Question

A retailer bought \(20 \mathrm{~kg}\) tea at a discount of \(10 \% .\) Besides \(1 \mathrm{~kg}\) tea was freely offered to him by the wholesaler at the purchase of \(20 \mathrm{~kg}\) tea. Now he sells all the tea at the marked price to a customer. What is profit percentage of retailer? (a) \(30 \%\) (b) \(12 \%\) (c) \(16.66 \%\) (d) none of these

Short answer

Answer: The profit percentage of the retailer is 16.66%.

Step-by-step solution

Calculate the cost price of tea per kg

Firstly, find the cost price per kg of tea before discount. Let it be \(x\) dollars per kg. Then, the cost price of 20 kg tea would be \(20x\) dollars.

Apply the discount to the cost price

Given a discount of 10%, we need to calculate the cost price after applying this discount. The cost price after discount can be calculated as \(20x \cdot (1 - 0.10)\) which equals \(18x\) dollars.

Calculate the effective cost price per kg after receiving free 1 kg

Now, besides the purchased 20 kg tea, the retailer also receives 1 kg for free. In total, the retailer has 21 kg of tea. The effective cost price per kg of tea can be calculated by dividing the cost price after discount by the total weight of tea (21 kg): \(\frac{18x}{21}\) dollars per kg.

Calculate the selling price of tea per kg

The retailer sells the tea at the marked price. The marked price per kg (before discount) was \(x\) dollars per kg. So, the selling price of tea per kg will be \(x\) dollars.

Calculate the profit per kg

Now, subtract the effective cost price, which is \(\frac{18x}{21}\) dollars, from the selling price, which is \(x\) dollars: \(x - \frac{18x}{21}\).

Calculate profit percentage

The profit percentage can be calculated using the following formula: $$ Profit\,Percentage = \frac{Profit}{Cost\,Price} \cdot 100. $$ So, the profit percentage = \(\frac{x - \frac{18x}{21}}{\frac{18x}{21}} \cdot 100\). Now simplify the expression: $$ \frac{(21x-18x)}{18x} \cdot 100 = \frac{3x}{18x} \cdot 100 = \frac{3}{18} \cdot 100 \rightarrow \frac{1}{6} \cdot 100 = 16.66 \%. $$ Hence, the profit percentage of the retailer is \(16.66 \%\). The correct answer is (c) \(16.66 \%\).

Key concepts

Effective Cost Price

Understanding the concept of effective cost price is crucial in scenarios where additional benefits, like free goods, are offered. In our exercise, the retailer received not just a discount but an additional 1 kg of tea for free when buying 20 kg. This means that while the retailer paid for only 20 kg, they ended up with 21 kg.

This kind of scenario changes the effective cost price per unit. To find the effective cost price per kg, you calculate the total amount spent (in this case, after getting a 10% discount) and divide it by the total quantity received.

Here's how it works:

• You start with the discounted total cost of 20 kg of tea, which was calculated as \(18x\) dollars.
• Then, divide this cost by the 21 kg of tea the retailer actually has.
• This results in an effective cost price of \(\frac{18x}{21}\) dollars per kg.

By lowering the per-unit cost, thanks to the free kg, the retailer is positioned to make a better profit when charging the customer per kg for the marked price of tea.

Discount Calculation

Discounts are a common strategy used in retail to encourage purchasing and can have a substantial impact on cost calculations. In this exercise, the retailer got a discount of 10% on the price of the tea.

The calculation of this discount might seem straightforward, but understanding it is crucial. Here's how it unfolds:

• If the tea costs \(x\) dollars per kg, the total cost for 20 kg before any discounts is \(20x\) dollars.
• With a 10% discount, the reduction in price is calculated as 10% of \(20x\), which equals \(2x\) dollars.
• The new cost after applying the discount therefore becomes \(20x - 2x = 18x\) dollars.

Understanding how to apply discounts properly ensures retailers can price their offerings competitively while also preserving margins, leading to improved profitability.

Selling Price Calculation

The selling price is essential for determining the profit made from each unit sold. In the context of this exercise, the selling price is set at the marked price before any discounts, since the retailer sells the tea at that rate.

Let's break down how that affects the sales and profits:

• The marked price was \(x\) dollars per kg, and this is the price at which the tea is sold to the customer.
• There's no additional discount offered to the customer in this scenario, so the selling price remains \(x\) dollars per kg.
• Hence, the profit per kg becomes the difference between this selling price \(x\) and the effective cost price of \(\frac{18x}{21}\) calculated earlier.

Understanding and implementing the correct selling price is vital as it directly influences the retailer's profit percentage and ensures the business remains sustainable.