The average height of a class of 50 students is 170 cm. What will be the average height (in cm, rounded off to the nearest integer) of the rest of the students if the average height of 15 of the students is 180 cm?
166
Understanding the Average Height Problem
This question asks us to find the average height of a group of students remaining after a smaller group is separated from a larger class. We are given the total number of students and their average height, as well as the number of students in a subset and their average height. To solve this, we need to use the concept of average and total sum.
Key Concepts: Average and Total Sum
The average of a set of values is calculated by dividing the sum of all values by the number of values. Mathematically:
\[ \text{Average} = \frac{\text{Sum of values}}{\text{Number of values}} \]
From this formula, we can also find the total sum of values if we know the average and the number of values:
\[ \text{Sum of values} = \text{Average} \times \text{Number of values} \]
In this problem, the "values" are the heights of the students, and the "number of values" is the number of students.
Step-by-Step Calculation to Find Average Height
Let's break down the problem into smaller steps to find the average height of the remaining students.
- Find the total height of all students in the class.
- Find the total height of the subset of 15 students.
- Find the number of students remaining after removing the subset.
- Find the total height of the remaining students.
- Calculate the average height of the remaining students.
- Round the final average height to the nearest integer.
Detailed Calculation Steps
Step 1: Total Height of All 50 Students
We know the total number of students is 50 and their average height is 170 cm.
Using the formula \( \text{Sum} = \text{Average} \times \text{Number} \):
Total height of 50 students = \( 170 \text{ cm} \times 50 \)
\[ \text{Total height (50 students)} = 8500 \text{ cm} \]
Step 2: Total Height of the 15 Students
We are given that 15 students have an average height of 180 cm.
Using the formula \( \text{Sum} = \text{Average} \times \text{Number} \):
Total height of 15 students = \( 180 \text{ cm} \times 15 \)
\[ \text{Total height (15 students)} = 2700 \text{ cm} \]
Step 3: Number of Remaining Students
The total number of students is 50, and 15 students are considered as a subset.
Number of remaining students = Total students - Number of students in the subset
Number of remaining students = \( 50 - 15 \)
\[ \text{Number of remaining students} = 35 \]
Step 4: Total Height of Remaining Students
The total height of all 50 students is the sum of the total height of the 15 students and the total height of the remaining 35 students.
Total height (50 students) = Total height (15 students) + Total height (remaining 35 students)
So, Total height (remaining 35 students) = Total height (50 students) - Total height (15 students)
Total height (remaining 35 students) = \( 8500 \text{ cm} - 2700 \text{ cm} \)
\[ \text{Total height (remaining 35 students)} = 5800 \text{ cm} \]
Step 5: Average Height of Remaining Students
Now we have the total height of the remaining 35 students (5800 cm) and the number of remaining students (35).
Using the formula \( \text{Average} = \frac{\text{Sum of values}}{\text{Number of values}} \):
Average height of remaining 35 students = \( \frac{\text{Total height (remaining 35 students)}}{\text{Number of remaining students}} \)
Average height = \( \frac{5800 \text{ cm}}{35} \)
Let's perform the division:
\[ \frac{5800}{35} = \frac{1160}{7} \approx 165.714... \]
Step 6: Rounding to the Nearest Integer
The question asks for the average height rounded off to the nearest integer.
The calculated average height is approximately 165.714 cm. Since the first decimal digit is 7 (which is 5 or greater), we round up the integer part.
Rounded average height = 166 cm.
This calculated average height of 166 cm matches one of the given options.
| Group | Number of Students | Average Height (cm) | Total Height (cm) |
|---|---|---|---|
| All Students | 50 | 170 | \(170 \times 50 = 8500\) |
| Subset Students | 15 | 180 | \(180 \times 15 = 2700\) |
| Remaining Students | \(50 - 15 = 35\) | \(5800 / 35 \approx 165.71\) | \(8500 - 2700 = 5800\) |
| Remaining Students (Rounded Avg) | 35 | 166 (Rounded) | 5800 |
Revision Table: Average Height Calculation
| Concept | Value | Calculation/Notes |
|---|---|---|
| Total Students | 50 | Given |
| Total Average Height | 170 cm | Given |
| Total Height (All) | 8500 cm | \(50 \times 170\) |
| Subset Students | 15 | Given |
| Subset Average Height | 180 cm | Given |
| Total Height (Subset) | 2700 cm | \(15 \times 180\) |
| Remaining Students | 35 | \(50 - 15\) |
| Total Height (Remaining) | 5800 cm | \(8500 - 2700\) |
| Average Height (Remaining) | \(5800/35 \approx 165.71\) cm | Calculated |
| Average Height (Rounded) | 166 cm | Rounded to nearest integer |
Additional Information on Average Calculation
Understanding averages is fundamental in data analysis. The average (or mean) gives us a single value that represents the center of a dataset. When dealing with multiple groups or subsets within a larger group, we can use the property that the total sum of values for the whole group is equal to the sum of the total values for each subgroup.
For example, if you have two groups of students, Group A and Group B, and you know the number of students and average height for each group, you can find the average height of the combined group by:
- Calculating the total height for Group A.
- Calculating the total height for Group B.
- Adding these totals to get the total height for the combined group.
- Adding the number of students in Group A and Group B to get the total number of students.
- Dividing the total height by the total number of students.
This problem is a variation where you know the total group and one subset, and you need to find the characteristics of the other subset (the remaining students). The principle of using total sums remains the same.
Remember that rounding to the nearest integer means if the decimal part is 0.5 or greater, you round up; otherwise, you round down. In our calculation, 165.71... has a decimal part of 0.71..., so we round up to 166.
Important Questions from Average
Average of 40 numbers is 71, if the number 100 replaced by 140, then average is increased by
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