Profit, Loss and Discount

Overview

Profit, Loss and Discount form the backbone of commercial mathematics at the primary level. This topic tests a student's ability to apply percentage concepts to real-world buying and selling scenarios—skills essential for both daily life and competitive exams. In Bihar TET Paper I, questions from this area are straightforward but require careful attention to terminology and calculation.

The topic connects directly with percentage and ratio concepts you have already studied. Examiners typically frame questions around finding profit or loss percentage, calculating selling price when profit/loss is given, or determining the final price after applying discounts. Mastery here builds a foundation for more advanced problems in Simple Interest and Data Handling.

Students must be comfortable converting between fractions, decimals and percentages, and should practice identifying what is being asked—whether it is an absolute amount or a percentage figure. Speed and accuracy in basic arithmetic are crucial.

---

Key Concepts

• **Cost Price (CP):** The price at which an article is purchased or manufactured. This is your baseline for all profit/loss calculations.

• **Selling Price (SP):** The price at which an article is sold to a customer. Compare SP with CP to determine profit or loss.

• **Profit (Gain):** When SP > CP, the seller earns a profit. Profit = SP − CP.

• **Loss:** When SP < CP, the seller incurs a loss. Loss = CP − SP.

• **Profit Percentage:** Always calculated on Cost Price. Profit% = (Profit / CP) × 100.

• **Loss Percentage:** Also calculated on Cost Price. Loss% = (Loss / CP) × 100.

• **Marked Price (MP):** The price written on the label or tag before any discount is given. Also called "list price."

• **Discount:** A reduction offered on the Marked Price. Discount = MP − SP. Discount% = (Discount / MP) × 100.

---

Formulas / Key Facts

| Concept | Formula | |---------|---------| | Profit | Profit = SP − CP | | Loss | Loss = CP − SP | | Profit % | Profit% = (Profit / CP) × 100 | | Loss % | Loss% = (Loss / CP) × 100 | | SP when Profit% given | SP = CP × (100 + Profit%) / 100 | | SP when Loss% given | SP = CP × (100 − Loss%) / 100 | | CP when SP and Profit% given | CP = SP × 100 / (100 + Profit%) | | CP when SP and Loss% given | CP = SP × 100 / (100 − Loss%) | | Discount | Discount = MP − SP | | Discount % | Discount% = (Discount / MP) × 100 | | SP after Discount | SP = MP × (100 − Discount%) / 100 |

**Key Facts to Remember:** 1. Profit and Loss percentages are always calculated on CP, never on SP. 2. Discount percentage is always calculated on MP, never on CP or SP. 3. If there is no discount, then MP = SP. 4. Two successive discounts of a% and b% are equivalent to a single discount of: a + b − (a × b)/100 percent.

---

Worked Examples

### Example 1: Finding Profit and Profit Percentage **Problem:** A shopkeeper buys a toy for ₹400 and sells it for ₹480. Find the profit and profit percentage.

**Solution:**

• CP = ₹400, SP = ₹480
• Profit = SP − CP = 480 − 400 = ₹80
• Profit% = (Profit / CP) × 100 = (80 / 400) × 100 = 20%

**Answer:** Profit = ₹80, Profit% = 20%

---

### Example 2: Finding SP when Loss% is given **Problem:** A book was bought for ₹250. It was sold at a loss of 12%. Find the selling price.

**Solution:**

• CP = ₹250, Loss% = 12%
• SP = CP × (100 − Loss%) / 100
• SP = 250 × (100 − 12) / 100 = 250 × 88 / 100 = ₹220

**Answer:** SP = ₹220

---

### Example 3: Discount Calculation **Problem:** A shirt has a marked price of ₹600. The shopkeeper offers a discount of 15%. Find the discount amount and the selling price.

**Solution:**

• MP = ₹600, Discount% = 15%
• Discount = MP × Discount% / 100 = 600 × 15 / 100 = ₹90
• SP = MP − Discount = 600 − 90 = ₹510

**Answer:** Discount = ₹90, SP = ₹510

---

### Example 4: Combined Problem (CP, Discount, Profit) **Problem:** A trader marks his goods 25% above the cost price and then offers a 10% discount. Find his profit or loss percentage.

**Solution:**

• Let CP = ₹100
• MP = CP + 25% of CP = 100 + 25 = ₹125
• Discount = 10% of MP = 10% of 125 = ₹12.50
• SP = MP − Discount = 125 − 12.50 = ₹112.50
• Profit = SP − CP = 112.50 − 100 = ₹12.50
• Profit% = (12.50 / 100) × 100 = 12.5%

**Answer:** Profit% = 12.5%

---

Common Mistakes

1. **Calculating Profit% on SP instead of CP**

• *Wrong thinking:* "Profit is ₹50 and SP is ₹250, so Profit% = 50/250 × 100 = 20%."
• *Correct fix:* Always use CP in the denominator. If CP = ₹200, then Profit% = 50/200 × 100 = 25%.

2. **Confusing Marked Price with Cost Price**

• *Wrong thinking:* Applying discount percentage directly on CP.
• *Correct fix:* Discount is always given on MP (the label price), not on what the shopkeeper paid.

3. **Forgetting to convert percentage to decimal or fraction during calculation**

• *Wrong thinking:* Writing SP = 400 × 120 instead of 400 × 120/100.
• *Correct fix:* Always divide by 100 when using percentages in multiplication.

4. **Adding successive discounts directly**

• *Wrong thinking:* Two discounts of 10% and 20% equal 30% total discount.
• *Correct fix:* Apply discounts one after another, or use the formula: Effective discount = 10 + 20 − (10 × 20)/100 = 28%.

5. **Sign errors in loss problems**

• *Wrong thinking:* Using (100 + Loss%) when finding SP at a loss.
• *Correct fix:* For loss, always use (100 − Loss%).

---

Quick Reference

• **Profit/Loss base = CP; Discount base = MP.**
• SP at Profit: SP = CP × (100 + P%) / 100.
• SP at Loss: SP = CP × (100 − L%) / 100.
• Discount always reduces MP to get SP.
• No profit, no loss means SP = CP.
• Two successive discounts of a% and b% = (a + b − ab/100)%.