In a class of 40 students, ratio of students who like badminton to those who like cricket is `3:8 and ` the ratio of students who like tennis to those who like cricket is `2:5.` Which is the more loved sport in the class after cricket ?

Answer

To solve the problem step by step, we will analyze the given ratios and find the number of students who like each sport.

### Step 1: Understand the Ratios
We are given two ratios:
1. The ratio of students who like badminton to those who like cricket is \(3:8\).
2. The ratio of students who like tennis to those who like cricket is \(2:5\).

### Step 2: Assign Variables
Let:
- The number of students who like badminton be \(3x\).
- The number of students who like cricket be \(8x\).
- The number of students who like tennis be \(2y\).
- The number of students who like cricket (again) be \(5y\).

### Step 3: Equate the Cricket Values
Since both ratios include cricket, we can equate the values of cricket:
\[ 8x = 5y \]

### Step 4: Solve for One Variable
From the equation \(8x = 5y\), we can express \(y\) in terms of \(x\):
\[ y = \frac{8x}{5} \]

### Step 5: Find Total Students
The total number of students in the class is 40. Therefore, we can express the total number of students as:
\[ 3x + 8x + 2y = 40 \]

Substituting \(y\) from Step 4 into the equation:
\[ 3x + 8x + 2\left(\frac{8x}{5}\right) = 40 \]
\[ 11x + \frac{16x}{5} = 40 \]

### Step 6: Clear the Fraction
To eliminate the fraction, multiply the entire equation by 5:
\[ 5(11x) + 16x = 200 \]
\[ 55x + 16x = 200 \]
\[ 71x = 200 \]

### Step 7: Solve for \(x\)
Now we can solve for \(x\):
\[ x = \frac{200}{71} \]

### Step 8: Calculate the Number of Students
Now we can find the number of students for each sport:
1. **Cricket**:
\[ 8x = 8 \times \frac{200}{71} = \frac{1600}{71} \approx 22.54 \] (approximately 23 students)
2. **Badminton**:
\[ 3x = 3 \times \frac{200}{71} = \frac{600}{71} \approx 8.45 \] (approximately 8 students)
3. **Tennis**:
Substitute \(y\) back to find tennis:
\[ y = \frac{8x}{5} = \frac{8 \times \frac{200}{71}}{5} = \frac{320}{71} \approx 4.51 \] (approximately 4 students)
\[ 2y = 2 \times \frac{320}{71} = \frac{640}{71} \approx 9 \] (approximately 9 students)

### Step 9: Conclusion
From our calculations:
- Students who like cricket: approximately 23
- Students who like badminton: approximately 8
- Students who like tennis: approximately 9

Thus, after cricket, the sport that is more loved is **tennis**.