a trader has 50 kg of rice, a part of which he sells at 14% profit and rest at 6% loss. on the whole his loss is 4% . what is the quantity sold at 14% profit and that at 6% loss?

June 28, 2026

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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