**Given:**
- Class intervals: 0-15, 15-30, 30-45, 45-60, 60-75, 75-90
- Frequencies: 6, 7, x, 15, 10, y
- Sum of fi: 51
- Mode: 55

**To find:**
- Values of x and y

**Solution:**

***Step 1: Calculation of the median class***

We can find the median class by using the formula:

Median class = L + (n/2 - F) * (width)

Where,
- L = lower limit of the median class
- n = total number of observations (sum of all fi)
- F = cumulative frequency of the class preceding the median class
- width = width of the class interval

Using the given data, we can calculate the median class as follows:

n = 6 + 7 + x + 15 + 10 + y = 38 + x + y

F = 6 + 7 + x = 13 + x

width = 15 - 0 = 15

Median class = L + (n/2 - F) * (width)

Let the median class be 45-60. Then,

45 + (38/2 - 13)/15 = 45 + 1/3

The median class is 45-60.

***Step 2: Calculation of the median***

We can find the median by using the formula:

Median = L + ((n/2) - CF) * (width)/f

Where,
- L = lower limit of the median class
- n = total number of observations (sum of all fi)
- CF = cumulative frequency of the class preceding the median class
- width = width of the class interval
- f = frequency of the median class

Using the given data, we can calculate the median as follows:

n = 6 + 7 + x + 15 + 10 + y = 38 + x + y

CF = 6 + 7 + x = 13 + x

width = 15 - 0 = 15

f = 15

Median = L + ((n/2) - CF) * (width)/f

Let the median class be 45-60. Then,

Median = 45 + ((38/2) - 13) * 15/15 = 45 + 12.5 = 57.5

The median is 57.5.

***Step 3: Calculation of the mode class***

We can find the mode class by using the formula:

Mode class = L + ((f1 - f0)/(2*f1 - f0 - f2)) * (width)

Where,
- L = lower limit of the mode class
- f1 = frequency of the mode class
- f0 = frequency of the class preceding the mode class
- f2 = frequency of the class succeeding the mode class
- width = width of the class interval

Using the given data, we can calculate the mode class as follows:

f0 = 7
f1 = 15
f2 = 10
width = 15 - 0 = 15

Mode class = L + ((f1 - f0)/(2*f1 - f0