The average monthly precipitation for​ October, November, and December is 3.34 in. If 2.05 in. falls in October and 3.13 in. falls in​ November, how many inches must fall in December so that the average monthly precipitation for these months exceeds 3.34 ​in.?

To find out how many inches of precipitation must fall in December to ensure the average for October, November, and December exceeds 3.34 inches, we'll first set up the equation for the average.

### Let:
- \( x \) be the precipitation in December.

### Given:
- The average precipitation for the three months is \( 3.34 \) inches.
- October's precipitation is \( 2.05 \) inches.
- November's precipitation is \( 3.13 \) inches.

### Average Calculation:
The average monthly precipitation for October, November, and December is:
\[
\text{Average} = \frac{\text{October} + \text{November} + \text{December}}{3}
\]

Substitute the given values and solve for \( x \):
\[
\frac{2.05 + 3.13 + x}{3} > 3.34
\]

### Solving the inequality:
1. Multiply both sides by 3 to eliminate the denominator:
   \[
   2.05 + 3.13 + x > 3 \times 3.34
   \]
   \[
   2.05 + 3.13 + x > 10.02
   \]

2. Combine the known values on the left side:
   \[
   5.18 + x > 10.02
   \]

3. Subtract 5.18 from both sides:
   \[
   x > 10.02 - 5.18
   \]
   \[
   x > 4.84
   \]

### Conclusion:
To ensure the average monthly precipitation exceeds 3.34 inches, more than 4.84 inches of precipitation must fall in December.

Would you like more details or have any questions? Here are some related questions you might find useful:

1. What would the exact average be if exactly 4.84 inches fell in December?
2. How would the calculation change if November's precipitation was 3.5 inches instead of 3.13 inches?
3. If only 4 inches fell in December, what would the average be?
4. How much precipitation would be needed in December to bring the average to 4 inches?
5. What would the total precipitation be for the three months if December receives 5 inches?

**Tip:** When working with averages, multiplying both sides by the number of elements helps eliminate the fraction, simplifying the problem.

Learn how to calculate the amount of precipitation needed in December to ensure the average monthly precipitation for October, November, and December exceeds 3.34 inches. This problem involves basic arithmetic and inequality solving, suitable for Grades 6-8.

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