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class 9 ganita manjari chapter 10
📖 Textbook Reference: 10.1.1 Average of Averages
When combining data from two separate collections, you must use the weighted average method if the collections have different sizes.
First, find the total sum of values in each collection by multiplying its average by its size.
Weighted Average Formula:
The average score of students on a test in Section A is 72 and that of students in Section B is 76. What is the combined average of both the sections given that Section A has 30 students and Section B has 25 students?
1
When calculating the combined average of two groups with different sizes, we must use the weighted average formula.
2
The formula is $\frac{an + bm}{n + m}$, where $a$ and $b$ are the respective averages, and $n$ and $m$ are the sizes of the two groups.
1
Calculate total marks for Section A: Multiply 72 by 30.
2
Calculate total marks for Section B: Multiply 76 by 25.
3
Add both totals to get the grand total of marks. Then, divide by the total number of students ($30 + 25 = 55$).
!
The "Midpoint" Error: Do not calculate $\frac{72 + 76}{2} = 74$. Because Section A has more students (30) than Section B (25), the final average will be "pulled" slightly closer to Section A's average (72).
Step 1: Calculate Total Scores
Total score of Section A = $72 \times 30 = 2160$
Total score of Section B = $76 \times 25 = 1900$
Grand Total = $2160 + 1900 = 4060$
Step 2: Calculate Combined Average
Total students = $30 + 25 = 55$
Combined Average = $\frac{\text{Grand Total}}{\text{Total Students}}$
$= \frac{4060}{55}$
$= 73.818...$
Ans: The combined average is 73.82.
In a corporate office, there are 40 junior employees with an average daily working time of 8 hours. There are 10 senior executives with an average daily working time of 9.5 hours. What is the average daily working time for the entire office?
1
Use the weighted average formula to combine the two groups based on their sizes.
1
Total hours for juniors = $40 \times 8$.
2
Total hours for executives = $10 \times 9.5$.
3
Find the total sum of hours and divide by the total number of people ($40 + 10 = 50$).
!
There are 4 times as many juniors as executives. The final average will be heavily anchored near the 8-hour mark. If your answer is higher than 8.5, you made an error!
Step-by-Step Calculation
Total junior hours = $40 \times 8 = 320$
Total executive hours = $10 \times 9.5 = 95$
Grand Total = $320 + 95 = 415$ hours
Combined Average = $\frac{415}{50}$
$= 8.3$
Ans: The average working time is 8.3 hours.
In the problem from Exercise 10.1, why does the combined average (73.82) fall closer to Section A's average (72) than Section B's average (76)?
A. Because 72 is numerically lower than 76.
B. Because Section A has a larger number of students, giving it a higher "weight".
C. Because 73.82 is not a whole number.
D. Because Section A's total marks are subtracted from Section B.
1
A weighted average will always be mathematically pulled toward the data group that possesses the larger "weight" (quantity/size).
1
Compare the sizes of the classes. Section A has 30 students, while Section B has 25 students.
!
Option A makes no statistical sense. The value of the average itself does not cause "pull"—only the frequency (number of items) creates the weight.
Logical Explanation
Since Section A has 30 students and Section B only has 25, Section A makes up a larger proportion of the entire 55-student group.
Because there are more students pulling the average toward 72, the final average ends up below the exact midpoint (74), landing at 73.82.
Ans: Option (B)
If Section A had exactly 25 students instead of 30, what would the combined average be?
A. $73.82$
B. $74$
C. $75$
D. $76$
1
If the sizes (weights) of two groups are exactly equal, the weighted average simplifies perfectly into a simple arithmetic mean.
Logical Breakdown
If Section A and Section B both had 25 students, they would have equal weight in the mixture equation.
Therefore, the combined average would sit directly in the middle of 72 and 76.
$\frac{72 + 76}{2} = 74$.
Ans: Option (B)
