Here is a problem that was posed by Mahāvīrāchārya in Gaṇita sāra saṅgraha (c. 850 CE).
The price of 9 citrons and 7 fragrant wood-apples taken together is 107; and the price of 7 citrons and 9 fragrant wood-apples taken together is 101.
O mathematician, tell me quickly the price of each citron and of each fragrant wood-apple.
(Hint: The given problem can be modelled as
9x + 7y = 107
7x + 9y = 101
Can you figure out a way of solving these equations without directly using elimination or the substitution method? What is special in this pair of equations? The x and y coefficients are interchanged in the 2 equations.
What will happen if we add the pair of equations? What will happen if we subtract the pair of equations? Can the resulting equations be solved to find the values of x and y?)
Answer: Adding: 16(x + y) = 208 ⇒ x + y = 13. Subtracting: 2(x − y) = 6 ⇒ x − y = 3. So a citron costs 8 and a wood-apple 5.
Step-by-step solution
Idea: The coefficients are swapped (9, 7 and 7, 9). Adding the equations makes both coefficients 16; subtracting makes them 2 and −2. Both results are very simple equations in x + y and x − y.
- Let a citron cost x and a wood-apple y: 9x + 7y = 107 … (1), 7x + 9y = 101 … (2). Special feature: the coefficients of x and y are interchanged.1 mark
- Add (1) and (2): 16x + 16y = 208 ⇒ x + y = 13 … (3).1 mark
- Subtract (2) from (1): 2x − 2y = 6 ⇒ x − y = 3 … (4).1 mark
- Yes, these can be solved at once: (3) + (4): 2x = 16 ⇒ x = 8; y = 13 − 8 = 5. Citron: 8, wood-apple: 5.1 mark
Each citron costs 8 and each fragrant wood-apple costs 5.
Check: 9(8) + 7(5) = 72 + 35 = 107 ✓; 7(8) + 9(5) = 56 + 45 = 101 ✓.
Answer to write in the exam
9x + 7y = 107 … (1); 7x + 9y = 101 … (2)
(1) + (2): 16x + 16y = 208 ⇒ x + y = 13 … (3)
(1) − (2): 2x − 2y = 6 ⇒ x − y = 3 … (4)
(3) + (4): 2x = 16 ⇒ x = 8; y = 5
∴ Citron = 8, wood-apple = 5
Common mistakes that cost marks
- Subtracting the wrong way and getting x − y = −3, which swaps the prices.
- Forgetting to divide by 16 after adding.
How this can come in the exam
Short answer (3 marks)
Solve 37x + 43y = 123 and 43x + 37y = 117.
Show answer
Add: 80(x + y) = 240 ⇒ x + y = 3 (1 mark). Subtract: −6x + 6y = 6 ⇒ y − x = 1 (1 mark). y = 2, x = 1 (1 mark).Try one yourself
Solve 11x + 13y = 37 and 13x + 11y = 35.
Show answer
Add: x + y = 3; subtract: y − x = 1 ⇒ x = 1, y = 2.
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