This question asks us to calculate a specific ratio based on student examination results provided in a table. The table details the performance of students in Class IX across four sections (A, B, C, and D) in Science and Mathematics examinations.

We need to find the ratio of:

• The number of students who failed in Science but passed in Mathematics from section B and section D together.
• To the number of students of section A that passed in both subjects.

Calculating the Number of Students for the Ratio

First, let's identify the required numbers from the provided table:

• Number of students who failed in Science but passed in Mathematics in Section B: Look at the row "Failed in Science but passed in Mathematics" and the column "Section B". The number is 12.
• Number of students who failed in Science but passed in Mathematics in Section D: Look at the row "Failed in Science but passed in Mathematics" and the column "Section D". The number is 15.
• Number of students who passed in both subjects in Section A: Look at the row "Passed in both" and the column "Section A". The number is 63.

Step-by-Step Ratio Calculation

Now, let's calculate the required values for the ratio:

1. Find the total number of students who failed in Science but passed in Mathematics from section B and section D together:

   Number of students (Section B + Section D) = (Failed in Science but passed in Mathematics in Section B) + (Failed in Science but passed in Mathematics in Section D)

   Number of students (Section B + Section D) = 12 + 15 = 27

2. The number of students of section A that passed in both subjects is already found: 63.
3. Form the ratio:

   Ratio = (Number of students from Section B and D who failed in Science but passed in Mathematics) : (Number of students from Section A who passed in both)

   Ratio = 27 : 63

4. Simplify the ratio:

   To simplify the ratio 27 : 63, find the greatest common divisor (GCD) of 27 and 63. Both numbers are divisible by 9.

   \( \frac{27}{63} = \frac{27 \div 9}{63 \div 9} = \frac{3}{7} \)

   So, the simplified ratio is 3 : 7.

Final Student Ratio Result

The ratio of the number of students who failed in Science but passed in Mathematics of section B and section D together to the number of students of section A that passed in both subjects is 3 : 7.

Revision Table: Key Data Points

Additional Information on Ratio Calculation

A ratio is a way to compare two or more quantities. It shows how many times one value is contained in another. Ratios can be written in several ways, such as a:b, or \( \frac{a}{b} \). When working with ratios, it is often helpful to simplify them to their simplest form by dividing all parts of the ratio by their greatest common divisor (GCD). In this problem, we compared two groups of students based on their performance in Science and Mathematics exams to find their ratio.