NCERT Solutions for Class 9 Maths Chapter 10 — Exercise Set 10.1#
Weighted mean: if values $x_1, x_2, \ldots$ come with weights (sizes) $w_1, w_2, \ldots$,
$$\text{Weighted mean} = \dfrac{w_1x_1 + w_2x_2 + \cdots}{w_1 + w_2 + \cdots}$$To combine averages of groups of different sizes, weight each average by the size of its group.
1. 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? #
Solution
Total marks in Section A $= 72 \times 30 = 2160$, and in Section B $= 76 \times 25 = 1900$.
$$\text{Combined average} = \dfrac{2160 + 1900}{30 + 25}$$$$\text{Combined average} = \dfrac{4060}{55}$$
$$\text{Combined average} \approx 73.82$$
∴ The combined average is about 73.82. (It is closer to 72 than to 76, because Section A has more students.)
2. A farmer mixes three equal quantities of fertilisers. The first one contains $\dfrac{1}{10}$ nitrogen, the second contains $\dfrac{9}{50}$ nitrogen, and the third contains $\dfrac{3}{60}$ nitrogen. What is the fraction of nitrogen in the mixture? #
Solution
The quantities are equal, so all three weights are equal and we take the ordinary average:
$$\text{Nitrogen} = \dfrac{1}{3}\left(\dfrac{1}{10} + \dfrac{9}{50} + \dfrac{3}{60}\right)$$$$\text{Nitrogen} = \dfrac{1}{3}(0.10 + 0.18 + 0.05)$$
$$\text{Nitrogen} = \dfrac{0.33}{3}$$
$$\text{Nitrogen} = 0.11$$
∴ The mixture is $\dfrac{11}{100}$ nitrogen.
3. (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. 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. #
Solution
The purity of the mixture is the weighted mean of the purities, weighted by the amounts:
$$\text{Purity} = \dfrac{9 \times 12 + 5 \times 10 + 17 \times 11}{9 + 5 + 17}$$$$\text{Purity} = \dfrac{108 + 50 + 187}{31}$$
$$\text{Purity} = \dfrac{345}{31}$$
$$\text{Purity} \approx 11.13 \text{ varṇa}$$
∴ The combined gold has a purity of $\dfrac{345}{31}$ ≈ 11.13 varṇa.
4. 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. #
Solution
The months have different numbers of days — May 31, June 30, July 31 — so each average is weighted by its number of days:
$$\text{Combined average} = \dfrac{3.5 \times 31 + 10 \times 30 + 8.7 \times 31}{31 + 30 + 31}$$Its value is:
$$\text{Combined average} = \dfrac{108.5 + 300 + 269.7}{92}$$$$\text{Combined average} = \dfrac{678.2}{92}$$
$$\text{Combined average} \approx 7.37 \text{ mm per day}$$
(The simple average $\dfrac{3.5 + 10 + 8.7}{3} = 7.4$ mm is close here, but it ignores that June has one day fewer.)
5. 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.
$$\text{Concentration} = \dfrac{500 + 2000 + 4500}{600}$$
$$\text{Concentration} = \dfrac{7000}{600}$$
$$\text{Concentration} \approx 11.67\%$$
(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.
$$\text{Concentration} = \dfrac{1500 + 2000 + 1500}{600}$$
$$\text{Concentration} = \dfrac{5000}{600}$$
$$\text{Concentration} \approx 8.33\%$$
The same three strengths give different results, because in (i) most of the liquid is the strong 15% kadha, and in (ii) most of it is the weak 5% one.