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Discount

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Introduction to Discount

Imagine you are shopping for clothes, electronics, or groceries, and you see a red sticker saying "20% off" or "Flat Rs.500 Discount." This means the price you pay is less than the original or listed price. This reduction in price is known as a discount. In mathematics, especially for entrance exams, understanding discounts is crucial because many questions test your ability to quickly calculate price reductions, compare prices, or find profit and loss after discounts.

Discount helps both consumers and sellers: consumers get goods at a lower price, while sellers attract more customers. Learning to handle discounts efficiently also sharpens your problem-solving skills.

Definition of Discount and Its Components

Discount is the amount by which the marked price of an item is reduced before selling it to the customer. It is expressed mostly in two ways:

  • As a fixed amount in currency (e.g., Rs.200 off)
  • As a percentage of the marked price (e.g., 20% off)

Let's understand the key terms involved:

  • Marked Price (M): The original listed price on the product's tag or sticker before discount.
  • Selling Price (S): The price at which the product is actually sold after discount.
  • Discount Amount (D): The difference between the marked price and the selling price.

Mathematically, these relate as:

Marked Price (M) Discount (D) Selling Price (S)

Here, the discount amount D is the amount by which M decreases to become S. In formula form,

Discount = Marked Price - Selling Price, or simply,

\(D = M - S\)

Because discounts are often given in percentage terms for easier communication, it's important to connect discount with percentages.

Key Formulas for Discount

Below are essential formulas needed for solving discount problems. Each formula uses:

  • M: Marked Price (in INR)
  • S: Selling Price after discount (in INR)
  • D: Discount amount (in INR)
  • d: Discount percentage (%), expressed as a number between 0 and 100
Formula Name Formula Variables Use
Discount Amount \( D = M - S \) \(D\): Discount amount
\(M\): Marked price
\(S\): Selling price		 Find the discount value in INR if marked and selling prices are known.
Selling Price \( S = M - D \) \(S\): Selling price
\(M\): Marked price
\(D\): Discount amount		 Calculate the final price after discount amount is given.
Discount Percentage \( \text{Discount \%} = \frac{D}{M} \times 100 \) \(D\): Discount amount
\(M\): Marked price		 Find what percent discount was offered given the amount.
Selling Price using Discount \% \( S = M \times \left(1 - \frac{d}{100}\right) \) \(S\): Selling price
\(M\): Marked price
\(d\): Discount percentage		 Calculate selling price directly from discount percentage.
Successive Discount (Two rates) \( \text{Effective Discount \%} = d_1 + d_2 - \frac{d_1 \times d_2}{100} \) \(d_1, d_2\): First and second discount percentages Find total discount when two discounts apply one after another.

Discount Amount

D = M - S

Discount in rupees equals marked price minus selling price

D = Discount amount in INR
M = Marked price in INR
S = Selling price in INR

Selling Price using Discount Percentage

\[S = M \times \left(1 - \frac{d}{100}\right)\]

Selling price after discount percentage is applied

S = Selling price in INR
M = Marked price in INR
d = Discount percentage

Worked Examples

Example 1: Finding Discount Amount Easy
A jacket is marked at Rs.2000, but it is sold for Rs.1800 during a sale. What is the discount amount?

Step 1: Identify the given values:

  • Marked price, \(M = 2000\) INR
  • Selling price, \(S = 1800\) INR

Step 2: Use the formula \( D = M - S \) to find the discount.

So, \( D = 2000 - 1800 = 200 \) INR

Answer: The discount amount is Rs.200.

Example 2: Selling Price Calculation Easy
A television is marked at Rs.1500 and is offered at a 15% discount. Find the selling price.

Step 1: Given:

  • Marked price, \(M = 1500\) INR
  • Discount percentage, \(d = 15\%\)

Step 2: Use the formula \( S = M \times \left(1 - \frac{d}{100}\right) \)

Calculate:

\( S = 1500 \times \left(1 - \frac{15}{100}\right) = 1500 \times 0.85 = 1275 \) INR

Answer: The selling price after 15% discount is Rs.1275.

Example 3: Finding Discount Percentage Medium
An article marked at Rs.2500 is sold with a discount of Rs.375. What is the discount percentage?

Step 1: Identify the values:

  • Marked price, \(M = 2500\) INR
  • Discount amount, \(D = 375\) INR

Step 2: Use the formula for discount percentage:

\( d = \frac{D}{M} \times 100 = \frac{375}{2500} \times 100 \)

Calculate:

\( d = 0.15 \times 100 = 15\% \)

Answer: The discount percentage is 15%.

Example 4: Successive Discounts Hard
A shop offers two successive discounts of 10% and 20% on a marked price of Rs.3000. What is the final price payable by the customer?

