Question

A shopkeeper offers a 22% discount on the marked price of chairs. He gives 13 chairs to a customer at the discounted price of 12 chairs. If he still makes a profit of 26%, what is the marked price (MP) of a single chair, assuming its cost price (CP) is Rupees 100?

Hint:

In problems with multiple discounts (e.g., percentage discount plus a quantity offer like 'buy X, pay for Y'), always calculate the total cost of all items given to the customer and the total money received. This simplifies finding the effective profit or loss.

Answer 1

Correct Answer: 175

Step 1: Understanding the Question:
The problem involves a transaction where a shopkeeper sells chairs with two types of discounts: a direct percentage discount and a quantity-based offer. Despite these offers, the shopkeeper makes a profit. We need to find the original marked price (MP) of one chair, given its cost price (CP). We will analyze the transaction for the entire batch of 13 chairs that are sold.
Step 2: Key Formula or Approach:
We will use the fundamental formulas of profit and loss:
1. Total Selling Price (SP): Total Cost Price (CP) + Total Profit
\[ SP_{Total} = CP_{Total} \times (1 + \frac{\text{Profit %}}{100}) \] 2. Discounted Price: Marked Price (MP) - Discount
\[ SP_{Discounted} = MP \times (1 - \frac{\text{Discount %}}{100}) \] The core idea is to find the total cost of the goods sold, calculate the total revenue (selling price) based on the profit, and then work backward using the discount information to find the marked price.
Step 3: Detailed Explanation:
Let's break down the transaction based on the given information. The unit of transaction is 13 chairs.
Part A: Calculate the Total Cost Price (CP)
We are given that the CP of one chair is Rupees 100.
The shopkeeper sells 13 chairs.
\[ \text{Total CP for 13 chairs} = 13 \times 100 = Rupees 1300 \] Part B: Calculate the Total Selling Price (SP)
The shopkeeper makes a profit of 26% on this cost.
\[ \text{Profit} = 26% \text{ of } 1300 = \frac{26}{100} \times 1300 = Rupees 338 \] The total selling price (revenue) for the 13 chairs is:
\[ SP_{Total} = \text{Total CP} + \text{Profit} = 1300 + 338 = Rupees 1638 \] Part C: Relate SP to MP
The problem states that the customer receives 13 chairs but pays the discounted price for only 12 chairs.
So, the total revenue of Rupees 1638 is the price of 12 chairs after the discount.
Let's find the effective discounted selling price of one chair (\(SP_{eff}\)).
\[ 12 \times SP_{eff} = 1638 \] \[ SP_{eff} = \frac{1638}{12} = Rupees 136.5 \] This effective price (Rupees 136.5) is the price of one chair after a 22% discount on its marked price (MP).
Using the discount formula:
\[ SP_{eff} = MP \times (1 - \frac{\text{Discount %}}{100}) \] \[ 136.5 = MP \times (1 - \frac{22}{100}) \] \[ 136.5 = MP \times (1 - 0.22) \] \[ 136.5 = MP \times 0.78 \] Now, we can solve for MP:
\[ MP = \frac{136.5}{0.78} = \frac{13650}{78} = 175 \] So, the marked price of a single chair is Rupees 175.
Step 4: Final Answer:
The marked price (MP) of a single chair is Rupees 175.