Solution:

To find the missing frequency (f), we can start by analyzing the given information and using the concept of the mode.

Given information:
- Class intervals: 120-130, 130-140, 140-150, 150-160, 160-170-180
- Frequencies: 2, 8, 12, f, 8, 7
- Mode: 154

To find the missing frequency (f), we need to determine the class interval where the mode lies. Since the mode is given as 154, it must fall within the class interval 150-160.

Let's calculate the cumulative frequency for each class interval to identify the position of the mode.

Step 1: Calculate the cumulative frequency

Class Interval Frequency Cumulative Frequency
120-130 2 2
130-140 8 10
140-150 12 22
150-160 f 22 + f
160-170-180 8 22 + f + 8 = 30 + f

Step 2: Determine the position of the mode

The mode is the value that occurs most frequently. In this case, the mode is 154. Since the mode lies within the class interval 150-160, the cumulative frequency for this interval must be greater than the cumulative frequency for the previous interval (140-150) and the next interval (160-170-180).

So, we have the inequality: 22 + f > 12 and 22 + f > 8

Simplifying the inequalities: f > 12 and f > 8

Therefore, the missing frequency (f) must be greater than both 12 and 8.

Step 3: Determine the value of f

Since the mode is the value that occurs most frequently, we can assume that the missing frequency (f) is equal to the difference between the cumulative frequency of the mode and the cumulative frequency of the previous interval.

So, we have: f = Cumulative Frequency of the mode - Cumulative Frequency of the previous interval
= 22 + f - 12
= 10 + f

Simplifying the equation: f - f = 10
0 = 10

This implies that the equation is inconsistent, and there is no value of f that satisfies the given conditions.

Hence, there is no valid value of f that satisfies the given data.

Summary:

- The missing frequency (f) cannot be determined as the equation becomes inconsistent.
- The given data suggests that the mode is 154 and it falls within the class interval 150-160.
- However, without any additional information or constraints, we cannot determine the exact value of the missing frequency (f).