Tea worth Rs. 126 per kg and Rs. 135 per kg are mixed with a third variety in…

Concept: The rule of alligation finds the ratio (or an unknown price) when two ingredients of different prices are mixed to give a specified mean price: if a cheaper ingredient priced x₁ and a dearer ingredient priced x₂ are mixed to give mean price m, then (quantity of cheaper) : (quantity of dearer) = (x₂ − m) : (m − x₁). When three ingredients are combined in stages, first average any two mixed in equal proportion, then apply alligation between that average and the remaining ingredient using their combined ratio.

Application:

1. The first two varieties (Rs. 126 per kg and Rs. 135 per kg) are mixed in equal proportion (1 : 1), so their average price = (126 + 135) / 2 = Rs. 130.50 per kg.

2. The overall ratio of the three varieties is 1 : 1 : 2, so the combined quantity of the first two varieties : quantity of the third variety = (1 + 1) : 2 = 2 : 2 = 1 : 1.

3. Let the price of the third variety be Rs. x per kg. Applying the rule of alligation between the average price Rs. 130.50 and Rs. x, with mean price Rs. 153, in the ratio 1 : 1:

4. (x − 153) / (153 − 130.50) = 1 / 1

5. x − 153 = 22.50

6. x = 175.50

Cross-check: Using the direct weighted-average method with weights 1 : 1 : 2 (total 4 parts): (126 × 1 + 135 × 1 + x × 2) / 4 = 153 ⇒ 261 + 2x = 612 ⇒ x = 175.50, confirming the value found above.