In what ratio must a merchant mix two varieties of oils worth Rs. 60/kg and Rs. 65/kg, so that by selling the mixture at Rs. 68.20/kg, he may gain 10%?

Step 1: Understand the selling scenario

• Selling Price (SP) of mixture = Rs. 68.20 per kg
• Merchant makes a profit of 10%

Step 2: Calculate the Cost Price (CP) of the mixture

Use the profit formula: SP = CP × (1 + Profit%)

68.20 = CP × (1 + 10/100)
68.20 = CP × 1.10
CP = 68.20 / 1.10 = Rs. 62 per kg

Step 3: Use Alligation Rule

We are mixing two oils at Rs. 60 and Rs. 65 per kg to get a mixture worth Rs. 62 per kg

Cheaper         : Rs. 60
Dearer          : Rs. 65
Mean Price      : Rs. 62

Apply alligation:
               60        65
                 \      /
                  \    /
                   62
                  /    \
                 /      \
           65 - 62 = 3   62 - 60 = 2

Step 4: Get the ratio

Required ratio = 3 : 2 (higher price : lower price)

Answer: 3 : 2

The merchant must mix the two oils in the ratio 3:2 (Rs. 65 oil : Rs. 60 oil) to gain 10% by selling at Rs. 68.20/kg.

This problem demonstrates the use of the Alligation Method — a quick technique in quantitative aptitude to find the ratio of mixing two ingredients when the mean value is known. It's commonly asked in banking, SSC, and other competitive exams, and can also be applied to real-life mixing of commodities to achieve a target cost with a desired profit margin.