In what ratio should rice costing Rs. 90 per kg be mixed with rice costing Rs.…

CONCEPT

When a mixture is sold at a gain, first remove the gain from the selling price to find the cost price of the mixture. In alligation, the ratio of the dearer item to the cheaper item equals the opposite differences from the required mean cost.

APPLICATION

1. Selling price of the mixture = Rs. 86.25 per kg and gain = 15%.

2. Cost price of the mixture = 86.25 / 1.15 = Rs. 75 per kg.

3. Dearer rice costs Rs. 90 per kg and cheaper rice costs Rs. 63 per kg, so the required mean cost Rs. 75 lies between them.

4. By alligation, dearer : cheaper = (75 - 63) : (90 - 75) = 12 : 15 = 4 : 5.

CROSS-CHECK

Using 4 : 5, the weighted cost is (4 × 90 + 5 × 63) / 9 = (360 + 315) / 9 = Rs. 75 per kg. Adding 15% gives 75 × 1.15 = Rs. 86.25, matching the given selling price.

CONTRAST

• 5 : 4 gives a higher weighted cost because it puts more quantity on the costlier rice.

• 3 : 5 gives a lower weighted cost because it puts too little quantity on the costlier rice for this target mean.

• 2 : 3 also keeps the weighted cost below the required mean for the stated selling-price condition.

RESULT

The required ratio of Rs. 90 rice to Rs. 63 rice is 4 : 5.