Percentage Applications: Profit, Loss, and Discount

TL;DR

You'll learn how percentages are used to calculate profit and loss on sales, and how discounts reduce prices. Understanding these concepts helps you make sense of everyday financial transactions and sales. We'll cover the formulas and provide examples to make it clear.

1. The Mental Model

Think of profit, loss, and discount as different ways money changes hands based on an original price. Profit means you gain money, loss means you lose money, and a discount means you pay less than the original price. All these are expressed as a percentage of the original amount.

2. The Core Material

When we talk about profit, loss, or discount, we're always comparing a change in value to an original value, and then expressing that comparison as a percentage. The "original value" is crucial – it's your base for all calculations.

Profit and Loss

Cost Price (CP): This is the price at which an item is bought.
Selling Price (SP): This is the price at which an item is sold.

• Profit: Occurs when the Selling Price (SP) is greater than the Cost Price (CP).
  • Profit = SP - CP
  • Profit Percentage = (Profit / CP) * 100%
• Loss: Occurs when the Selling Price (SP) is less than the Cost Price (CP).
  • Loss = CP - SP
  • Loss Percentage = (Loss / CP) * 100%

Notice that both profit percentage and loss percentage are always calculated based on the Cost Price.

Example: Profit

You buy a shirt for \$20 (CP) and sell it for \$25 (SP).
* Profit = \$25 - \$20 = \$5
* Profit Percentage = (\$5 / \$20) * 100% = 0.25 * 100% = 25%

Example: Loss

You buy a book for \$30 (CP) and sell it for \$24 (SP).
* Loss = \$30 - \$24 = \$6
* Loss Percentage = (\$6 / \$30) * 100% = 0.20 * 100% = 20%

Discount

Marked Price (MP): This is the original price displayed on an item, often called the list price or sticker price.
Discount: This is the reduction offered on the Marked Price.
Selling Price (SP): After the discount, this is the price the customer actually pays.

• Discount Amount = MP - SP
• Discount Percentage = (Discount Amount / MP) * 100%

The discount percentage is always calculated based on the Marked Price.

Example: Discount

A jacket has a Marked Price of \$80. It's offered with a 15% discount.
* Discount Amount = 15% of \$80 = 0.15 * \$80 = \$12
* Selling Price (what you pay) = MP - Discount Amount = \$80 - \$12 = \$68

Alternatively, if there's a 15% discount, you're paying 100% - 15% = 85% of the Marked Price.
* SP = 85% of \$80 = 0.85 * \$80 = \$68

Here's a diagram showing how these concepts relate:

graph TD
    A["Initial Purchase (Cost Price)"] --> B["Sale Price (Selling Price)"];
    B --> |SP > CP| C["Profit"];
    B --> |SP < CP| D["Loss"];
    E["Original Display (Marked Price)"] --> F["Reduced Price (Selling Price)"];
    F --> |MP > SP| G["Discount"];
    C --> H["Profit % Calculation (based on CP)"];
    D --> I["Loss % Calculation (based on CP)"];
    G --> J["Discount % Calculation (based on MP)"];

3. Worked Example

Let's say a shopkeeper buys a bicycle for \$150. They want to make a 20% profit. However, it doesn't sell quickly, so they decide to offer a 10% discount on its marked price to attract buyers. What is the marked price, and what is the final selling price after the discount?

Step 1: Calculate the desired Selling Price (SP) for a 20% profit.
* Cost Price (CP) = \$150
* Desired Profit = 20% of CP = 0.20 * \$150 = \$30
* Desired Selling Price (SP) = CP + Profit = \$150 + \$30 = \$180

Step 2: Determine the Marked Price (MP) given the 10% discount to reach the desired SP.
We know that the final Selling Price (\$180) is *after* a 10% discount on the Marked Price. This means \$180 represents 90% (100% - 10%) of the Marked Price.
* Let MP be the Marked Price.
* 0.90 * MP = \$180
* MP = \$180 / 0.90
* MP = \$200

So, the shopkeeper should mark the bicycle at \$200 to be able to give a 10% discount and still achieve a 20% profit based on their cost. The final selling price after the discount will be \$180.

4. Key Takeaways

• Profit and Loss percentages are always calculated relative to the Cost Price (CP).
• Discount percentages are always calculated relative to the Marked Price (MP).
• Profit increases the original price, while loss and discount decrease it.
• You can work backward from a final price to find an original price using percentages.
• A discount often leads to a lower selling price, which might still be profitable if the marked price was set high enough.
• Understand whether a percentage is "of" the original amount or "added to/subtracted from" it.

Common Mistakes to Avoid

• Confusing CP and MP: Don't calculate profit/loss on the Marked Price, or discount on the Cost Price.
• Wrong base for percentage: Always ask yourself, "percentage of what?" Is it of CP, SP, or MP?
• Adding/subtracting percentages directly: You can't just add a 20% profit and subtract a 10% discount from the same base number unless that number is the starting point for both calculations.
• Not converting percentages to decimals: Remember that 20% is 0.20 in calculations.

5. Now Try It

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