Average of the weight of 150 students of a school is 60 kg. If the average weight of the boys is 52 kg and the average weight of the girls is 30 kg, then what will be the respective ratio of the total weight of boys and the total weight of girls?

Options

A. `5:43`
B. `13:2`
C. `3:49`
D. `4:1`

Answer

To solve the problem step by step, we need to find the respective ratio of the total weight of boys and girls based on the given averages and total number of students.

### Step 1: Understand the given data
- Total number of students = 150
- Average weight of all students = 60 kg
- Average weight of boys = 52 kg
- Average weight of girls = 30 kg

### Step 2: Calculate the total weight of all students
The total weight of all students can be calculated using the average weight:
\[ \text{Total weight of all students} = \text{Average weight} \times \text{Total number of students} = 60 \, \text{kg} \times 150 = 9000 \, \text{kg} \]

### Step 3: Set up the equations for boys and girls
Let the number of boys be \( b \) and the number of girls be \( g \).
From the problem, we know:
\[ b + g = 150 \]

### Step 4: Use the average weights to express total weights
The total weight of boys and girls can be expressed as:
\[ \text{Total weight of boys} = 52b \]
\[ \text{Total weight of girls} = 30g \]

### Step 5: Set up the equation for total weight
From the total weight of all students, we can write:
\[ 52b + 30g = 9000 \]

### Step 6: Solve the system of equations
Now we have the two equations:
1. \( b + g = 150 \)
2. \( 52b + 30g = 9000 \)

From the first equation, we can express \( g \) in terms of \( b \):
\[ g = 150 - b \]

Substituting this into the second equation:
\[ 52b + 30(150 - b) = 9000 \]
\[ 52b + 4500 - 30b = 9000 \]
\[ 22b + 4500 = 9000 \]
\[ 22b = 9000 - 4500 \]
\[ 22b = 4500 \]
\[ b = \frac{4500}{22} = 204.54 \text{ (not possible, must be an integer)} \]

### Step 7: Use the rule of alligation
Instead, we can use the rule of alligation to find the ratio of boys to girls based on their average weights:
- Average weight of all students = 60 kg
- Average weight of boys = 52 kg
- Average weight of girls = 30 kg

Using the rule of alligation:
\[ \text{Difference between average of all and average of boys} = 60 - 52 = 8 \]
\[ \text{Difference between average of all and average of girls} = 60 - 30 = 30 \]

### Step 8: Find the ratio of boys to girls
The ratio of the number of boys to girls is given by the inverse of these differences:
\[ \text{Ratio of boys to girls} = 30 : 8 = 15 : 4 \]

### Step 9: Calculate total weight ratio
Now, we can find the total weight of boys and girls:
\[ \text{Total weight of boys} = 52b \]
\[ \text{Total weight of girls} = 30g \]
Using the ratio of boys to girls:
- Let \( b = 15k \) and \( g = 4k \) for some \( k \).

Then:
\[ b + g = 15k + 4k = 19k = 150 \implies k = \frac{150}{19} \]

### Step 10: Calculate total weights
Now, substituting back:
\[ \text{Total weight of boys} = 52(15k) = 780k \]
\[ \text{Total weight of girls} = 30(4k) = 120k \]

### Step 11: Find the ratio of total weights
The ratio of total weights of boys to girls:
\[ \frac{780k}{120k} = \frac{780}{120} = \frac{13}{2} \]

### Final Answer
Thus, the respective ratio of the total weight of boys to the total weight of girls is:
\[ \text{Ratio} = 13 : 2 \]