Read the following passage and answer the questions the follow. In a family of five persons A, B, C, D and E, each and everyone loves one another very much. Their birthdays are in different months and on different dates. A remembers that his birthday is between 25th and 30th, of B it is between 20th and 25th, of C it is between 10th and 20th, of D it is between 5th and 10th and of E it is between 1st to 5th of the month, the sum of the date of birth is defined as the addition of the date and the month, for example 12th January will be written as 12/1 and will add to a sum of the date of 13. (Between 25th and 30th includes both 25 and 30).
If the date of birth of four of them are prime numbers, then find the maximum average of the sum of their dates of birth.

Options

A. 27.2
B. 26.4
C. 28
D. None of these

Answer

To solve the problem step by step, let's analyze the information provided and calculate the maximum average of the sum of the dates of birth.

### Step 1: Identify the date ranges for each person
- **A**: Between 25th and 30th (inclusive)
- **B**: Between 20th and 25th (inclusive)
- **C**: Between 10th and 20th (inclusive)
- **D**: Between 5th and 10th (inclusive)
- **E**: Between 1st and 5th (inclusive)

### Step 2: Determine the prime numbers in each range
- **A**: Possible dates: 25, 26, 27, 28, 29, 30 (Prime: 29)
- **B**: Possible dates: 20, 21, 22, 23, 24, 25 (Prime: 23)
- **C**: Possible dates: 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 (Prime: 11, 13, 17, 19)
- **D**: Possible dates: 5, 6, 7, 8, 9, 10 (Prime: 5, 7)
- **E**: Possible dates: 1, 2, 3, 4, 5 (Prime: 2, 3)

### Step 3: Choose the maximum prime dates for A, B, C, D, and E
To maximize the average, we will select the highest prime numbers from each range:
- **A**: 29
- **B**: 23
- **C**: 19 (highest prime from C's range)
- **D**: 7 (highest prime from D's range)
- **E**: 5 (highest prime from E's range)

### Step 4: Calculate the sum of the dates of birth
Now we will calculate the sum of the selected dates:
- Sum = 29 (A) + 23 (B) + 19 (C) + 7 (D) + 5 (E)

Calculating the sum:
- Sum = 29 + 23 + 19 + 7 + 5 = 83

### Step 5: Calculate the average
To find the average, we divide the total sum by the number of persons (5):
- Average = Total Sum / Number of Persons = 83 / 5 = 16.6

### Conclusion
The maximum average of the sum of their dates of birth is **16.6**.

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