Average of the height of 251 students of a school is 48. If the average of the height of the boys Is 51 and the average of the height of the girls is 29, then what will be the ratio of the total height of boys and the total height of girls respectively?

Options

A. `323 : 29`
B. `31 : 441`
C. `29 : 323`
D. `441 : 23`

Answer

To solve the problem, we need to find the ratio of the total height of boys to the total height of girls given the average heights and the total number of students.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- Total number of students = 251
- Average height of all students = 48
- Average height of boys = 51
- Average height of girls = 29

2. **Calculate the Total Height of All Students**:
\[ \text{Total height of all students} = \text{Average height} \times \text{Total number of students} \]
\[ = 48 \times 251 = 12048 \]

3. **Set Up the Allegation Method**:
- Let the number of boys be \( b \) and the number of girls be \( g \).
- We know that \( b + g = 251 \).

4. **Use the Allegation to Find the Ratio of Boys to Girls**:
- The average height of boys is 51, and the average height of girls is 29.
- The difference between the average height of boys and the overall average:
\[ 51 - 48 = 3 \]
- The difference between the overall average and the average height of girls:
\[ 48 - 29 = 19 \]
- The ratio of boys to girls can be found using the differences:
\[ \text{Ratio of boys to girls} = \frac{19}{3} \]

5. **Express the Number of Boys and Girls in Terms of a Variable**:
- Let \( b = 19x \) and \( g = 3x \).
- Then, from the total number of students:
\[ 19x + 3x = 251 \]
\[ 22x = 251 \implies x = \frac{251}{22} \]

6. **Calculate the Total Height of Boys and Girls**:
- Total height of boys:
\[ \text{Total height of boys} = b \times \text{Average height of boys} = 19x \times 51 \]
- Total height of girls:
\[ \text{Total height of girls} = g \times \text{Average height of girls} = 3x \times 29 \]

7. **Substituting \( x \)**:
- Total height of boys:
\[ = 19 \times \frac{251}{22} \times 51 = \frac{19 \times 251 \times 51}{22} \]
- Total height of girls:
\[ = 3 \times \frac{251}{22} \times 29 = \frac{3 \times 251 \times 29}{22} \]

8. **Finding the Ratio of Total Heights**:
- The ratio of total height of boys to total height of girls:
\[ \text{Ratio} = \frac{19 \times 251 \times 51}{3 \times 251 \times 29} \]
- Cancelling \( 251 \):
\[ = \frac{19 \times 51}{3 \times 29} \]

9. **Final Calculation**:
- Calculate \( 19 \times 51 = 969 \) and \( 3 \times 29 = 87 \):
\[ \text{Ratio} = \frac{969}{87} \]
- Simplifying gives:
\[ = 11.14 \text{ (approximately)} \]

### Final Ratio:
Thus, the ratio of the total height of boys to the total height of girls is approximately \( 11:1 \).