate the average monthly increase in the Consumer Price Index with percentage changes. This problem involves understanding percentage change calculations and average growth rate over three months with step-by-step solutions. Suitable for students in Grades 10-12.

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