Study the following information carefully and answer the given questions.
In a school 4 students A, B, C and D appeared in an English examination which is consist of 15 short and 4 long questions are asked. Each of short question carry 5 marks and each of long questions carry 10 mark and for a spelling error 0.5 mark is deducted. For not attempting a question, zero marks is awarded. In the English exam, A answered (x + 6) short questions, x long questions and did (2x + 10) spelling errors. Ratio of number of spelling errors made by A and B is 8 : 3. Total marks scored by D is 86. Ratio of short and long questions answered by B In the English exam is 5 : 1 and he scored total 102 marks. C scored 35 marks more in short questions than long questions except spelling errors and the number of spelling errors made by C is equal to the difference of the number of spelling errors made by A and B. Number of short questions answered by D is average of that answered by A and B. Percentage of number of long questions answered among all the question answered by D is `20%`. Average of the number of spelling errors made by all four students A, B, C and D is 10. Each student attempts integral number of short as well as long questions.
Ratio of the total number of questions answered by B and that by D is

Options

A. `4 : 3`
B. `4 : 5`
C. ` 6 : 5`
D. Cannot be determined

Explanation

To solve the problem step by step, we will analyze the information given about the students A, B, C, and D and derive the necessary equations to find the required ratio of the total number of questions answered by B and D.

### Step 1: Define Variables
Let:
- \( x \) = number of long questions answered by A
- \( x + 6 \) = number of short questions answered by A
- \( 2x + 10 \) = number of spelling errors made by A

### Step 2: Spelling Errors of B
The ratio of the number of spelling errors made by A and B is given as \( 8:3 \). Therefore, we can express the number of spelling errors made by B as:
\[ \text{Spelling errors of B} = \frac{3}{8} \times (2x + 10) \]

### Step 3: Total Marks for B
B scored a total of 102 marks. The ratio of short to long questions answered by B is \( 5:1 \). Let the number of long questions answered by B be \( y \). Then the number of short questions answered by B is \( 5y \).

The total marks scored by B can be expressed as:
\[ \text{Marks by B} = (5y \times 5) + (y \times 10) - \text{Spelling errors of B} \]
Substituting the spelling errors:
\[ 102 = (5y \times 5) + (y \times 10) - \frac{3}{8} \times (2x + 10) \]

### Step 4: Total Marks for D
D scored a total of 86 marks. The number of short questions answered by D is the average of those answered by A and B:
\[ \text{Short questions by D} = \frac{(x + 6) + 5y}{2} \]
The percentage of long questions answered by D is 20%. Therefore, if \( S_D \) is the total number of questions answered by D:
\[ \text{Long questions by D} = 0.2 \times S_D \]
And the total number of questions answered by D can be expressed as:
\[ S_D = \text{Short questions by D} + \text{Long questions by D} \]

### Step 5: Spelling Errors of C
C scored 35 marks more in short questions than in long questions. The number of spelling errors made by C is equal to the difference of the spelling errors made by A and B:
\[ \text{Spelling errors of C} = (2x + 10) - \frac{3}{8}(2x + 10) \]

### Step 6: Average Spelling Errors
The average of the number of spelling errors made by all four students A, B, C, and D is 10:
\[ \frac{(2x + 10) + \frac{3}{8}(2x + 10) + \text{Spelling errors of C} + \text{Spelling errors of D}}{4} = 10 \]

### Step 7: Solve the Equations
We will solve the equations derived from the total marks and spelling errors for A, B, C, and D to find the values of \( x \) and \( y \).

### Step 8: Calculate Total Questions Answered
Once we have the values of \( x \) and \( y \), we can calculate:
- Total questions answered by B: \( 5y + y = 6y \)
- Total questions answered by D: \( \text{Short questions by D} + \text{Long questions by D} \)

### Step 9: Find the Ratio
Finally, we can find the ratio of the total number of questions answered by B and D:
\[ \text{Ratio} = \frac{6y}{\text{Total questions answered by D}} \]