Finding the Cost Price of the Washing Machine: Discount and Profit Calculation

Let's break down this problem step-by-step to find the cost price of the washing machine. We are given the selling price after a discount and the profit percentage if no discount is applied. We need to use these pieces of information to work backwards to the cost price.

Understanding the Terms

• Marked Price (MP): The original price tagged on the washing machine before any discount.
• Discount: A reduction offered on the Marked Price.
• Selling Price (SP): The price at which the washing machine is actually sold after the discount.
• Cost Price (CP): The price at which the shopkeeper bought the washing machine.
• Profit: The gain made when the Selling Price is more than the Cost Price. Profit percentage is calculated on the Cost Price.

Step 1: Calculate the Marked Price (MP)

We know the washing machine was sold for Rs. 29,750 after a 15% discount on the Marked Price. Let the Marked Price be MP.

The selling price (SP) is the Marked Price minus the discount. The discount is 15% of the MP, which can be written as $\frac{15}{100} \times \text{MP}$.

So, the formula for selling price after a discount is:

$\text{SP} = \text{MP} - (\text{Discount \% of MP})$

$\text{SP} = \text{MP} - \frac{15}{100} \times \text{MP}$

$\text{SP} = \text{MP} \times \left(1 - \frac{15}{100}\right)$

$\text{SP} = \text{MP} \times \left(\frac{100 - 15}{100}\right)$

$\text{SP} = \text{MP} \times \frac{85}{100}$

We are given SP = Rs. 29,750.

$29750 = \text{MP} \times \frac{85}{100}$

To find MP, we rearrange the formula:

$\text{MP} = 29750 \times \frac{100}{85}$

Let's calculate the value:

$\text{MP} = 29750 \times \frac{20}{17}$ (Dividing 100 and 85 by 5)

Divide 29750 by 17:

$29750 \div 17 = 1750$

Now, multiply 1750 by 20:

$\text{MP} = 1750 \times 20 = 35000$

So, the Marked Price of the washing machine is Rs. 35,000.

Step 2: Calculate the Cost Price (CP)

The problem states that if no discount is allowed, the shopkeeper gains 12%. If no discount is allowed, the selling price is equal to the Marked Price (MP).

So, in this scenario, the Selling Price (SP) = MP = Rs. 35,000.

This selling price (Rs. 35,000) results in a 12% profit on the Cost Price (CP). The formula for selling price with profit is:

$\text{SP} = \text{CP} + (\text{Profit \% of CP})$

$\text{SP} = \text{CP} + \frac{12}{100} \times \text{CP}$

$\text{SP} = \text{CP} \times \left(1 + \frac{12}{100}\right)$

$\text{SP} = \text{CP} \times \left(\frac{100 + 12}{100}\right)$

$\text{SP} = \text{CP} \times \frac{112}{100}$

We know SP = Rs. 35,000 in this case.

$35000 = \text{CP} \times \frac{112}{100}$

To find CP, we rearrange the formula:

$\text{CP} = 35000 \times \frac{100}{112}$

Let's simplify the fraction $\frac{100}{112}$ by dividing both numerator and denominator by 4:

$\frac{100 \div 4}{112 \div 4} = \frac{25}{28}$

So, $\text{CP} = 35000 \times \frac{25}{28}$

Now, divide 35000 by 28:

$35000 \div 28$

We can simplify $35000/28 = (7 \times 5000) / (7 \times 4) = 5000/4 = 1250$.

Now, multiply 1250 by 25:

$\text{CP} = 1250 \times 25$

$1250 \times 25 = 1250 \times (20 + 5) = (1250 \times 20) + (1250 \times 5) = 25000 + 6250 = 31250$

So, the Cost Price of the washing machine is Rs. 31,250.

Summary of Calculation

The final answer is the cost price we calculated.

The cost price of the washing machine is Rs. 31,250.

Revision Table: Key Concepts in Profit and Loss

Additional Information: Relationship Between MP, SP, CP, Discount, and Profit

This problem highlights the important relationships between Marked Price, Selling Price, and Cost Price under scenarios involving discounts and profits.

• Discount is always calculated on the Marked Price. It reduces the price from MP to SP.
• Profit or Loss is always calculated on the Cost Price. It determines the relationship between CP and SP.
• When a discount is applied, the Selling Price (SP) is less than the Marked Price (MP).
• When there is a profit, the Selling Price (SP) is greater than the Cost Price (CP).
• If an item is sold without a discount, then Selling Price = Marked Price.
• Understanding these relationships and knowing which base price (MP or CP) percentages are calculated on is crucial for solving profit and loss problems with discounts.
• In such problems, you often move between SP and MP using discount information and then between SP and CP using profit/loss information.