Step 1: Given:

  • Marked price, \(M = 3000\) INR
  • First discount, \(d_1 = 10\%\)
  • Second discount, \(d_2 = 20\%\)

Step 2: Calculate the effective discount percentage using the formula:

\[ d_{\text{effective}} = d_1 + d_2 - \frac{d_1 \times d_2}{100} = 10 + 20 - \frac{10 \times 20}{100} = 30 - 2 = 28\% \]

Step 3: Calculate selling price after successive discounts:

\[ S = M \times \left(1 - \frac{d_{\text{effective}}}{100}\right) = 3000 \times \left(1 - \frac{28}{100}\right) = 3000 \times 0.72 = 2160 \]

Answer: Final price payable is Rs.2160 after two successive discounts.

Example 5: Discount and Profit-Loss Combination Hard
A shopkeeper offers a 20% discount on the marked price and still makes a profit of 10%. If the marked price is Rs.1000, find the cost price.

Step 1: Known values:

  • Marked price, \(M = 1000\) INR
  • Discount = 20%
  • Profit = 10%

Step 2: Calculate selling price \(S\):

\[ S = M \times \left(1 - \frac{20}{100}\right) = 1000 \times 0.8 = 800 \text{ INR} \]

Step 3: Since profit is 10%, selling price is 110% of cost price \(C\). So:

\[ S = C \times \left(1 + \frac{10}{100}\right) = 1.1 \times C \]

Step 4: Equate and solve for cost price \(C\):

\[ 800 = 1.1 \times C \implies C = \frac{800}{1.1} \approx 727.27 \]

Answer: The cost price of the article is approximately Rs.727.27.

Formula Bank

Discount Amount
\[ D = M - S \]
where: \(D\) = Discount amount in INR, \(M\) = Marked price in INR, \(S\) = Selling price in INR
Selling Price
\[ S = M - D \]
where: \(S\) = Selling price in INR, \(M\) = Marked price in INR, \(D\) = Discount amount in INR
Discount Percentage
\[ \text{Discount \%} = \frac{D}{M} \times 100 \]
where: \(D\) = Discount amount in INR, \(M\) = Marked price in INR
Selling Price using Discount Percentage
\[ S = M \times \left(1 - \frac{d}{100}\right) \]
where: \(S\) = Selling price in INR, \(M\) = Marked price in INR, \(d\) = Discount percentage
Successive Discount
\[ \text{Effective Discount \%} = d_1 + d_2 - \frac{d_1 \times d_2}{100} \]
where: \(d_1, d_2\) = First and second discount percentages

Tips & Tricks

Tip: Convert percentage discount into decimal by dividing by 100 for quicker multiplication.
When to use: When calculating selling price directly from marked price and discount percentage.
Tip: For successive discounts, use the effective discount formula instead of applying each discount one after the other separately.
When to use: When two or more discounts may be applied sequentially, this saves time.
Tip: Always remember that discount amount is in currency (INR), not percentage.
When to use: To avoid confusion between discount amount and discount percentage.
Tip: Break complex word problems into simple pieces by first identifying marked price (M), discount (D), and selling price (S).
When to use: Useful in entrance exam multi-step problems.
Tip: Practice mental math with common discount percentages like 10%, 15%, 20%, 25%, 50% for faster calculations.
When to use: For quick approximations or solving questions within limited time.

Common Mistakes to Avoid

❌ Confusing discount amount (INR) with discount percentage (%)
✓ Clearly identify whether the given value is an amount or a percentage before using formulas
Why: Because similar terms may mislead students, causing calculation errors.
❌ Using cost price instead of marked price in discount calculations
✓ Always use marked price (listed price) as the base for discount calculations
Why: Marked price and cost price differ; mixing them leads to wrong selling price or discount values.
❌ Adding successive discount percentages directly instead of using the effective discount formula.
✓ Use the formula: \( d_{\text{effective}} = d_1 + d_2 - \frac{d_1 \times d_2}{100} \)
Why: Percent discounts apply on the reduced price, not the original price, so simple addition overestimates the discount.
❌ Subtracting discount percentage directly from marked price to find selling price (e.g., \( S = M - d \) ).
✓ Convert discount percentage to amount first, or use \( S = M \times \left(1 - \frac{d}{100}\right) \)
Why: Discount percentage is a percent, not a rupee value, so direct subtraction is incorrect.
❌ Misreading word problems and mixing variables (e.g., confusing cost price and selling price)
✓ Read the problem carefully and underline key values, ensure correct assignment of variables before calculations
Why: Accurate interpretation is essential to apply formulas properly and arrive at correct answers.
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