In what ratio must a grocer mix two varieties of tea worth ₹ 60 a kg and ₹ 65 a kg so that by selling the mixture at ₹ 68.20 a kg he may gain 10%?

### Concept & Alligation with Cost Price The Rule of Alligation must ALWAYS be applied to the Cost Price (CP), not the Selling Price (SP). If SP and profit percentage are given, first calculate the CP of the mixture. $$ \text{CP} = \frac{\text{SP} \times 100}{100 + \text{Profit \%}} $$ ### Step-by-Step Solution 1. Find the CP of the mixture: - Selling Price (SP) = ₹ 68.20 - Gain = $10\%$ - Mean CP = $\frac{68.20 \times 100}{110} = \frac{6820}{110} = \text{₹ } 62$ 2. Apply the alligation method using CPs: - CP of 1st variety = ₹ 60 - CP of 2nd variety = ₹ 65 - Mean CP = ₹ 62 3. Calculate the ratio: - Quantity of 1st : Quantity of 2nd = $(65 - 62) : (62 - 60)$ - Ratio = $3 : 2$ ### Exam Strategy & Shortcut Quickly deduce the CP: $110\%$ of CP = $68.20 \implies \text{CP} = 62$. Then do mentally: difference from 65 is 3, difference from 60 is 2. Thus, the ratio is $3 : 2$. ### Common Pitfall Applying alligation directly to the Selling Price (68.20) alongside the Cost Prices (60 and 65). This is a fatal error and will yield an incorrect result. Always harmonize the units/types to Cost Price. ### Final Answer Therefore, the correct answer is **3 : 2**.