Think and Reflect
Q1. Why did Method 1 not work?
Answer:
Method 1 does not work because it gives equal importance to the average heights of seniors and juniors, even though there are 8 seniors and only 3 juniors. The combined average must take the number of students in each group into account.
Exercise Set 10.1
Q1. The average score of students on a test in Section A is 72 and that of students in Section B is 76. What is the combined average of both the sections given that Section A has 30 students and Section B has 25 students?
Answer:
Given: Average of Section A = 72, number of students = 30;
Average of Section B = 76, number of students = 25.
Formula: Combined average = (Sum of values in both sections)/(Total number of students)
Combined average = (72 × 30 + 76 × 25)/(30 + 25)
= (2160 + 1900)/55 = 4060/55 = 73.82
Final answer: 73.82 marks (approximately).
Q2. A farmer mixes three equal quantities of fertilisers. The first one contains 1/10 nitrogen, the second contains 9/50 nitrogen, and the third contains 3/60 nitrogen. What is the fraction of nitrogen in the mixture?
Answer:
Since the three quantities are equal, the fraction of nitrogen in the mixture is the arithmetic mean of the three fractions.
Fraction of nitrogen = (1/10 + 9/50 + 3/60)/3
= (30/300 + 54/300 + 15/300)/3
= (99/300)/3 = 99/900 = 11/100
Final answer: 11/100.
Q3. (Śrīdharācārya, Pāṭīgaṇita, c. 750 CE) In ancient India, Varṇa was the measure of gold purity. A purity of 16 varṇa meant pure gold; in general, a purity of k varṇa meant that the gold-alloy was k/16 gold and the rest impurities. (Now the term used is karat; 16 Varna = 24 karat.) Suppose a goldsmith melts together three pieces of gold: 9 units at 12 varṇa, 5 units at 10 varṇa, and 17 units at 11 varṇa. Find the purity in varṇa of the combined gold.
Answer:
Formula: Weighted average = (Sum of products of quantities and purities)/(Total quantity)
Purity = (9 × 12 + 5 × 10 + 17 × 11)/(9 + 5 + 17)
= (108 + 50 + 187)/31 = 345/31 ≈ 11.13
Final answer: Approximately 11.13 varṇa.
Q4. The average rainfall per day in the months of May, June, and July in a certain location are 3.5 mm, 10 mm and 8.7 mm respectively. Write an expression that gives their combined average.
Answer:
May, June and July have 31, 30 and 31 days respectively.
Combined average rainfall per day = (3.5 × 31 + 10 × 30 + 8.7 × 31)/(31 + 30 + 31)
Final answer: (108.5 + 300 + 269.7)/92 mm per day.
Q5. Calculate the concentration of spice mix in these two scenarios. (i) A 100 mL kashayam/kadha with 5% spice mix, a 200 mL one with 10% spice mix, and a 300 mL one with 15% spice mix are combined. (ii) A 300 mL kashayam/kadha with 5% spice mix, a 200 mL one with 10% spice mix, and a 100 mL one with 15% spice mix are mixed.
Answer:
(i) Concentration = (100 × 5 + 200 × 10 + 300 × 15)/(100 + 200 + 300)
= (500 + 2000 + 4500)/600 = 7000/600 ≈ 11.67%
Final answer (i): 11.67% approximately.
(ii) Concentration = (300 × 5 + 200 × 10 + 100 × 15)/(300 + 200 + 100)
= (1500 + 2000 + 1500)/600 = 5000/600 ≈ 8.33%
Final answer (ii): 8.33% approximately.