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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: The combined average is calculated using the weighted mean formula:
Combined Average = $\frac{(72 \times 30) + (76 \times 25)}{30 + 25}$
= $\frac{2160 + 1900}{55} = \frac{4060}{55} \approx 73.81$
A farmer mixes three equal quantities of fertilisers. The first one contains $\frac{1}{10}$ nitrogen, the second contains $\frac{9}{50}$ nitrogen, and the third contains $\frac{3}{60}$ nitrogen. What is the fraction of nitrogen in the mixture?
Answer: Since the quantities are equal, we find the simple average of their concentrations:
Average = $\frac{\frac{1}{10} + \frac{9}{50} + \frac{3}{60}}{3} = \frac{\frac{10}{100} + \frac{18}{100} + \frac{5}{100}}{3} = \frac{\frac{33}{100}}{3} = \frac{11}{100}$ (or 11%).
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: Using the weighted mean:
Purity = $\frac{(9 \times 12) + (5 \times 10) + (17 \times 11)}{9 + 5 + 17}$
= $\frac{108 + 50 + 187}{31} = \frac{345}{31} \approx 11.13$ varṇa.
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: The months of May, June, and July have 31, 30, and 31 days respectively.
Expression: $\frac{(3.5 \times 31) + (10 \times 30) + (8.7 \times 31)}{31 + 30 + 31}$
Calculate the concentration of spice mix in these two scenarios:
(i) A 100 mL kashayam with 5% spice, 200 mL with 10% spice, and 300 mL with 15% spice.
(ii) A 300 mL kashayam with 5% spice, 200 mL with 10% spice, and 100 mL with 15% spice.
Answer:
(i) Concentration = $\frac{(100 \times 5) + (200 \times 10) + (300 \times 15)}{100 + 200 + 300} = \frac{500 + 2000 + 4500}{600} = \frac{7000}{600} \approx 11.67\%$
(ii) Concentration = $\frac{(300 \times 5) + (200 \times 10) + (100 \times 15)}{300 + 200 + 100} = \frac{1500 + 2000 + 1500}{600} = \frac{5000}{600} \approx 8.33\%$
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