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class 9 ganita manjari chapter 10
📖 Textbook Reference: 10.1.3 Custom Weights
The weighted mean is used whenever parts of data require different levels of importance or weightage.
The weights assigned to the values in any data indicate the relative importance/emphasis that each value gets with respect to the others.
Here, $x_1, x_2, x_3$ are the scores, and $W_1, W_2, W_3$ are the specific ratio weights assigned to them.
Example 6: Rehmat’s marks in Maths are as follows: 60% in internal tests, 64% in the project, 73% in the final exam. The annual percentage score is calculated by combining the internals, project, and final exam in the ratio 3 : 2 : 5. What is Rehmat’s annual score in Maths?
1
The weighted mean allows us to assign different levels of importance to different data points.
2
Here, the ratio 3 : 2 : 5 means the internal tests have a weight of 3, the project a weight of 2, and the final exam a heavily emphasized weight of 5.
1
Multiply each percentage score by its corresponding ratio weight: $(60 \times 3)$, $(64 \times 2)$, and $(73 \times 5)$.
2
Add these products together to find the total weighted sum.
3
Divide the total sum by the sum of the ratio weights ($3 + 2 + 5$).
!
The Simple Average Mistake: A common error is completely ignoring the weights and just dividing by 3: $\frac{60+64+73}{3} = 65.6\%$. This treats the internal test as equally important to the final exam, which is incorrect based on the school's given ratio!
Step-by-Step Calculation
Given Data & Weights:
Internal: $x_1 = 60\%$, Weight $w_1 = 3$
Project: $x_2 = 64\%$, Weight $w_2 = 2$
Final: $x_3 = 73\%$, Weight $w_3 = 5$
Summing the weights:
Total Weight = $3 + 2 + 5 = 10$
Applying the weighted mean formula:
$\text{Annual Score} = \frac{(60 \times 3) + (64 \times 2) + (73 \times 5)}{10}$
$\text{Annual Score} = \frac{180 + 128 + 365}{10}$
$\text{Annual Score} = \frac{673}{10} = 67.3\%$
Ans: Rehmat's annual score is 67.3%.
Anjali's scores in English are: 80% in Speaking, 70% in Listening, and 90% in Writing. The overall grade is calculated using the weightage ratio 1 : 2 : 2. What is her final overall percentage?
1
This uses the exact same weighted mean formula as Rehmat's problem, just with different percentages and ratio weights.
1
Multiply each score by its weight: $(80 \times 1) + (70 \times 2) + (90 \times 2)$.
2
Divide by the total sum of the weights: $1 + 2 + 2 = 5$.
!
Make sure you divide by the sum of the weights ($5$), not by the number of subjects ($3$)!
Step-by-Step Calculation
Using the weighted average formula:
$\text{Numerator} = (80 \times 1) + (70 \times 2) + (90 \times 2)$
$= 80 + 140 + 180 = 400$
Sum of the weights:
$\text{Denominator} = 1 + 2 + 2 = 5$
$\text{Final Score} = \frac{400}{5} = 80\%$
Ans: Anjali's final score is 80%.
In Example 6, what would Rehmat's score be if the school decided to give equal weightage to all three assessments?
A. 67.3%
B. 65.6%
C. 73.0%
D. 60.0%
1
"Equal weightage" means finding the simple arithmetic mean (adding the scores and dividing by the count of the scores).
Logical Explanation
If all modes of assessment had an equal weightage, the annual score is obtained by calculating the arithmetic mean of $60\%, 64\%,$ and $73\%$.
$\text{Arithmetic Mean} = \frac{60 + 64 + 73}{3} = \frac{197}{3} \approx 65.6\%$.
Ans: Option (B)
Why is Rehmat's weighted annual score (67.3%) higher than the simple average (65.6%) of his marks?
A. Because the final exam has the lowest score.
B. Because weights always artificially inflate the final average.
C. Because his highest score (73%) carries the highest weight (5), pulling the average up.
D. Because he scored lowest on the internal tests.
1
A weighted average will always be "pulled" toward the values that have the largest weights.
Logical Breakdown
Rehmat's scores are $60\%, 64\%,$ and $73\%$.
His highest score is $73\%$ (Final Exam).
The weights assigned are $3, 2,$ and $5$. The highest weight ($5$) is assigned to the Final Exam.
Because his best performance is given the most importance, it heavily influences the final result, dragging the average higher than a simple mean.
Ans: Option (C)
