In what ratio must a grocer mix two varieties of pulses costing ₹ 15 and ₹ 20 per kg respectively so as to get a mixture worth ₹ 16.50 per kg ?

### Concept & Rule of Alligation The Rule of Alligation allows us to find the ratio in which two ingredients at given prices must be mixed to produce a mixture at a desired mean price. $$ \frac{\text{Quantity of cheaper}}{\text{Quantity of dearer}} = \frac{\text{CP of dearer} - \text{Mean price}}{\text{Mean price} - \text{CP of cheaper}} $$ ### Step-by-Step Solution 1. Identify the cost price (CP) of the two varieties: - CP of 1st kind (cheaper) = ₹ 15 - CP of 2nd kind (dearer) = ₹ 20 2. Identify the desired mean price of the mixture = ₹ 16.50 3. Apply the alligation cross-method: - Quantity of cheaper : Quantity of dearer = $(20 - 16.50) : (16.50 - 15)$ - Quantity ratio = $3.50 : 1.50$ 4. Simplify the ratio: - $3.50 : 1.50 = 35 : 15 = 7 : 3$ ### Exam Strategy & Shortcut Write the values in a cross format: 15 20 16.50 3.50 1.50 The ratio is directly $3.50 : 1.50$ which simplifies to $7 : 3$. ### Common Pitfall A common mistake is reversing the ratio by subtracting in the wrong order or matching the wrong quantity with the price. Always remember the cross-pattern maps the difference on the opposite side. ### Final Answer Therefore, the correct answer is **7 : 3**.