A grocer buys rice at Rs. 1.80/kg and Rs. 1.20/kg. In what proportion should they be mixed so that selling the mixture at Rs. 1.75/kg gives a 25% gain?

Introduction / Context:
We must determine the cost price (CP) target of the mixture given a selling price and desired profit. Then we use alligation to find the mixing ratio between cheaper and dearer varieties that achieves that CP.

Given Data / Assumptions:

• Prices: Rs. 1.80/kg (dearer) and Rs. 1.20/kg (cheaper).
• Selling price (SP) of mixture = Rs. 1.75/kg.
• Desired gain = 25% ⇒ CP of mixture = SP / 1.25.

Concept / Approach:
First compute mixture CP: CP_mix = 1.75 / 1.25 = Rs. 1.40. Then apply alligation: For prices a (dearer), c (cheaper), and mean m, the ratio (quantity at a) : (quantity at c) = (m − c) : (a − m).

Step-by-Step Solution:
CP_mix = 1.75 / 1.25 = 1.40. Using alligation with a = 1.80, c = 1.20, m = 1.40: Ratio (dearer : cheaper) = (1.40 − 1.20) : (1.80 − 1.40) = 0.20 : 0.40 = 1 : 2. Therefore, rice at Rs. 1.80 : Rs. 1.20 should be mixed in 1 : 2.

Verification / Alternative check:
Weighted average with 1 part at 1.80 and 2 parts at 1.20 gives CP = (1.80*1 + 1.20*2)/3 = (1.80 + 2.40)/3 = 4.20/3 = 1.40, as required.

Why Other Options Are Wrong:
2 : 1, 3 : 2, and 3 : 4 do not average to a CP of Rs. 1.40, hence they will not yield 25% gain at Rs. 1.75/kg.

Common Pitfalls:
Using SP in the alligation instead of CP_mix, or reversing the ratio formula. Derive CP first, then apply alligation properly.

Final Answer:
1 : 2