The ratio of the number of boys to that of girls in a school is 3:4. Two-third of the number of boys are cricket players and the rest are football players. One-fourth of the number of girls are badminton players and the rest are handball players. Which game has the greatest number of participants if the total strength of the school is 1400?

Answer

To solve the problem step by step, we will follow the information given in the question and calculate the number of boys, girls, and participants in each game.

### Step 1: Determine the total number of boys and girls
The ratio of boys to girls is given as 3:4. The total strength of the school is 1400.

1. **Calculate the total parts in the ratio:**
\[ \text{Total parts} = 3 + 4 = 7 \]

2. **Calculate the number of boys:**
\[ \text{Number of boys} = \frac{3}{7} \times 1400 = 600 \]

3. **Calculate the number of girls:**
\[ \text{Number of girls} = \frac{4}{7} \times 1400 = 800 \]

### Step 2: Calculate the number of cricket and football players among boys
We know that two-thirds of the boys are cricket players.

1. **Calculate the number of cricket players:**
\[ \text{Cricket players} = \frac{2}{3} \times 600 = 400 \]

2. **Calculate the number of football players:**
\[ \text{Football players} = \text{Total boys} - \text{Cricket players} = 600 - 400 = 200 \]

### Step 3: Calculate the number of badminton and handball players among girls
We know that one-fourth of the girls are badminton players.

1. **Calculate the number of badminton players:**
\[ \text{Badminton players} = \frac{1}{4} \times 800 = 200 \]

2. **Calculate the number of handball players:**
\[ \text{Handball players} = \text{Total girls} - \text{Badminton players} = 800 - 200 = 600 \]

### Step 4: Compare the number of participants in each game
Now we will summarize the number of participants in each game:

- Cricket players: 400
- Football players: 200
- Badminton players: 200
- Handball players: 600

### Conclusion
The game with the greatest number of participants is **Handball**, with **600 players**.

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