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class 9 ganita manjari chapter 10
📖 Textbook Reference: Historical Application of Weighted Average
The concept of weighted arithmetic mean was used by Indian mathematicians like Śrīdharācārya (c. 750 CE) to estimate the purity of gold-alloys.
In ancient India, Varṇa was the measure of gold purity (similar to today's karat system). A purity of 16 varṇa meant pure gold.
Historical Note: The weighted mean now plays a crucial role across mathematics and statistics, allowing us to combine data of different sizes into one number to give an overall accurate picture.
(Śrīdharācārya, Pātīgaṇita, c. 750 CE) In ancient India, Varṇa was the measure of gold purity.
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.
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.
1
This historical problem is solved using the exact same Weighted Average formula we use today! The "weights" are the units of gold, and the "values" are the Varṇa purities.
2
Multiply each mass by its purity to find the "total pure units", then divide by the total mass of the new alloy.
1
Total pure units = $(9 \times 12) + (5 \times 10) + (17 \times 11)$.
2
Total mass of alloy = $9 + 5 + 17$.
3
Divide the total pure units by the total mass to find the final Varṇa.
!
Simple Average Trap: Do not calculate $\frac{12 + 10 + 11}{3} = 11$. Since the masses are not equal (9, 5, and 17 units), the final purity must be weighted heavier towards the 17-unit block!
Step 1: Calculate Total Purity Units
Piece 1: $9 \text{ units} \times 12 = 108$
Piece 2: $5 \text{ units} \times 10 = 50$
Piece 3: $17 \text{ units} \times 11 = 187$
Total Purity = $108 + 50 + 187 = 345$
Step 2: Calculate Total Mass
Total Mass = $9 + 5 + 17 = 31\text{ units}$
Step 3: Final Purity Calculation
$\text{Final Varṇa} = \frac{\text{Total Purity}}{\text{Total Mass}}$
$= \frac{345}{31}$
$= 11 \frac{4}{31} \approx 11.13$
Ans: The combined purity is $\frac{345}{31}$ varṇa.
A modern jeweller melts 10 grams of 18-karat gold, 20 grams of 22-karat gold, and 10 grams of 24-karat gold to make a necklace. Find the purity of the new alloy in karats.
1
This uses the exact same logic as the Varṇa problem. Today we use "Karat", but the mathematical principle of weighted averages remains unchanged.
1
Numerator: $(10 \times 18) + (20 \times 22) + (10 \times 24)$.
2
Denominator: $10 + 20 + 10 = 40$ grams.
!
Look at the masses. 20 grams of the alloy is 22-karat. Since this forms half the total mass ($20/40$), the final answer will be heavily anchored towards 22.
Step-by-Step Calculation
Total Purity Units = $180 + 440 + 240 = 860$
Total Mass = $10 + 20 + 10 = 40\text{ grams}$
Final Karat = $\frac{860}{40}$
$= \frac{86}{4}$
$= 21.5$
Ans: The purity of the alloy is 21.5 karats.
In the main exercise question, why would simply taking the average of 12, 10, and 11 varṇa yield the wrong answer?
A. Because 11 varṇa is a prime number.
B. Because the pieces of gold have unequal weights (9, 5, and 17 units).
C. Because varṇa is an ancient measurement system.
D. Because 16 varṇa is required for pure gold.
1
A simple average assumes every component has the exact same weight or influence. The moment components have different sizes, a weighted average must be used.
1
Compare the quantities: 9 units, 5 units, and 17 units. Are they equal?
!
Options C and D state true facts from the text, but they have absolutely nothing to do with the mathematical reason for using a weighted average.
Logical Explanation
Because the 11-varṇa piece is much larger (17 units) than the 10-varṇa piece (5 units), it contributes more to the final purity. A simple average ignores this mass difference.
Ans: Option (B)
If the goldsmith had instead melted three pieces of gold that were all exactly 10 units each (with purities of 12, 10, and 11 varṇa), what would the combined purity be?
A. 10 varṇa
B. 11 varṇa
C. 12 varṇa
D. 33 varṇa
1
When combining components that have exactly equal masses, the "Equal Quantities" shortcut applies. You can find the simple arithmetic mean.
Mathematical Application
Since all three pieces would weigh 10 units, they have equal weight.
Therefore: $\text{Average} = \frac{12 + 10 + 11}{3} = \frac{33}{3} = 11$.
Ans: Option (B)
