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To solve this problem, we need to compute both the **three-month moving average** and the **five-month moving average** for the sales data, specifically from **June to December**.

### **Given Sales Data:**
- January: 300
- February: 600
- March: 1,000
- April: 2,000
- May: 4,000
- June: 7,000
- July: 22,000
- August: 44,000
- September: 58,000
- October: 65,000

### **1. Three-Month Moving Average:**
To calculate the three-month moving average, we take the average of the sales for the three months preceding each target month. 

**June:**
- Average of March, April, May: \(\frac{1000 + 2000 + 4000}{3} = \frac{7000}{3} \approx 2333.33\)

**July:**
- Average of April, May, June: \(\frac{2000 + 4000 + 7000}{3} = \frac{13000}{3} \approx 4333.33\)

**August:**
- Average of May, June, July: \(\frac{4000 + 7000 + 22000}{3} = \frac{33000}{3} = 11000\)

**September:**
- Average of June, July, August: \(\frac{7000 + 22000 + 44000}{3} = \frac{73000}{3} \approx 24333.33\)

**October:**
- Average of July, August, September: \(\frac{22000 + 44000 + 58000}{3} = \frac{124000}{3} \approx 41333.33\)

### **2. Five-Month Moving Average:**
For the five-month moving average, we average the sales for the five months preceding each target month.

**June:**
- Average of January, February, March, April, May: \(\frac{300 + 600 + 1000 + 2000 + 4000}{5} = \frac{7900}{5} = 1580\)

**July:**
- Average of February, March, April, May, June: \(\frac{600 + 1000 + 2000 + 4000 + 7000}{5} = \frac{14600}{5} = 2920\)

**August:**
- Average of March, April, May, June, July: \(\frac{1000 + 2000 + 4000 + 7000 + 22000}{5} = \frac{36000}{5} = 7200\)

**September:**
- Average of April, May, June, July, August: \(\frac{2000 + 4000 + 7000 + 22000 + 44000}{5} = \frac{79000}{5} = 15800\)

**October:**
- Average of May, June, July, August, September: \(\frac{4000 + 7000 + 22000 + 44000 + 58000}{5} = \frac{135000}{5} = 27000\)

### **Summary of Forecasts:**

#### **Three-Month Moving Average:**
- June: 2333.33
- July: 4333.33
- August: 11000
- September: 24333.33
- October: 41333.33

#### **Five-Month Moving Average:**
- June: 1580
- July: 2920
- August: 7200
- September: 15800
- October: 27000

Would you like further details or have any questions?

### **Related Questions:**
1. How does the moving average method help in forecasting sales?
2. What are the limitations of using a simple moving average for forecasting?
3. Can the moving average method be applied to data with a clear seasonal pattern?
4. What is the difference between a three-month and a five-month moving average in terms of sensitivity to recent changes?
5. How would the inclusion of more data points affect the moving average calculation?

### **Tip:** 
The more periods you include in a moving average, the smoother the forecast becomes, but it may be less sensitive to recent changes in the data.

Forecast the June to December demand by three-month and five-month moving averages.

Learn how to forecast sales demand from June to December using three-month and five-month moving averages. This exercise involves calculating averages based on historical sales data, making it ideal for students studying statistics and forecasting methods in Grades 9-12.

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