Step-by-step solutions for Class 9 Mathematics Ganita Manjari Part 1I Chapter 10 Exercise Set 10.1 (Questions 1–5). Covers combined averages, average of averages, weighted mean, mixtures, concentrations, and historic gold purity problems.
When combining two or more groups with different averages and group sizes, each group's average must be weighted by its size. The combined average is the total sum divided by the total count.
When quantities with different concentrations or values are combined, each quantity acts as a weight. The combined concentration is the sum of (weight × concentration) divided by the total weight.
When all component quantities are equal (w₁ = w₂ = w₃), the weights cancel out, and the mixture concentration equals the simple arithmetic average of the concentrations.
To find the combined daily average over several months, each month's daily average must be multiplied by the exact number of days in that month (e.g. May = 31, June = 30, July = 31).
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?
For two collections with averages a and b, and sizes n and m, respectively, the combined average is:
Section A: a = 72, n = 30
Section B: b = 76, m = 25
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?
The three quantities of fertilisers are equal. Therefore, we can use the weighted-mean formula with equal weights.
For quantities with concentrations x₁, x₂, x₃ and respective quantities w₁, w₂, w₃:
Since the quantities are equal, let:
The concentrations of nitrogen are:
x₁ = 110, x₂ = 950, x₃ = 360
The LCM of 10, 50, 60 is 300.
(Śrīdharācārya, Pāṭigaṇita, 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.
The quantities and purities of the three pieces are different. Therefore, we use the weighted-mean formula.
For quantities w₁, w₂, w₃ having purities x₁, x₂, x₃, respectively:
First piece: Quantity w₁ = 9 units, purity x₁ = 12 varṇa.
Second piece: Quantity w₂ = 5 units, purity x₂ = 10 varṇa.
Third piece: Quantity w₃ = 17 units, purity x₃ = 11 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.
The combined average is the total rainfall divided by the total number of days.
Using the weighted-mean formula:
First, calculate the total rainfall in each month:
May:
June:
July:
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.
The general formula for the concentration of a mixture is:
Here, w₁, w₂, w₃ are the volumes in mL, and x₁, x₂, x₃ are the numerical percentage values.
w₁ = 100, w₂ = 200, w₃ = 300
x₁ = 5, x₂ = 10, x₃ = 15
Answer (i):
w₁ = 300, w₂ = 200, w₃ = 100
x₁ = 5, x₂ = 10, x₃ = 15
Answer (ii):
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