The mean weight of the students in a certain class is 60 kg. The mean weight of the boys from the class is 70 kg and that of the girls is 55 kg. What is the ratio of the number of boys to that of girls ?

Options

A. `2:1`
B. `1:2`
C. `1:4`
D. `4:1`

Answer

To find the ratio of the number of boys to the number of girls in the class, we can follow these steps:

### Step 1: Define Variables
Let:
- \( n \) = total number of students in the class
- \( n_1 \) = number of boys
- \( n_2 \) = number of girls

### Step 2: Write the Mean Weight Equations
From the problem, we know:
- The mean weight of all students is 60 kg, so:
\[ \text{Total weight of all students} = 60n \]
- The mean weight of boys is 70 kg, so:
\[ \text{Total weight of boys} = 70n_1 \]
- The mean weight of girls is 55 kg, so:
\[ \text{Total weight of girls} = 55n_2 \]

### Step 3: Set Up the Equation for Total Weight
The total weight of all students can also be expressed as the sum of the total weights of boys and girls:
\[ 60n = 70n_1 + 55n_2 \]

### Step 4: Set Up the Equation for Total Students
The total number of students is the sum of the number of boys and girls:
\[ n = n_1 + n_2 \]

### Step 5: Solve the Equations
From the second equation, we can express \( n_2 \) in terms of \( n \) and \( n_1 \):
\[ n_2 = n - n_1 \]

Substituting \( n_2 \) into the first equation:
\[ 60n = 70n_1 + 55(n - n_1) \]
Expanding this gives:
\[ 60n = 70n_1 + 55n - 55n_1 \]
Combining like terms:
\[ 60n = (70n_1 - 55n_1) + 55n \]
\[ 60n = 15n_1 + 55n \]

### Step 6: Isolate \( n_1 \)
Now, we can isolate \( n_1 \):
\[ 60n - 55n = 15n_1 \]
\[ 5n = 15n_1 \]
Dividing both sides by 15:
\[ n_1 = \frac{5n}{15} = \frac{n}{3} \]

### Step 7: Find \( n_2 \)
Using \( n_2 = n - n_1 \):
\[ n_2 = n - \frac{n}{3} = \frac{3n}{3} - \frac{n}{3} = \frac{2n}{3} \]

### Step 8: Find the Ratio of Boys to Girls
Now we can find the ratio of the number of boys to the number of girls:
\[ \text{Ratio} = \frac{n_1}{n_2} = \frac{\frac{n}{3}}{\frac{2n}{3}} = \frac{1}{2} \]

### Final Answer
The ratio of the number of boys to the number of girls is:
\[ \text{Ratio of boys to girls} = 1 : 2 \]

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