The mean weight of all the students in a certain class is 60 kg. The mean weight of the boys from the class is 70 kg, while that of the girls is 55 kg. What is the ratio of number of boys to that of girls?
(a)`2:1`
(b)`1:2`
(c)`1:4`
(d)`4:1`

Options

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

Answer

To solve the problem, we will use the concept of weighted averages to find the ratio of the number of boys to the number of girls in the class.

### Step 1: Define the variables
Let:
- \( b \) = number of boys
- \( g \) = number of girls

### Step 2: Set up the equations based on the mean weights
We know the following:
- The mean weight of all students is 60 kg.
- The mean weight of boys is 70 kg.
- The mean weight of girls is 55 kg.

Using the formula for the mean, we can express the total weight of boys and girls:
\[ \text{Total weight of boys} = 70b \]
\[ \text{Total weight of girls} = 55g \]
The total weight of all students can be expressed as:
\[ \text{Total weight of all students} = 70b + 55g \]
Since the mean weight of all students is 60 kg, we can write:
\[ \frac{70b + 55g}{b + g} = 60 \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 70b + 55g = 60(b + g) \]

### Step 4: Expand the equation
Expanding the right side:
\[ 70b + 55g = 60b + 60g \]

### Step 5: Rearrange the equation
Rearranging the terms gives:
\[ 70b - 60b = 60g - 55g \]
\[ 10b = 5g \]

### Step 6: Simplify the ratio
Dividing both sides by 5:
\[ 2b = g \]
This implies:
\[ \frac{b}{g} = \frac{1}{2} \]

### Step 7: Write the ratio of boys to girls
Thus, the ratio of the number of boys to the number of girls is:
\[ b:g = 1:2 \]

### Conclusion
The answer is (b) \( 1:2 \).

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