Commercial Mathematics

Overview

Commercial Mathematics forms a significant portion of the WB TET Paper II Mathematics section, testing your ability to apply arithmetic concepts to real-world financial scenarios. This topic bridges abstract number operations with practical situations involving buying, selling, borrowing and lending — skills every upper-primary teacher must convey clearly to students.

Expect 3–5 questions from this topic, typically presented as word problems requiring you to calculate profit percentages, find selling prices after discount, or compare simple and compound interest amounts. Mastery here demands not just formula recall but the ability to quickly identify what quantity is being asked and which base value (cost price, marked price, principal) to use for percentage calculations.

The concepts interlink tightly: discount problems use the same percentage logic as profit/loss, and compound interest builds upon simple interest. A student who grasps the underlying percentage framework can handle all variations confidently.

Key Concepts

• **Cost Price (CP)** is the price at which an article is purchased; **Selling Price (SP)** is the price at which it is sold. Profit or loss is always calculated on CP.

• **Profit** occurs when SP > CP; **Loss** occurs when SP < CP. Profit% and Loss% are always expressed as a percentage of CP, never of SP.

• **Marked Price (MP)** is the price displayed on an article before any discount. Discount is calculated on MP, not on CP.

• **Successive discounts** are applied one after another on the reduced price, not added together on the original marked price.

• **Simple Interest (SI)** is calculated on the original principal only, remaining constant each year for a fixed rate.

• **Compound Interest (CI)** is calculated on the principal plus accumulated interest, so interest "earns interest" in subsequent periods.

• **The difference between CI and SI for 2 years** equals P × (R/100)², a frequently tested shortcut.

• **Effective rate** in compound interest increases with more frequent compounding (half-yearly, quarterly).

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 made
• If a trader uses false weights: Gain% = (Error / True Value − Error) × 100

**Discount**

• Discount = MP − SP
• Discount% = (Discount / MP) × 100
• SP = MP × (100 − Discount%) / 100
• For successive discounts d₁% and d₂%: Effective single discount% = d₁ + d₂ − (d₁ × d₂)/100

**Simple Interest**

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

**Compound Interest**

• A = P × (1 + R/100)ⁿ — for annual compounding, n = number of years
• CI = A − P
• For half-yearly compounding: A = P × (1 + R/200)^(2n)
• For quarterly compounding: A = P × (1 + R/400)^(4n)
• CI − SI for 2 years = P × (R/100)²
• CI − SI for 3 years = P × (R/100)² × (3 + R/100)

Worked Examples

**Example 1 — Profit and Loss** A shopkeeper buys an article for ₹800 and sells it for ₹920. Find the profit percentage.

Solution:

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

**Example 2 — Successive Discounts** A shirt has a marked price of ₹500. Two successive discounts of 10% and 20% are offered. Find the selling price.

Solution:

• After first discount (10%): SP₁ = 500 × (90/100) = ₹450
• After second discount (20% on ₹450): SP₂ = 450 × (80/100) = ₹360
• Final Selling Price = ₹360

Alternatively, effective discount = 10 + 20 − (10 × 20)/100 = 28% SP = 500 × (72/100) = ₹360

**Example 3 — Simple vs Compound Interest** Find the difference between CI and SI on ₹5,000 at 10% per annum for 2 years.

Solution:

• SI = 5000 × 10 × 2 / 100 = ₹1,000
• CI: A = 5000 × (1 + 10/100)² = 5000 × 1.21 = ₹6,050
• CI = 6050 − 5000 = ₹1,050
• Difference = 1050 − 1000 = ₹50

Shortcut verification: Difference = P × (R/100)² = 5000 × (10/100)² = 5000 × 0.01 = ₹50 ✓

**Example 4 — Half-Yearly Compounding** Find the amount when ₹8,000 is invested at 12% per annum compounded half-yearly for 1 year.

Solution:

• Half-yearly rate = 12/2 = 6%
• Number of half-years = 2
• A = 8000 × (1 + 6/100)² = 8000 × 1.1236 = ₹8,988.80

Common Mistakes

• **Calculating profit% on SP instead of CP** → Always remember: profit and loss percentages are computed on Cost Price. The base is what you paid, not what you received.

• **Adding successive discounts directly** → Two discounts of 10% and 20% do not equal 30%. Apply them sequentially or use the effective discount formula.

• **Confusing marked price with cost price** → Discount is on MP; profit/loss is on CP. These are different base values. Read the problem carefully to identify which price is given.

• **Using annual formula for half-yearly/quarterly problems** → When compounding is not annual, adjust both the rate (divide) and time (multiply) accordingly before applying the formula.

• **Forgetting to subtract principal when finding CI** → The compound interest formula gives the Amount. CI = Amount − Principal. Many students report the amount as CI.

• **Ignoring "per annum" when time is in months** → Convert months to years (divide by 12) before substituting into SI or CI formulas.

Quick Reference

• Profit% = (SP − CP) / CP × 100 — always on CP
• Discount% = (MP − SP) / MP × 100 — always on MP
• SI = PRT / 100; CI = P(1 + R/100)ⁿ − P
• CI − SI (2 years) = P × (R/100)²
• Successive discounts a% and b%: Net = a + b − ab/100
• Half-yearly compounding: halve the rate, double the time periods