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class 9 ganita manjari chapter 10
📖 Textbook Reference: 10.1.3 Custom Weights
The weighted mean is also 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.
Crucial Rule: Before applying custom ratio weights (like 3:4:5), the data points being weighted must be on an equal scale. You cannot accurately apply weights to a score out of 50 alongside a score out of 100 without converting them to percentages or common fractions first.
Savitri's marks in Kashmiri are as follows: 35 out of 50 in internal tests, 44 out of 60 in the project, and 80 out of 100 in the final exam. The annual percentage score is calculated by combining the internals, project, and final exam in the ratio 3 : 4 : 5. Which of the following expression(s) gives her annual score (as a percentage) in Kashmiri?
Analyze the following expressions:
(i) $$\frac{35 \times 3 + 44 \times 4 + 80 \times 5}{3+4+5}$$
(ii)$$\frac{(\frac{35}{100}) \times 3 + (\frac{44}{100}) \times 4 + (\frac{80}{100}) \times 5}{3+4+5}$$
(iii)$$\frac{(\frac{35}{50}) \times 3 + (\frac{44}{60}) \times 4 + (\frac{80}{100}) \times 5}{3+4+5} \times 100$$
(iv)$$\frac{(\frac{35}{50} \times 100) \times 3 + (\frac{44}{60} \times 100) \times 4 + (\frac{80}{100} \times 100) \times 5}{3+4+5}$$
(ii)
(iii)
(iv)
1
When applying ratio weights (like 3:4:5) to data, the data points must be on the exact same mathematical scale.
2
You cannot directly combine raw marks that are out of 50 with raw marks that are out of 60. You must convert them to fractions (out of 1) or percentages (out of 100) first!
1
Look at option (i). It multiplies the raw scores $35, 44,$ and $80$ by the weights. Is this fair if they have different maximum possible marks?
2
Look at option (ii). It divides every score by 100. Was the internal exam out of 100? No!
3
Look at options (iii) and (iv). They correctly divide the raw score by its actual maximum marks ($\frac{35}{50}$, $\frac{44}{60}$, $\frac{80}{100}$) before applying weights.
!
The Raw Score Trap: Many students mistakenly choose Option (i) because it resembles the standard formula $\frac{w_1x_1 + w_2x_2 + w_3x_3}{w_1+w_2+w_3}$. But $x_1, x_2, x_3$ MUST be standardized first! If you use Option (i), a score of 40/50 would be treated identically to 40/100, which ruins the true average.
Evaluating the Expressions
To find an accurate annual percentage, the scores must be converted to uniform fractions or percentages before applying weights.
Option (i) is Incorrect.
It uses raw marks ($35, 44, 80$) which are out of different base totals. This does not represent an equal scale.
Option (ii) is Incorrect.
It arbitrarily divides everything by $100$ ($\frac{35}{100}$, $\frac{44}{100}$). The internal was out of $50$, not $100$.
Option (iii) is Correct.
It correctly converts each score to a fraction ($\frac{35}{50}$, $\frac{44}{60}$), applies the weights ($3, 4, 5$), divides by the sum of weights, and multiplies the final result by $100$ to get a percentage.
Option (iv) is Correct.
It converts each score into a percentage first (e.g., $\frac{35}{50} \times 100$), then applies the weights, and divides by the sum of the weights.
Ans: Expressions (iii) and (iv) are mathematically correct.
A gymnast receives scores from three judges: Judge A gives 8 out of 10, Judge B gives 15 out of 20, and Judge C gives 45 out of 50. The competition weights the judges' scores in the ratio 2 : 3 : 5 respectively. What is the gymnast's final percentage score?
1
Convert every raw score into a percentage out of 100 first, exactly like Option (iv) in the main question.
2
Multiply each percentage by its assigned weight, sum them up, and divide by the total sum of the weights.
1
Judge A percentage: $\frac{8}{10} \times 100 = 80\%$. Weight = $2$.
2
Judge B percentage: $\frac{15}{20} \times 100 = 75\%$. Weight = $3$.
3
Judge C percentage: $\frac{45}{50} \times 100 = 90\%$. Weight = $5$.
4
Sum of weights = $2 + 3 + 5 = 10$. Calculate the final average!
!
Do not add the raw scores ($8 + 15 + 45$) and try to multiply by the weights. That completely ignores the different scales (10, 20, 50) the judges used!
Step-by-Step Calculation
1. Convert scores to percentages:
Judge A = $80\%$
Judge B = $75\%$
Judge C = $90\%$
2. Apply Weights ($2, 3, 5$):
$\text{Final Score} = \frac{(80 \times 2) + (75 \times 3) + (90 \times 5)}{2 + 3 + 5}$
$\text{Final Score} = \frac{160 + 225 + 450}{10}$
$\text{Final Score} = \frac{835}{10} = 83.5\%$
Ans: The gymnast's final score is 83.5%.
Why is mathematical expression (ii) in the main question completely incorrect for Savitri's marks?
A. Because it uses fractions instead of decimals.
B. Because it falsely divides all the scores by 100 instead of their actual maximum marks (50, 60).
C. Because the denominator of the main fraction is wrong.
D. Because you cannot mathematically apply weights to fractions.
1
A fraction or percentage represents a part out of a whole. You must divide the "part" (marks scored) by the *actual* "whole" (maximum marks) to get a true ratio.
Logical Explanation
Expression (ii) writes the internal marks as $\frac{35}{100}$.
However, Savitri scored $35$ out of $50$, not $35$ out of $100$. This mistake artificially lowers her score significantly.
To combine them fairly, they must be divided by their respective base totals ($\frac{35}{50}$, $\frac{44}{60}$) as shown in expressions (iii) and (iv).
Ans: Option (B)
What would be the total sum of the weights (the denominator) if a new "Viva" assessment was added with a ratio weight of 2 to Savitri's existing 3:4:5 ratio?
A. 10
B. 12
C. 14
D. 4
1
In any weighted average formula $\frac{w_1x_1 + w_2x_2 + w_3x_3 + \dots}{w_1+w_2+w_3 + \dots}$, the denominator is always the sum of all the applied weights.
Denominator Calculation
The original weights given in the problem are $3, 4,$ and $5$.
If a new weight of $2$ is added, the new ratio becomes $3:4:5:2$.
The new denominator is the sum of these weights: $3 + 4 + 5 + 2 = 14$.
Ans: Option (C)
