A lump of two metals weighing 18 g is worth Rs. 87 but if their weight is…

Concept: When two components at different rates are mixed, and swapping their weights changes the total value, adding the original and swapped value equations cancels the individual weights and leaves only the combined rate of one unit of each component -- this pins down an unknown rate directly. Once both rates are known, the standard Alligation Rule finds the ratio in which the two components must be mixed to produce a given mean rate, by cross-subtracting each rate from the mean.

Applying this to the given lump:

1. Let the weight of the Rs. 6.70/g metal be x g, so the other metal weighs (18 − x) g.

2. Original value: x(6.70) + (18 − x)(p) = 87, where p is the unknown price per gram of the other metal.

3. Swapped value: (18 − x)(6.70) + x(p) = 78.60.

4. Adding the two equations: 18(6.70) + 18(p) = 165.60, so 6.70 + p = 9.20, giving p = Rs. 2.50 per gram.

5. Mean price of the whole 18 g lump = 87/18 per gram.

6. By the Alligation Rule, quantity(6.70-metal) : quantity(2.50-metal) = (mean − 2.50) : (6.70 − mean) = 5 : 4.

7. Since the two parts total 9 and the whole lump is 18 g, the weight of the Rs. 2.50 metal = (4/9) × 18 = 8 g.

Cross-check: with 10 g of the Rs. 6.70 metal and 8 g of the Rs. 2.50 metal, the original value is 10(6.70) + 8(2.50) = 67 + 20 = Rs. 87, and the swapped value is 8(6.70) + 10(2.50) = 53.60 + 25 = Rs. 78.60 -- both match the values given in the question, confirming the weight of the other metal is 8 g.