Understanding Average Height Calculation

This question asks us to find the combined average height of students from two different classes, given the number of students and the average height for each class separately. To do this, we first need to find the total height of all students in each class and then sum them up. We also need to find the total number of students from both classes. Finally, the combined average height is the total height divided by the total number of students.

Step-by-Step Calculation of Combined Average Height

Here's how we can break down the problem:

1. Calculate Total Height for Class 8 Students

We are given:

• Number of students in Class 8 = 20
• Average height of Class 8 students = 152 cm

The total height of students in Class 8 is the number of students multiplied by their average height.

Total Height (Class 8) $$= \text{Number of students} \times \text{Average height}$$

Total Height (Class 8) $$= 20 \times 152 \text{ cm}$$

Total Height (Class 8) $$= 3040 \text{ cm}$$

2. Calculate Total Height for Class 9 Students

We are given:

• Number of students in Class 9 = 15
• Average height of Class 9 students = 168 cm

The total height of students in Class 9 is the number of students multiplied by their average height.

Total Height (Class 9) $$= \text{Number of students} \times \text{Average height}$$

Total Height (Class 9) $$= 15 \times 168 \text{ cm}$$

Total Height (Class 9) $$= 2520 \text{ cm}$$

3. Calculate Total Number of Students

The total number of students is the sum of students from both classes.

Total Students $$= \text{Students in Class 8} + \text{Students in Class 9}$$

Total Students $$= 20 + 15$$

Total Students $$= 35$$

4. Calculate Combined Total Height

The combined total height is the sum of the total heights of Class 8 and Class 9 students.

Combined Total Height $$= \text{Total Height (Class 8)} + \text{Total Height (Class 9)}$$

Combined Total Height $$= 3040 \text{ cm} + 2520 \text{ cm}$$

Combined Total Height $$= 5560 \text{ cm}$$

5. Calculate Combined Average Height

The combined average height is the combined total height divided by the total number of students.

Combined Average Height $$= \frac{\text{Combined Total Height}}{\text{Total Students}}$$

Combined Average Height $$= \frac{5560 \text{ cm}}{35}$$

Let's perform the division:

$$ \frac{5560}{35} = \frac{1112}{7} \approx 158.857 \text{ cm} $$

6. Round to the Nearest cm

The question asks for the average height to the nearest cm. Rounding 158.857 cm to the nearest whole number gives 159 cm.

Therefore, the average height of the students of both classes to the nearest cm is 159 cm.

Revision Table: Key Calculations for Average Height

Combined Average Height $$= \frac{5560}{35} \approx 158.86$$ cm

Rounded to the nearest cm: 159 cm

Additional Information on Combined Average Height

When calculating the average of combined groups with different numbers of items (like students), this is sometimes referred to as a weighted average. The average height of each class is weighted by the number of students in that class.

The formula for a weighted average when combining two groups is:

$$ \text{Weighted Average} = \frac{(n_1 \times A_1) + (n_2 \times A_2)}{n_1 + n_2} $$

Where:

• $$n_1$$ = Number of items in Group 1 (students in Class 8)
• $$A_1$$ = Average of Group 1 (average height of Class 8)
• $$n_2$$ = Number of items in Group 2 (students in Class 9)
• $$A_2$$ = Average of Group 2 (average height of Class 9)

Using this formula for our problem:

$$ \text{Combined Average Height} = \frac{(20 \times 152) + (15 \times 168)}{20 + 15} $$

$$ \text{Combined Average Height} = \frac{3040 + 2520}{35} $$

$$ \text{Combined Average Height} = \frac{5560}{35} $$

$$ \text{Combined Average Height} \approx 158.857 \text{ cm} $$

Rounding this to the nearest cm gives 159 cm.