1. **Problem statement:** A shopkeeper claims to sell goods at a 5% loss but uses a false weight and actually gains 15%. We need to find the actual weight used for 1 kg of goods, rounded to one decimal place. 2. **Let the actual weight used be $x$ kg.** 3. The shopkeeper claims a 5% loss, so the selling price (SP) is 95% of the cost price (CP) per kg. 4. However, by using a false weight $x$ instead of 1 kg, he gains 15%, so the effective SP for $x$ kg is 115% of CP for 1 kg. 5. Set up the equation: $$\text{SP for } x \text{ kg} = 1 \times \text{CP} \times 1.15$$ But SP per kg is 0.95 CP, so: $$x \times 0.95 \times \text{CP} = 1.15 \times \text{CP}$$ 6. Cancel CP from both sides: $$x \times 0.95 = 1.15$$ 7. Solve for $x$: $$x = \frac{1.15}{0.95} = 1.2105$$ 8. This means the shopkeeper uses 1.2105 kg of goods but charges for 1 kg. 9. To find the actual weight used for 1 kg claimed, invert this: $$\text{Actual weight} = \frac{1}{1.2105} = 0.8261 \text{ kg} = 826.1 \text{ g}$$ **Final answer:** The actual weight used is approximately **826.1 g**.