Profit, Loss and Interest

Overview

Profit, Loss and Interest forms a core quantitative topic in TN TET Mathematics, appearing consistently in both Paper I and Paper II. This topic tests your ability to apply percentage concepts to real-world commercial transactions—buying and selling goods, lending and borrowing money. Mastery here builds the foundation for teaching students how mathematics connects to everyday life, from shopkeepers calculating margins to banks computing loan repayments.

For TN TET, expect 2–4 questions from this area. Problems typically involve straightforward calculations of profit/loss percentages, finding cost or selling price, and computing simple or compound interest for given periods. The pedagogy angle also matters—you should understand how to teach these concepts using real-life examples and help students avoid common calculation errors.

Success requires memorising key formulas, understanding their derivations, and practising quick mental calculations since exam time is limited.

Key Concepts

• **Cost Price (CP)** is the amount paid to acquire an item; **Selling Price (SP)** is the amount received when selling it. The relationship between these determines profit or loss.

• **Profit occurs when SP > CP**; the gain equals SP − CP. **Loss occurs when SP < CP**; the loss equals CP − SP.

• **Profit and Loss percentages are always calculated on Cost Price**, not Selling Price—this is the standard convention unless stated otherwise.

• **Marked Price (MP)** is the listed/labelled price; **Discount** is the reduction from MP. So SP = MP − Discount.

• **Simple Interest (SI)** is calculated only on the original principal—interest does not earn further interest.

• **Compound Interest (CI)** is calculated on principal plus accumulated interest—"interest on interest" effect causes faster growth.

• **The difference between CI and SI for 2 years** equals SI for one year multiplied by the rate, divided by 100. This shortcut saves time.

• **Successive discounts** are not additive—two discounts of 10% and 20% do not equal 30%. Apply them sequentially.

Formulas / Key Facts

**Profit and Loss:**

• Profit = SP − CP
• Loss = CP − SP
• Profit % = (Profit / CP) × 100
• Loss % = (Loss / CP) × 100
• SP = CP × (100 + Profit%) / 100 ... when profit is made
• SP = CP × (100 − Loss%) / 100 ... when loss is incurred
• CP = SP × 100 / (100 + Profit%) ... finding CP from SP and profit%
• CP = SP × 100 / (100 − Loss%) ... finding CP from SP and loss%

**Discount:**

• Discount = MP − SP
• Discount % = (Discount / MP) × 100
• SP = MP × (100 − Discount%) / 100

**Simple Interest:**

• SI = (P × R × T) / 100
• Amount (A) = P + SI = P(1 + RT/100)
• Where P = Principal, R = Rate % per annum, T = Time in years

**Compound Interest:**

• A = P(1 + R/100)^n ... for annual compounding
• CI = A − P = P[(1 + R/100)^n − 1]
• For half-yearly compounding: A = P(1 + R/200)^(2n)
• For quarterly compounding: A = P(1 + R/400)^(4n)

**Useful shortcut for 2 years:**

• CI − SI = P × (R/100)²

Worked Examples

**Example 1: Finding Profit Percentage** A shopkeeper buys a chair for ₹800 and sells it for ₹920. Find the profit percentage.

*Solution:*

• CP = ₹800, SP = ₹920
• Profit = SP − CP = 920 − 800 = ₹120
• Profit % = (120 / 800) × 100 = 15%

**Example 2: Simple Interest Calculation** Find the simple interest on ₹5000 at 8% per annum for 3 years.

*Solution:*

• P = 5000, R = 8, T = 3
• SI = (P × R × T) / 100 = (5000 × 8 × 3) / 100
• SI = 120000 / 100 = ₹1200
• Amount = 5000 + 1200 = ₹6200

**Example 3: Compound Interest for 2 Years** Find the compound interest on ₹10,000 at 10% per annum for 2 years, compounded annually.

*Solution:*

• P = 10000, R = 10, n = 2
• A = P(1 + R/100)^n = 10000 × (1 + 10/100)² = 10000 × (1.1)²
• A = 10000 × 1.21 = ₹12,100
• CI = A − P = 12100 − 10000 = ₹2100

*Verification using shortcut:*

• SI for 2 years = (10000 × 10 × 2) / 100 = ₹2000
• CI − SI = P × (R/100)² = 10000 × (0.1)² = ₹100
• CI = SI + 100 = 2000 + 100 = ₹2100 ✓

**Example 4: Successive Discounts** A shirt with marked price ₹500 is sold at successive discounts of 10% and 20%. Find the selling price.

*Solution:*

• After first discount (10%): SP₁ = 500 × (90/100) = ₹450
• After second discount (20%): SP = 450 × (80/100) = ₹360
• Final SP = ₹360

*Note:* Single equivalent discount is not 30%, but 28%. (500 − 360 = 140; 140/500 × 100 = 28%)

Common Mistakes

• **Calculating profit/loss % on SP instead of CP** → Always use CP as the base for profit/loss percentages unless the question specifically asks otherwise.

• **Adding successive discounts directly** (thinking 10% + 20% = 30%) → Apply discounts one after another on the reduced price; the combined effect is always less than the sum.

• **Using years in months without converting** → If time is given as 18 months, convert to 1.5 years before applying SI/CI formulas.

• **Forgetting to subtract principal when finding CI** → The formula A = P(1 + R/100)^n gives Amount, not CI. Always compute CI = A − P.

• **Confusing half-yearly rate with annual rate** → For half-yearly compounding, halve the rate and double the time period; don't just substitute values directly.

Quick Reference

• Profit % = (Profit / CP) × 100 — always on CP, never SP.
• SI = PTR / 100 — memorise as "Principal × Time × Rate divided by 100."
• CI for 2 years exceeds SI by P(R/100)² — quick check for 2-year problems.
• Successive discounts: apply sequentially, never add directly.
• Amount in CI = P(1 + R/100)^n — remember to subtract P to get CI.
• When time is in months, divide by 12 to convert to years.