The question asks us to calculate the percentage decrease in the number of students participating in Chess from Class A when the number of students not participating in both Chess and Carrom from Class A is increased by a certain percentage. The ratio of students participating in Chess to Carrom remains the same for Class A.

Understanding the Data for Class A

Let's first extract the initial data provided for Class A from the table:

• Total number of students in Class A = 420
• Number of students without participation in both games from Class A = 119
• Ratio of students participating in Chess to Carrom in Class A = 4 ∶ 3

Calculating Initial Number of Participants

The students who are not participating in both games are subtracted from the total number of students to find the number of students who participate in at least one game (either Chess or Carrom, or possibly both, but based on the structure of such problems, "remaining students play either chess or carrom" usually implies mutually exclusive participation categories based on the ratio given, meaning those playing 'either chess or carrom' are the ones counted in the ratio). We will assume 'remaining students play either chess or carrom' means the total number of students minus those playing neither, are split into Chess and Carrom players according to the given ratio.

• Total students participating in at least one game (Chess or Carrom) = Total Students - Students without participation in both
• Initial students participating in at least one game = \(420 - 119 = 301\)

This group of 301 students is divided between Chess and Carrom participants in the ratio 4 ∶ 3. The total ratio parts are \(4 + 3 = 7\).

• Initial number of students participating in Chess from Class A = \(\frac{4}{7} \times 301\)
• Initial number of students participating in Chess from Class A = \(4 \times 43\) (since \(301 \div 7 = 43\))
• Initial number of students participating in Chess from Class A = 172

We can also find the initial Carrom participants: \(\frac{3}{7} \times 301 = 3 \times 43 = 129\). Note that \(172 + 129 = 301\), which matches the total participants.

Calculating the New Number of Non-Participants

The number of students who do not participate in both Chess and Carrom from A is increased by \(\frac{300}{17}\)%. Let's calculate the increase amount.

• Percentage increase = \(\frac{300}{17}\)%
• Increase amount = Initial non-participants \(\times\) Percentage increase
• Increase amount = \(119 \times \left(\frac{300}{17} \div 100\right)\)
• Increase amount = \(119 \times \frac{300}{1700}\)
• Increase amount = \(119 \times \frac{3}{17}\)
• Since \(119 = 17 \times 7\), Increase amount = \( (17 \times 7) \times \frac{3}{17} = 7 \times 3 = 21\)

Now, let's find the new number of students who do not participate in both games.

• New number of students without participation in both games = Initial non-participants + Increase amount
• New number of students without participation in both games = \(119 + 21 = 140\)

Calculating the New Number of Participants

The total number of students in Class A remains 420. With the increased number of non-participants, the number of students participating in at least one game changes.

• New total students participating in at least one game = Total Students - New students without participation in both
• New total students participating in at least one game = \(420 - 140 = 280\)

The ratio of students participating in Chess to Carrom remains the same for A (4 ∶ 3). The new total participants (280) are split according to this ratio.

• Total ratio parts = \(4 + 3 = 7\)
• New number of students participating in Chess from Class A = \(\frac{4}{7} \times 280\)
• New number of students participating in Chess from Class A = \(4 \times 40\) (since \(280 \div 7 = 40\))
• New number of students participating in Chess from Class A = 160

Calculating the Percentage Decrease

The initial number of Chess participants was 172, and the new number is 160. The decrease in the number of Chess participants is:

• Decrease = Initial Chess participants - New Chess participants
• Decrease = \(172 - 160 = 12\)

Now, we calculate the percentage decrease relative to the initial number of Chess participants.

• Percentage Decrease = \(\frac{\text{Decrease}}{\text{Initial Chess participants}} \times 100\%\)
• Percentage Decrease = \(\frac{12}{172} \times 100\%\)
• We can simplify the fraction \(\frac{12}{172}\) by dividing both numerator and denominator by their greatest common divisor, which is 4.
• \(\frac{12 \div 4}{172 \div 4} = \frac{3}{43}\)
• Percentage Decrease = \(\frac{3}{43} \times 100\%\)
• Percentage Decrease = \(\frac{300}{43}\) %

Thus, the number of students participating in Chess from A is decreased by \(\frac{300}{43}\) percent.

Summary of Calculation Steps

1. Find the initial number of students participating in at least one game in Class A.
2. Use the Chess:Carrom ratio to find the initial number of Chess participants.
3. Calculate the increase in the number of students not participating in both games.
4. Calculate the new number of students not participating in both games.
5. Find the new number of students participating in at least one game.
6. Use the same Chess:Carrom ratio to find the new number of Chess participants.
7. Calculate the decrease in Chess participants.
8. Calculate the percentage decrease in Chess participants relative to the initial number.

Revision Table: Class A Participation Analysis

Let's review the key values calculated for Class A.

• Initial Non-Participants: 119
• Initial Total Participants (Chess + Carrom): 301
• Initial Chess Participants: 172
• Percentage Increase in Non-Participants: \(\frac{300}{17}\)%
• Increase in Non-Participants Amount: 21
• New Non-Participants: 140
• New Total Participants (Chess + Carrom): 280
• New Chess Participants: 160
• Decrease in Chess Participants: 12
• Percentage Decrease in Chess Participants: \(\frac{300}{43}\)%

Additional Information: Understanding Percentage Change

Percentage change is a way to express how much a quantity changes relative to its original value. The formula for percentage change is:

\(\text{Percentage Change} = \left(\frac{\text{New Value} - \text{Original Value}}{\text{Original Value}}\right) \times 100\%\)

If the new value is greater than the original value, the percentage change is positive, indicating a percentage increase. If the new value is less than the original value, the percentage change is negative, indicating a percentage decrease. In our calculation, we explicitly calculated the 'decrease' amount and then divided it by the 'initial value' to find the percentage decrease, which is equivalent to using the formula:

\(\text{Percentage Decrease} = \left(\frac{\text{Original Value} - \text{New Value}}{\text{Original Value}}\right) \times 100\%\)

Here, Original Chess Participants = 172 and New Chess Participants = 160. Decrease = \(172 - 160 = 12\).

Percentage Decrease = \(\frac{12}{172} \times 100\% = \frac{3}{43} \times 100\% = \frac{300}{43}\) %.

This confirms our result.