In what ratio must a grocer mix two varieties of rice costing ₹206 and ₹126…

Concept

The alligation rule finds the mixing ratio when two ingredients priced differently combine into a blend at a mean price. If a cheaper ingredient costs c per unit and a dearer ingredient costs d per unit, and they blend to a mean price m, then Qty(cheaper) : Qty(dearer) = (d − m) : (m − c). This holds because the extra amount earned on each unit of the dearer stock, (d − m), exactly offsets the shortfall on each unit of the cheaper stock, (m − c), so the blended revenue matches the mean price times the total quantity.

Application

1. Here the dearer rate is d = ₹206 per kg (first variety) and the cheaper rate is c = ₹126 per kg (second variety), matching the order given in the question.

2. The mean price of the mixture is m = ₹183 per kg.

3. By the alligation rule, cheaper : dearer = (d − m) : (m − c) = (206 − 183) : (183 − 126) = 23 : 57.

4. So dearer : cheaper = 57 : 23 — the ₹206 variety must be mixed with the ₹126 variety, in that order, in the ratio 57 : 23.

Cross-check

Weighted-average check: (206 × 57 + 126 × 23) ÷ (57 + 23) = (11,742 + 2,898) ÷ 80 = 14,640 ÷ 80 = ₹183 per kg, which matches the required mean price and confirms the ratio.

So the required ratio is 57 : 23.