A fan is listed at 10% loss. If the shopkeeper bought it 16.66% less and sold it for Rs 162 more, he would have gained 12.5%. Find the listed selling price of the fan.

Steps to Solve

1. Define Variables Let the listed selling price (SP) of the fan be ( x ). The cost price (CP) when sold at a 10% loss can be defined as: $$ CP = x \times \left(1 - 0.10\right) = 0.90x $$

2. Calculate the Purchase Price The shopkeeper bought the fan at 16.66% less than the listed selling price. Thus, the buying price (BP) is: $$ BP = x \times \left(1 - 0.1666\right) = 0.8334x $$

3. Setup the Selling Price with Gain If he sold it for Rs 162 more than the BP, the selling price would then be: $$ SP_{actual} = BP + 162 = 0.8334x + 162 $$

4. Express the Gain Condition He would have gained 12.5%, so we can set up the equation for the selling price: $$ SP_{actual} = BP \times \left(1 + 0.125\right) = 0.8334x \times 1.125 $$

5. Equate and Solve for x Now we equate both expressions for ( SP_{actual} ): $$ 0.8334x + 162 = 0.8334x \times 1.125 $$

6. Simplify the Equation Simplify the equation: $$ 162 = 0.8334x \times 1.125 - 0.8334x $$ $$ 162 = 0.8334x \times (1.125 - 1) $$ $$ 162 = 0.8334x \times 0.125 $$

7. Calculate x Solving for ( x ) gives: $$ x = \frac{162}{0.8334 \times 0.125} $$

8. Final Calculation Calculate the above expression.