Understanding the Problem: Washing Machine Pricing

This question asks us to find the marked price (MP) of a washing machine. We are given the cost price (CP), the percentage discount allowed on the marked price, and the percentage profit gained on the cost price.

Here's what we know:

• Cost Price (CP) = Rs. 6400
• Discount Percentage = 20% (on MP)
• Profit Percentage = 30% (on CP)
• We need to find the Marked Price (MP).

Steps to Calculate the Marked Price

To find the marked price, we first need to determine the selling price (SP) because both profit and discount are related to it. Profit is calculated on the cost price to get the selling price, and discount is calculated on the marked price to get the selling price.

1. Calculate the profit amount based on the cost price and profit percentage.
2. Calculate the selling price by adding the profit to the cost price.
3. Relate the selling price to the marked price and the discount percentage.
4. Use the selling price calculated in step 2 to find the marked price.

Detailed Calculation for Marked Price

Step 1: Calculate the Profit Amount

The dealer gains 30% on the cost price. $$ \text{Profit} = \text{Profit Percentage} \times \text{CP} $$ $$ \text{Profit} = 30\% \times 6400 $$ $$ \text{Profit} = \frac{30}{100} \times 6400 $$ $$ \text{Profit} = 0.30 \times 6400 $$ $$ \text{Profit} = 1920 $$ The profit amount is Rs. 1920.

Step 2: Calculate the Selling Price (SP)

The selling price is the cost price plus the profit. $$ \text{SP} = \text{CP} + \text{Profit} $$ $$ \text{SP} = 6400 + 1920 $$ $$ \text{SP} = 8320 $$ Alternatively, using the percentage: $$ \text{SP} = \text{CP} \times (1 + \text{Profit Percentage}) $$ $$ \text{SP} = 6400 \times (1 + \frac{30}{100}) $$ $$ \text{SP} = 6400 \times (\frac{100+30}{100}) $$ $$ \text{SP} = 6400 \times \frac{130}{100} $$ $$ \text{SP} = 64 \times 130 $$ $$ \text{SP} = 8320 $$ The selling price of the washing machine is Rs. 8320.

Step 3: Relate SP, MP, and Discount

The discount is allowed on the marked price. The selling price is the marked price minus the discount. $$ \text{SP} = \text{MP} - \text{Discount} $$ The discount is 20% of the marked price. $$ \text{Discount} = 20\% \times \text{MP} $$ $$ \text{Discount} = \frac{20}{100} \times \text{MP} $$ $$ \text{Discount} = 0.20 \times \text{MP} $$ So, the selling price can be expressed as: $$ \text{SP} = \text{MP} - 0.20 \times \text{MP} $$ $$ \text{SP} = \text{MP} \times (1 - 0.20) $$ $$ \text{SP} = \text{MP} \times 0.80 $$ Or, using fractions: $$ \text{SP} = \text{MP} \times (1 - \frac{20}{100}) $$ $$ \text{SP} = \text{MP} \times (\frac{100-20}{100}) $$ $$ \text{SP} = \text{MP} \times \frac{80}{100} $$ $$ \text{SP} = \text{MP} \times \frac{4}{5} $$

Step 4: Solve for Marked Price (MP)

We know the SP from Step 2 is Rs. 8320. We can use the relationship from Step 3: $$ \text{SP} = \text{MP} \times 0.80 $$ Substitute the value of SP: $$ 8320 = \text{MP} \times 0.80 $$ Now, solve for MP: $$ \text{MP} = \frac{8320}{0.80} $$ $$ \text{MP} = \frac{8320}{\frac{80}{100}} $$ $$ \text{MP} = 8320 \times \frac{100}{80} $$ $$ \text{MP} = 8320 \times \frac{10}{8} $$ $$ \text{MP} = \frac{83200}{8} $$ $$ \text{MP} = 10400 $$ The marked price of the washing machine is Rs. 10,400.

Summary of Calculations

Thus, the marked price of the washing machine is Rs. 10,400.

Revision Table: Key Concepts

Additional Information: Profit, Loss, and Discount

Understanding the relationship between Cost Price, Selling Price, Marked Price, Profit, Loss, and Discount is fundamental in commercial arithmetic problems. Here are some important points:

• Profit or Loss is always calculated on the Cost Price unless stated otherwise.
• Discount is always calculated on the Marked Price unless stated otherwise.
• The Selling Price is the bridge that connects the Cost Price (via profit/loss) and the Marked Price (via discount).
• If there is no discount, the Selling Price is equal to the Marked Price.
• If an item is sold at a profit, SP is greater than CP. If sold at a loss, SP is less than CP.
• If a discount is given, SP is less than MP.

These concepts are useful in various problems involving buying and selling goods, calculating percentages, and determining pricing strategies.