NCERT Solutions Class 9 Mathematics Ganita Manjari 2 Chapter 10 How Quantities Combine Understanding Data Exercise 10.2

Exercise Set 10.2

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) \( \frac{\left(\frac{35}{50} \times 100\right) \times 3 + \left(\frac{44}{60} \times 100\right) \times 4 + \left(\frac{80}{100} \times 100\right) \times 5}{3 + 4 + 5} \)
Answer:
The three marks are out of different maximums (50, 60, 100), so each must first be turned into a fraction or percentage of its own maximum; then the weighted mean with weights 3, 4, 5 is taken.
Percentages: internals 35/50 = 70%, project 44/60 ≈ 73.33%, final 80/100 = 80%.
Annual score = \( \frac{70 \times 3 + 73.33 \times 4 + 80 \times 5}{12} = \frac{210 + 293.33 + 400}{12} = \frac{903.33}{12} \approx 75.28 \)%.
(i) Incorrect. It uses raw marks out of different totals; its value 681 ÷ 12 = 56.75 is not her percentage.
(ii) Incorrect. It divides 35 and 44 by 100 although they are out of 50 and 60; its value 6.81 ÷ 12 ≈ 0.5675 is neither the right fraction nor a percentage.
(iii) Correct. Fractions of each maximum, weighted, then × 100: \( \frac{0.7 \times 3 + 0.7333 \times 4 + 0.8 \times 5}{12} \times 100 \approx 75.28\% \).
(iv) Correct. Percentages first, then the weighted mean: ≈ 75.28%.
Expressions (iii) and (iv) give her annual score, about 75.28%.

Teacher's Note:
1. Put all marks on one scale before applying weights.
2. (iii) and (iv) differ only in where the × 100 is applied.