Calculating Number of Boys Using Mean Weight

This problem involves finding the number of boys in a class given the total number of students, the overall mean weight, and the mean weights of boys and girls separately. We can solve this using the concept of weighted averages or by considering the total weight of all students.

Understanding the Problem Variables

Let's define the variables given in the question:

• Total number of students (\(\text{N}\)) = 150
• Overall mean weight of all students (\(\bar{W}\)) = 60 kg
• Mean weight of boys (\(\bar{W}_b\)) = 70 kg
• Mean weight of girls (\(\bar{W}_g\)) = 55 kg

We need to find the number of boys, which we can denote as \(\text{N}_b\). Let \(\text{N}_g\) be the number of girls.

Setting Up the Equations

We know that the total number of students is the sum of the number of boys and girls:

\(\text{N} = \text{N}_b + \text{N}_g\)

Substituting the given total number of students:

\(150 = \text{N}_b + \text{N}_g\)

This gives us a relationship between the number of boys and girls:

\(\text{N}_g = 150 - \text{N}_b\)

The total weight of all students is the overall mean weight multiplied by the total number of students:

\(\text{Total Weight} = \text{N} \times \bar{W} = 150 \times 60\)

\(\text{Total Weight} = 9000 \text{ kg}\)

The total weight of all students is also the sum of the total weight of boys and the total weight of girls.

\(\text{Total Weight of Boys} = \text{N}_b \times \bar{W}_b = \text{N}_b \times 70\)

\(\text{Total Weight of Girls} = \text{N}_g \times \bar{W}_g = \text{N}_g \times 55\)

So, we can write the equation for the total weight:

\(\text{N} \times \bar{W} = (\text{N}_b \times \bar{W}_b) + (\text{N}_g \times \bar{W}_g)\)

\(150 \times 60 = (\text{N}_b \times 70) + (\text{N}_g \times 55)\)

\(9000 = 70 \text{N}_b + 55 \text{N}_g\)

Solving for the Number of Boys

Now we have a system of two equations with two variables (\(\text{N}_b\) and \(\text{N}_g\)):

1. \(\text{N}_g = 150 - \text{N}_b\)
2. \(9000 = 70 \text{N}_b + 55 \text{N}_g\)

Substitute the first equation into the second equation to eliminate \(\text{N}_g\):

\(9000 = 70 \text{N}_b + 55 (150 - \text{N}_b)\)

Now, simplify and solve for \(\text{N}_b\):

\(9000 = 70 \text{N}_b + (55 \times 150) - (55 \times \text{N}_b)\)

Calculate \(55 \times 150\):

\(55 \times 150 = 8250\)

So the equation becomes:

\(9000 = 70 \text{N}_b + 8250 - 55 \text{N}_b\)

Combine the terms with \(\text{N}_b\):

\(9000 = (70 - 55) \text{N}_b + 8250\)

\(9000 = 15 \text{N}_b + 8250\)

Subtract 8250 from both sides:

\(9000 - 8250 = 15 \text{N}_b\)

\(750 = 15 \text{N}_b\)

Divide by 15 to find \(\text{N}_b\):

\(\text{N}_b = \frac{750}{15}\)

\(\text{N}_b = 50\)

So, the number of boys in the class is 50.

Verification

Let's check if this answer is consistent with the given information.

If \(\text{N}_b = 50\), then \(\text{N}_g = 150 - 50 = 100\).

Total weight of boys = \(\text{N}_b \times \bar{W}_b = 50 \times 70 = 3500\) kg

Total weight of girls = \(\text{N}_g \times \bar{W}_g = 100 \times 55 = 5500\) kg

Total weight of all students = \(3500 + 5500 = 9000\) kg

Overall mean weight = \(\frac{\text{Total Weight}}{\text{Total Students}} = \frac{9000}{150} = 60\) kg

This matches the given overall mean weight, confirming our calculation is correct.

The number of boys in the class is 50.

Revision Table: Mean Weight Calculation

Additional Information: Concepts Related to Mean Weight

The mean is a measure of central tendency, providing a single value that represents the typical value of a dataset. In this case, the mean weight gives us an idea of the average weight of students in the class or subgroups.

• Simple Mean: Calculated by summing all values and dividing by the number of values. This applies when calculating the mean weight for a homogenous group like only boys or only girls.
• Weighted Mean: Used when different values contribute differently to the total. In this problem, the overall mean weight is a weighted mean of the boys' mean weight and the girls' mean weight. The weights are the number of boys and the number of girls, as larger groups have a greater impact on the overall average.
• Total Sum Method: An alternative approach to weighted mean problems is to calculate the total sum (e.g., total weight) for the entire group and for each subgroup. The total sum of the subgroups must equal the total sum of the entire group. This is the method used in the step-by-step solution above (\(150 \times 60 = \text{N}_b \times 70 + \text{N}_g \times 55\)). This method often feels more intuitive for problems involving counts and averages of subgroups.

Understanding how to work with mean values and total sums for different groups within a population is a fundamental skill in statistics and data analysis.