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To solve for \( x \), let's determine the value step by step using the given information:

### Given Data:

- **Class Intervals**: 0-10, 10-20, 20-30, 30-40, 40-50, 50-60, 60-70
- **Frequencies**: 2, 3, \( x \), 6, 5, 3, 2
- **Median**: 35

### Step 1: Calculate the total frequency \( N \)

The total frequency \( N \) is the sum of all frequencies:

\[
N = 2 + 3 + x + 6 + 5 + 3 + 2 = 21 + x
\]

### Step 2: Determine the Median Class

The median is given as 35. The median class is the class interval in which the cumulative frequency reaches or exceeds \( \frac{N}{2} \).

\[
\frac{N}{2} = \frac{21 + x}{2}
\]

### Step 3: Calculate Cumulative Frequency

Let's list the cumulative frequencies:

- Cumulative frequency up to 0-10: 2
- Cumulative frequency up to 10-20: \( 2 + 3 = 5 \)
- Cumulative frequency up to 20-30: \( 5 + x = 5 + x \)
- Cumulative frequency up to 30-40: \( 5 + x + 6 = 11 + x \)
- Cumulative frequency up to 40-50: \( 11 + x + 5 = 16 + x \)
- Cumulative frequency up to 50-60: \( 16 + x + 3 = 19 + x \)
- Cumulative frequency up to 60-70: \( 19 + x + 2 = 21 + x \)

Since the median is 35, the median class must be 30-40 because the median class is the one where the cumulative frequency crosses \( \frac{N}{2} \).

### Step 4: Apply the Median Formula

The formula for the median is:

\[
\text{Median} = L + \left(\frac{\frac{N}{2} - F}{f}\right) \times h
\]

Where:
- \( L \) is the lower boundary of the median class = 30
- \( F \) is the cumulative frequency before the median class = \( 5 + x \)
- \( f \) is the frequency of the median class = 6
- \( h \) is the class width = 10
- \( \text{Median} \) = 35

Substituting the values:

\[
35 = 30 + \left(\frac{\frac{21 + x}{2} - (5 + x)}{6}\right) \times 10
\]

### Step 5: Solve the Equation

Simplify the equation:

\[
35 = 30 + \left(\frac{\frac{21 + x}{2} - 5 - x}{6}\right) \times 10
\]

\[
5 = \left(\frac{\frac{21 + x}{2} - 5 - x}{6}\right) \times 10
\]

\[
\frac{5}{10} = \frac{\frac{21 + x}{2} - 5 - x}{6}
\]

\[
\frac{1}{2} = \frac{\frac{21 + x}{2} - 5 - x}{6}
\]

\[
3 = \frac{21 + x - 10 - 2x}{2}
\]

\[
3 = \frac{11 - x}{2}
\]

\[
6 = 11 - x
\]

\[
x = 5
\]

### Final Answer

The value of \( x \) is 5.

Would you like further details on any part of this solution, or do you have any additional questions?

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Here are some related questions for further practice:

1. How would the calculation change if the median were 40 instead of 35?
2. How can you confirm the median class without calculating cumulative frequencies?
3. What is the effect of changing the class interval width on the median?
4. How do you determine the mode for the same frequency distribution?
5. What would be the mean of the given distribution for \( x = 5 \)?

**Tip:** Always check that the cumulative frequency for the median class encompasses \( \frac{N}{2} \), as this confirms the correct median class selection.

Learn how to calculate the median of a frequency distribution with unknown frequencies using a step-by-step solution approach. This problem involves determining the median class and applying the median formula to find the unknown frequency. Suitable for advanced high school students studying statistics.

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