A trader has 50 kg of rice, selling part at 14% profit and the rest at 6% loss, with an overall 4% loss. Let's break this down step by step to find the quantities sold at each rate. This is a classic profit-loss mixture problem solved using the alligation method , which finds ratios based on weighted averages of percentages.

Step-by-Step Solution

1. Understand Percentages Relative to Cost Price

Assume the cost price (CP) of 50 kg rice is 100 units for simplicity (percentages work regardless of actual CP).

  • Sold at 14% profit: Selling price (SP) = 114 units per same quantity.
  • Sold at 6% loss: SP = 94 units per same quantity.
  • Overall 4% loss: Total SP = 96 units for entire 50 kg.

Let x kg be sold at 14% profit, and (50 - x) kg at 6% loss.

2. Set Up the Equation

Total SP = (x × 1.14 × CP/kg) + ((50 - x) × 0.94 × CP/kg) = 0.96 × 50 × CP/kg Divide by CP/kg:
1.14x + 0.94(50 - x) = 48
1.14x + 47 - 0.94x = 48
0.20x = 1
x = 5 kg (at 6% loss) Thus, at 14% profit: 50 - 5 = 45 kg.

3. Alligation Method (Faster Visual Way)

     14% (Profit)
  18%  |     2%  
-------o--------  
     -4%    -6% (Loss)
  • Ratio (profit qty : loss qty) = 2 : 18 = 1 : 9.
  • Total parts = 10; each part = 50/10 = 5 kg.
  • Profit qty = 9 × 5 = 45 kg ; loss qty = 1 × 5 = 5 kg.

Verification Table

Quantity| Rate| SP per kg (CP=100)| Total SP
---|---|---|---
45 kg| +14%| 114| 45×114=5130
5 kg| -6%| 94| 5×94=470
50 kg| -4%| -| 5600 (96% of 5000 CP)

TL;DR: 45 kg at 14% profit, 5 kg at 6% loss. Matches across sources—reliable classic problem!

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