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 |