Commercial Mathematics

Ratio, Proportion, Percentage, Profit-Loss, Simple Interest and Compound Interest

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Overview

Commercial Mathematics forms a highly practical and frequently tested segment of the UTET Paper II Mathematics section. These concepts appear in everyday transactions—shopping, banking, business, and budgeting—making them essential for upper-primary students to understand and apply.

For UTET, expect questions that test both conceptual understanding and calculation speed. You must be comfortable converting between fractions, decimals, and percentages, and applying formulas to word problems involving profit-loss and interest calculations. Questions often combine multiple concepts (e.g., percentage increase followed by profit calculation), so fluency in each building block is critical.

Mastery here also supports the pedagogical goal of connecting mathematics to real-life contexts—a key NCF emphasis that UTET pedagogy questions may reference.

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Key Concepts

• **Ratio** expresses the relative size of two quantities of the same kind. Written as a:b or a/b, ratios have no units and should always be expressed in simplest form.

• **Proportion** states that two ratios are equal. If a:b = c:d, then a, b, c, d are in proportion, and the product of extremes equals the product of means (a × d = b × c).

• **Percentage** means "per hundred." It standardises comparison by expressing a number as a fraction of 100. Converting between fractions, decimals, and percentages is foundational.

• **Profit and Loss** measure the gain or loss in a transaction relative to the Cost Price (CP). Selling Price (SP) determines whether there is profit (SP > CP) or loss (SP < CP).

• **Discount** is a reduction on Marked Price (MP). The actual SP after discount differs from MP, and successive discounts do not simply add up.

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

• **Compound Interest (CI)** is calculated on principal plus accumulated interest. CI grows faster than SI over multiple periods.

• **The relationship P × R × T / 100** is the backbone formula; variations apply to CI depending on compounding frequency.

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Formulas / Key Facts

| Concept | Formula / Fact | |---------|----------------| | Ratio simplification | Divide both terms by their HCF | | Proportion (mean proportional) | If a:x = x:b, then x = √(a × b) | | Fraction → Percentage | Multiply by 100 | | Percentage → Fraction | Divide by 100 | | 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 | | Discount | Discount = MP − SP | | Discount % | Discount % = (Discount / MP) × 100 | | Simple Interest | SI = (P × R × T) / 100 | | Amount (SI) | A = P + SI | | Compound Interest (annual) | A = P × (1 + R/100)^n ; CI = A − P | | CI compounded half-yearly | Use R/2 and 2n instead of R and n |

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Worked Examples

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

**Solution:**

• Total parts = 2 + 3 = 5
• Value of one part = 1800 / 5 = ₹360
• A's share = 2 × 360 = ₹720
• B's share = 3 × 360 = ₹1,080

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### Example 2: Profit and Loss **Problem:** A shopkeeper buys an article for ₹400 and sells it for ₹460. Find profit %.

**Solution:**

• CP = ₹400, SP = ₹460
• Profit = 460 − 400 = ₹60
• Profit % = (60 / 400) × 100 = 15%

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### Example 3: Simple Interest **Problem:** Find SI on ₹5,000 at 8% per annum for 3 years.

**Solution:**

• SI = (P × R × T) / 100
• SI = (5000 × 8 × 3) / 100 = ₹1,200

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### Example 4: Compound Interest **Problem:** Find CI on ₹10,000 at 10% per annum for 2 years, compounded annually.

**Solution:**

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

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### Example 5: Successive Discounts **Problem:** A shirt marked at ₹500 is offered at successive discounts of 10% and 20%. Find SP.

**Solution:**

• After first discount: 500 × (90/100) = ₹450
• After second discount: 450 × (80/100) = ₹360
• Final SP = ₹360
• (Note: Successive 10% and 20% ≠ 30% discount)

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Common Mistakes

| Wrong Thinking | Correct Approach | |----------------|------------------| | Adding successive discounts directly (10% + 20% = 30%) | Apply each discount sequentially on the reduced price | | Calculating profit/loss % on SP instead of CP | Profit % and Loss % are always calculated on Cost Price | | Using SI formula for CI problems | For CI, use the power formula; interest is added to principal each period | | Confusing ratio order (writing b:a instead of a:b) | Maintain the order as stated in the problem; order matters | | Forgetting to convert time to years for interest | If time is in months, divide by 12; if in days, divide by 365 | | Applying annual rate directly for half-yearly compounding | Halve the rate and double the time period for half-yearly CI |

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Quick Reference

• **Ratio a:b in simplest form** → Divide both by HCF.
• **Percentage = (Part / Whole) × 100**
• **Profit % = (Profit / CP) × 100** — always on CP, never SP.
• **SI = PRT / 100** — interest stays constant each year.
• **CI formula:** A = P(1 + R/100)^n — interest compounds.
• **Successive discounts:** Apply one after another, not by addition.