Ex 10.2 Class 9 Maths Ganita Manjari Solutions | Sid Classes
Welcome to Sid Classes, your trusted destination for comprehensive academic resources! In this detailed study guide, we present complete, step-by-step solutions for the NCERT Class 9 Maths Ganita Manjari Chapter 10 Exercise 10.2 Weighted Averages and Percentage Calculations. Designed to reflect how top-scoring students write their solutions and reasoning in school examinations, this walkthrough breaks down weighted score expressions, academic performance evaluations, and percentage conversions entirely clear.
🔑 Key Concepts & Formulas:
- Weighted Average Percentage: When combining scores from assessments with different maximum marks and relative weights, each score must first be converted to its fractional (or percentage) form before multiplying by its respective weight.
- General Expression: \(\text{Weighted Percentage} = \frac{\sum \left(\left(\frac{\text{Obtained Marks}}{\text{Maximum Marks}}\right) \times \text{Weight}\right)}{\sum \text{Weights}} \times 100\).
NCERT Class 9 Maths Ganita Manjari Chapter 10 Exercise 10.2 Solutions
Question 1
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?
(i) \(\frac{35 \times 3 + 44 \times 4 + 80 \times 5}{3 + 4 + 5}\)
(ii) \(\frac{\left(\frac{35}{100}\right) \times 3 + \left(\frac{44}{100}\right) \times 4 + \left(\frac{80}{100}\right) \times 5}{3 + 4 + 5}\)
(iii) \(\frac{\left(\frac{35}{50}\right) \times 3 + \left(\frac{44}{60}\right) \times 4 + \left(\frac{80}{100}\right) \times 5}{3 + 4 + 5} \times 100\)
(iv) \(\frac{\left(\frac{35}{50} \times 100\right) \times 3 + \left(\frac{44}{60} \times 100\right) \times 4 + \left(\frac{80}{100} \times 100\right) \times 5}{3 + 4 + 5}\)
Solution:
To find the correct expression for Savitri's annual percentage score, we must ensure two conditions are met:
1. Each raw score must be converted into a true proportion or percentage by dividing by its respective maximum marks (\(50, 60,\) and \(100\)).
2. The weighted average of these proportions must be multiplied by \(100\) to yield a final percentage (or each individual score converted to a percentage first).
* Expression (i) is incorrect because it adds raw marks with different maximum denominators directly without normalization.
* Expression (ii) is incorrect because it divides raw marks by \(100\) instead of their actual maximum marks (\(50\) and \(60\)).
* Expression (iii) is correct because it computes the fractional score for each assessment (\(\frac{35}{50}, \frac{44}{60}, \frac{80}{100}\)), multiplies each by its respective weight (\(3, 4, 5\)), divides by the sum of weights (\(3 + 4 + 5\)), and multiplies the entire result by \(100\) to convert it into a percentage.
* Expression (iv) is also correct because it converts each score into an individual percentage first (\(\times 100\)), multiplies each by its respective weight, and divides by the total sum of weights.
Answer: Expressions (iii) and (iv) both correctly give her annual percentage score (approximately \(75.28\%\)).
Frequently Asked Questions (FAQs)
Q1: Why can't we simply add raw marks when calculating a weighted average?
A: Raw marks have different maximum possible values (e.g., out of 50 vs out of 100). They must first be standardized into fractions or percentages so that each component contributes fairly based on its weight.
Q2: How do ratio weights like \(3 : 4 : 5\) work in weighted averages?
A: The numbers represent the relative importance of each component. They act as multipliers for each score, and the sum of the ratio numbers (\(3 + 4 + 5 = 12\)) serves as the total divisor.
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