Commercial Mathematics

Overview

Commercial Mathematics forms a consistent and scoring segment of the PSTET Paper II Mathematics section. This topic tests your ability to apply arithmetic reasoning to everyday financial and business scenarios—skills that upper-primary students must develop for real-world numeracy. Questions typically involve direct calculation or word problems on ratio, proportion, percentage, profit-loss, and simple/compound interest.

Mastery here requires two things: (1) solid command of the formulas and their variations, and (2) the ability to translate word problems into mathematical expressions quickly. Since most questions are computational, accuracy and speed matter equally. Expect 3–5 questions from this cluster, often presented as practical situations involving shopkeepers, bank deposits, or mixing quantities.

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Key Concepts

• **Ratio** expresses a comparison between two quantities of the same unit. If A:B = 3:5, it means for every 3 parts of A, there are 5 parts of B. Ratios have no units.

• **Proportion** states that two ratios are equal. If A:B = C:D, then A × D = B × C (cross-multiplication rule). This is the foundation for solving "if-then" problems.

• **Percentage** means "per hundred." Converting a fraction to percentage: multiply by 100. Converting percentage to fraction: divide by 100.

• **Profit and Loss** are always calculated on **Cost Price (CP)**. Profit = SP − CP; Loss = CP − SP.

• **Marked Price (MP)** is the listed price; **Discount** is calculated on MP. Selling Price after discount = MP − Discount.

• **Simple Interest (SI)** is computed on the original principal only, remaining constant each year.

• **Compound Interest (CI)** is computed on the principal plus accumulated interest; the interest "compounds" each period.

• **Successive discounts or percentage changes** cannot be simply added; they must be applied stepwise or using the net-effect formula.

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Formulas / Key Facts

| Concept | Formula / Fact | |---------|----------------| | Ratio simplification | Divide both terms by their HCF | | Mean proportion of a and b | √(a × b) | | Proportion rule | If a:b = c:d, then ad = bc | | Percentage of a number | (Percentage × Number) / 100 | | Fraction to % | Fraction × 100 | | Profit % | (Profit / CP) × 100 | | Loss % | (Loss / CP) × 100 | | SP when Profit % given | SP = CP × (1 + P%/100) | | SP when Loss % given | SP = CP × (1 − L%/100) | | Discount % | (Discount / MP) × 100 | | SP after discount | MP × (1 − D%/100) | | Simple Interest | SI = (P × R × T) / 100 | | Amount (SI) | A = P + SI = P(1 + RT/100) | | Compound Interest (annual) | A = P × (1 + R/100)^T ; CI = A − P | | CI for 2 years shortcut | CI − SI (for 2 yrs) = SI × R / 100 | | Net effect of two successive changes x% and y% | x + y + (xy/100) % |

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Worked Examples

### Example 1 — Ratio and Proportion **Problem:** Divide ₹1,800 among A, B, C in the ratio 2:3:4.

**Solution:** Sum of ratio parts = 2 + 3 + 4 = 9

A's share = (2/9) × 1800 = ₹400 B's share = (3/9) × 1800 = ₹600 C's share = (4/9) × 1800 = ₹800

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### Example 2 — Profit, Loss and Discount **Problem:** A shopkeeper buys an article for ₹500, marks it at ₹700, and offers a 10% discount. Find the profit %.

**Solution:** CP = ₹500, MP = ₹700 Discount = 10% of 700 = ₹70 SP = 700 − 70 = ₹630 Profit = 630 − 500 = ₹130 Profit % = (130/500) × 100 = **26%**

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### Example 3 — Simple and Compound Interest **Problem:** Find the difference between CI and SI on ₹8,000 at 5% per annum for 2 years.

**Solution:** SI = (8000 × 5 × 2) / 100 = ₹800

For CI: A = 8000 × (1 + 5/100)² = 8000 × (1.05)² = 8000 × 1.1025 = ₹8,820 CI = 8820 − 8000 = ₹820

Difference = 820 − 800 = **₹20**

*Shortcut check:* Difference for 2 years = P × (R/100)² = 8000 × (5/100)² = 8000 × 0.0025 = ₹20 ✓

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Common Mistakes

| Wrong Thinking | Correct Fix | |----------------|-------------| | Calculating profit/loss % on Selling Price | Always use **Cost Price** as the base for profit/loss %. | | Adding successive discounts directly (e.g., 20% + 10% = 30%) | Apply them stepwise or use net formula: 20 + 10 − (20×10)/100 = 28% effective discount. | | Using SI formula when problem says "compounded annually" | Read carefully; CI requires the power formula A = P(1 + R/100)^T. | | Forgetting to convert time to years when rate is per annum | If time is in months, divide by 12; if in days, divide by 365. | | Confusing Marked Price with Cost Price in discount problems | Discount is always on **MP**; profit/loss is always on **CP**. |

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Quick Reference

• **Ratio has no unit**; both quantities must be in the same unit before forming the ratio.

• **Profit % and Loss % → base is CP; Discount % → base is MP.**

• **SI formula:** P × R × T / 100 — interest stays constant each year.

• **CI formula (annual):** A = P(1 + R/100)^T — interest grows each year.

• **Two successive % changes (x, y):** Net = x + y + xy/100 (use signs: increase +, decrease −).

• **CI − SI for 2 years** = P × (R/100)² — a quick check shortcut.