Average of the weight of 138 students of a school is 45 kg. If the average weight of the boys is 49 kg and the average weight of the girls is 25 kg, then what will be the respective ratio of the total weight of boys and the total weight of girls?
(a)`5:43`
(b)`49:5`
(c)`3:49`
(d)`7:1`

Options

A. `5:43`
B. `49:5`
C. `3:49`
D. `7:1`

Answer

To solve the problem, we need to find the respective ratio of the total weight of boys and girls in a school where the average weights are given. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the given data
- Total number of students = 138
- Average weight of all students = 45 kg
- Average weight of boys = 49 kg
- Average weight of girls = 25 kg

### Step 2: Calculate the total weight of all students
To find the total weight of all students, we multiply the average weight by the total number of students:
\[ \text{Total weight of all students} = \text{Average weight} \times \text{Total number of students} = 45 \, \text{kg} \times 138 = 6210 \, \text{kg} \]

### Step 3: Set up the equations for boys and girls
Let the number of boys be \( b \) and the number of girls be \( g \). We know:
\[ b + g = 138 \]

### Step 4: Express total weights in terms of boys and girls
The total weight of boys can be expressed as:
\[ \text{Total weight of boys} = b \times 49 \]
The total weight of girls can be expressed as:
\[ \text{Total weight of girls} = g \times 25 \]

### Step 5: Set up the equation for total weight
The total weight of all students can also be expressed as:
\[ b \times 49 + g \times 25 = 6210 \]

### Step 6: Solve the system of equations
We have two equations:
1. \( b + g = 138 \)
2. \( 49b + 25g = 6210 \)

From the first equation, we can express \( g \) in terms of \( b \):
\[ g = 138 - b \]

Substituting \( g \) into the second equation:
\[ 49b + 25(138 - b) = 6210 \]
Expanding this gives:
\[ 49b + 3450 - 25b = 6210 \]
Combining like terms:
\[ 24b + 3450 = 6210 \]
Subtracting 3450 from both sides:
\[ 24b = 6210 - 3450 = 2760 \]
Dividing by 24:
\[ b = \frac{2760}{24} = 115 \]

### Step 7: Calculate the number of girls
Using \( b = 115 \) in the equation \( g = 138 - b \):
\[ g = 138 - 115 = 23 \]

### Step 8: Calculate the total weights
Now we can calculate the total weight of boys and girls:
- Total weight of boys:
\[ \text{Total weight of boys} = 115 \times 49 = 5635 \, \text{kg} \]
- Total weight of girls:
\[ \text{Total weight of girls} = 23 \times 25 = 575 \, \text{kg} \]

### Step 9: Find the ratio of total weights
Now we find the ratio of the total weight of boys to the total weight of girls:
\[ \text{Ratio} = \frac{\text{Total weight of boys}}{\text{Total weight of girls}} = \frac{5635}{575} \]

To simplify this ratio:
\[ \text{Ratio} = \frac{5635 \div 575}{575 \div 575} = \frac{49}{5} \]

### Final Answer
Thus, the respective ratio of the total weight of boys to the total weight of girls is:
\[ \text{Ratio} = 49:5 \]