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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.
A higher weight "pulls" the average closer to its corresponding score. If two students have the same simple average, the student who scores higher in the heavily weighted category will win!
A physical fitness evaluation is being undertaken. The final marks are calculated by combining the marks for strength, flexibility, and agility in the ratio 4 : 5 : 6. Rashi has scored 60, 65, and 70. Keerthi has scored 55, 65, and 75 respectively.
(i) Find out whose total is more without calculating.
(ii) What are their final marks?
(i) Find out whose total is more without calculating.
(ii) What are their final marks?
1
A weighted average heavily favours the categories with the highest weights.
2
If two students have similar scores, you can determine the winner without doing full calculations by comparing their differences and multiplying those differences by the respective weights.
1
Part (i) Logic: Compare Keerthi directly to Rashi. Flexibility is tied (65 vs 65).
2
Keerthi lost $5$ marks in Strength. Strength has a weight of $4$. Impact = $-5 \times 4 = -20$.
3
Keerthi gained $5$ marks in Agility. Agility has a weight of $6$. Impact = $+5 \times 6 = +30$.
4
Part (ii) Calculation: Use $\frac{w_1x_1 + w_2x_2 + w_3x_3}{w_1+w_2+w_3}$ for both girls independently. Total weight = $4+5+6 = 15$.
!
The Simple Average Illusion: If you just look at the raw numbers, Rashi's simple average is $\frac{60+65+70}{3} = 65$, and Keerthi's simple average is $\frac{55+65+75}{3} = 65$. They look exactly tied! This is why part (i) exists—to prove that the weights break the tie.
Part (i): Logical Comparison
Comparing Keerthi's scores to Rashi's scores:
Flexibility: Both scored $65$ (Tie).
Strength (wt 4): Keerthi scored $5$ less. Impact = $-5 \times 4 = -20$.
Agility (wt 6): Keerthi scored $5$ more. Impact = $+5 \times 6 = +30$.
Net impact for Keerthi = $-20 + 30 = +10$.
Because Keerthi gained points in the category with the highest weight, her total must be higher.
Part (ii): Exact Calculations
Sum of Weights = $4 + 5 + 6 = 15$.
Rashi's Final Marks:
$= \frac{(60 \times 4) + (65 \times 5) + (70 \times 6)}{15}$
$= \frac{240 + 325 + 420}{15} = \frac{985}{15} \approx 65.67$
Keerthi's Final Marks:
$= \frac{(55 \times 4) + (65 \times 5) + (75 \times 6)}{15}$
$= \frac{220 + 325 + 450}{15} = \frac{995}{15} \approx 66.33$
Ans: Keerthi has a higher score. Rashi: 65.67, Keerthi: 66.33.
Two students, Amit and Bala, take a test with sections A, B, and C weighted in the ratio 2 : 3 : 5. Amit scores 80, 70, 90. Bala scores 90, 70, 80. Who has the higher overall score and what is it?
1
Use the logical comparison trick first! Amit and Bala both score exactly 240 total raw points. However, the weights drastically change the final outcome.
1
Compare Bala to Amit. Section B is tied.
2
Bala gained 10 points in Section A (weight 2) $\rightarrow +20$.
3
Bala lost 10 points in Section C (weight 5) $\rightarrow -50$. Net impact for Bala is $-30$. Amit must be the winner!
4
Calculate Amit's score: $\frac{(80\times2) + (70\times3) + (90\times5)}{10}$.
!
Assuming a student who scores 90 in the first section automatically wins. Section A only has a weight of 2, so it matters the least!
Step-by-Step Calculation
Total Weight = $2 + 3 + 5 = 10$.
Amit's Score:
$= \frac{(80 \times 2) + (70 \times 3) + (90 \times 5)}{10}$
$= \frac{160 + 210 + 450}{10} = \frac{820}{10} = 82\%$
Bala's Score:
$= \frac{(90 \times 2) + (70 \times 3) + (80 \times 5)}{10}$
$= \frac{180 + 210 + 400}{10} = \frac{790}{10} = 79\%$
Ans: Amit has the higher score (82%).
In the main problem, why do Rashi and Keerthi have the exact same simple average (65) despite having different scores?
A. Because the weights perfectly cancel out their score differences.
B. Because they both scored exactly 65 in flexibility.
C. Because their scores (60, 65, 70) and (55, 65, 75) are perfectly symmetric around the number 65.
D. Because they have the same weighted average.
1
A simple arithmetic mean completely ignores weights. It just looks at the raw numbers.
Logical Explanation
Rashi's simple average: $\frac{60 + 65 + 70}{3} = \frac{195}{3} = 65$.
Keerthi's simple average: $\frac{55 + 65 + 75}{3} = \frac{195}{3} = 65$.
Both sets of numbers sum to $195$ because they are perfectly symmetric sequences centered around $65$. The simple average treats a $55$ and a $75$ exactly the same as a $60$ and a $70$.
Ans: Option (C)
If the school completely reversed the ratio weights to 6 : 5 : 4 (Strength : Flexibility : Agility), who would have the higher score?
A. Rashi, because she scored higher in Strength, which now has the highest weight.
B. Keerthi, because her 75 in Agility will always beat Rashi's 70.
C. They would tie.
D. It cannot be determined without full calculations.
1
The category with the largest weight dictates who wins.
Logical Breakdown
If the weights are reversed to $6 : 5 : 4$, Strength becomes the most important category (weight 6), and Agility becomes the least important (weight 4).
Rashi scored $60$ in Strength, while Keerthi only scored $55$.
Since Rashi dominates the most heavily weighted category, her weighted average will be significantly pulled up, making her the winner.
Ans: Option (A)
