(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.
Step-by-step solution
To find: The purity, in varṇa, of the melted gold
Idea: Purity is a concentration of gold, so the combined purity is the weighted mean of the purities, with the weights being the amounts of gold-alloy.
- Pure gold in each piece: 9 × 1216, 5 × 1016, 17 × 1116 units. Total pure gold = 108 + 50 + 18716 = 34516 units.1 mark
- Total alloy = 9 + 5 + 17 = 31 units. Fraction of gold = 345/1631 = 34531 × 16.1 mark
- A fraction k16 means k varṇa, so the purity is k = 34531 = 11.129… ≈ 11.13 varṇa. (This is the weighted mean 9 × 12 + 5 × 10 + 17 × 119 + 5 + 17.)1 mark
Check: 11.13 lies between 10 and 12, close to 11 because the largest piece (17 units) is 11 varṇa ✓. In karat: 11.13 × 2416 ≈ 16.7 karat.
Answer to write in the exam
Purity = 9 × 12 + 5 × 10 + 17 × 119 + 5 + 17
= 108 + 50 + 18731 = 34531
∴ Purity of the combined gold ≈ 11.13 varṇa
Common mistakes that cost marks
- Averaging 12, 10 and 11 to get 11. The pieces have different weights (9, 5 and 17 units).
- Dividing by 3 (pieces) instead of 31 (units of gold-alloy).
- Pairing the wrong weight with a purity, such as 5 × 12 instead of 9 × 12.
How this can come in the exam
4 units of gold at 14 varṇa are melted with 4 units at 10 varṇa. The purity of the combined gold is
- 12 varṇa
- 24 varṇa
- 11 varṇa
- 13 varṇa
Show answer
(A) 12 varṇa
Equal amounts, so 14 + 102 = 12 varṇa.
A jeweller melts 2 g of 22-karat gold with 3 g of 18-karat gold. Find the purity of the new gold in karat.
Show answer
Purity = 2 × 22 + 3 × 182 + 3 (1 mark) = 44 + 545 = 985 = 19.6 karat (1 mark).Try one yourself
6 units of gold at 15 varṇa are melted with 2 units at 11 varṇa. Find the purity of the combined gold.
Show answer
6 × 15 + 2 × 118 = 90 + 228 = 1128 = 14 varṇa.
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