Arithmetic

Percentage, Ratio-Proportion, Profit-Loss and Interest

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Overview

Arithmetic forms the backbone of quantitative aptitude in AP TET Paper I and Paper II. This topic tests your ability to handle everyday numerical problems involving percentages, ratios, profit-loss calculations, and interest computations—skills essential for teaching mathematics at the primary and upper primary levels.

Questions from this section appear consistently in the Mathematics content portion of the exam. Examiners test not just calculation speed but also conceptual clarity—whether you understand what "25% of" truly means or how simple interest differs from compound interest. Mastery here also strengthens your pedagogy responses, as you'll need to explain these concepts to young learners.

Focus on building mental math shortcuts, understanding word-problem patterns, and avoiding common traps in percentage-to-fraction conversions and ratio manipulations.

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Key Concepts

• **Percentage means "per hundred"**: x% = x/100. Converting between percentages, fractions, and decimals is foundational (e.g., 25% = 1/4 = 0.25).

• **Ratio expresses a relationship between two quantities** in the same unit. Written as a:b or a/b. Ratios have no units; they compare magnitudes.

• **Proportion states that two ratios are equal**: If a:b = c:d, then a×d = b×c (cross-multiplication rule).

• **Cost Price (CP)** is what you pay; **Selling Price (SP)** is what you receive. Profit or loss is always calculated on CP unless stated otherwise.

• **Simple Interest (SI)** is calculated on the original principal only: SI = (P × R × T)/100.

• **Compound Interest (CI)** is calculated on principal plus accumulated interest. Amount A = P(1 + R/100)^T.

• **Successive percentages** don't simply add. A 10% increase followed by a 10% decrease does not return to the original—it results in a 1% net decrease.

• **Ratio of division** problems require understanding that if a sum S is divided in ratio a:b, the parts are S×a/(a+b) and S×b/(a+b).

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Formulas / Key Facts

| Concept | Formula / Fact | |---------|----------------| | Percentage to Fraction | x% = x/100 | | Finding x% of a number N | (x/100) × N | | Percentage Increase | [(New − Old)/Old] × 100 | | Percentage Decrease | [(Old − New)/Old] × 100 | | Ratio a:b as fraction | a/(a+b) and b/(a+b) of the whole | | Proportion (cross-multiply) | If a:b = c:d, then ad = bc | | Profit | SP − CP (when SP > CP) | | Loss | CP − SP (when CP > SP) | | Profit % | (Profit/CP) × 100 | | Loss % | (Loss/CP) × 100 | | SP when Profit % given | SP = CP × (100 + Profit%)/100 | | SP when Loss % given | SP = CP × (100 − Loss%)/100 | | Simple Interest | SI = (P × R × T)/100 | | Amount (SI) | A = P + SI = P(1 + RT/100) | | Compound Interest Amount | A = P(1 + R/100)^T | | CI | CI = A − P |

**Key Fraction-Percentage Equivalents (memorise)**:

• 1/2 = 50%, 1/3 ≈ 33.33%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%, 1/6 ≈ 16.67%

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Worked Examples

### Example 1: Percentage Problem **A number is increased by 20% and then decreased by 20%. Find the net percentage change.**

Step 1: Let original number = 100. Step 2: After 20% increase = 100 + 20 = 120. Step 3: After 20% decrease on 120 = 120 − (20% of 120) = 120 − 24 = 96. Step 4: Net change = 100 − 96 = 4 (decrease). Step 5: Net percentage change = (4/100) × 100 = **4% decrease**.

*Shortcut*: For successive changes of +a% and −a%, net change = −(a²/100)% = −(400/100)% = −4%.

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### Example 2: Ratio and Proportion **Divide ₹1800 between A and B in the ratio 2:3. How much does each get?**

Step 1: Total parts = 2 + 3 = 5. Step 2: A's share = (2/5) × 1800 = ₹720. Step 3: B's share = (3/5) × 1800 = ₹1080.

**Answer**: A gets ₹720, B gets ₹1080.

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### Example 3: Profit and Loss **A shopkeeper buys an article for ₹400 and sells it at a profit of 15%. Find the selling price.**

Step 1: Profit % = 15, CP = ₹400. Step 2: SP = CP × (100 + Profit%)/100 = 400 × 115/100 = 400 × 1.15 = **₹460**.

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### Example 4: Simple and Compound Interest **Find SI and CI on ₹5000 at 10% per annum for 2 years.**

**Simple Interest**: SI = (5000 × 10 × 2)/100 = ₹1000. Amount = 5000 + 1000 = ₹6000.

**Compound Interest**: A = 5000 × (1 + 10/100)² = 5000 × (1.1)² = 5000 × 1.21 = ₹6050. CI = 6050 − 5000 = ₹1050.

**Difference**: CI − SI = 1050 − 1000 = ₹50.

*Shortcut for 2 years*: Difference = P × (R/100)² = 5000 × 0.01 = ₹50.

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Common Mistakes

| Wrong Thinking | Correct Fix | |----------------|-------------| | "20% increase then 20% decrease gives 0% change" | Successive percentage changes multiply, not add. Use the formula: net = a + b + ab/100. Here: 20 − 20 − 4 = −4%. | | Calculating profit/loss percentage on SP | Profit% and Loss% are always calculated on **Cost Price** unless the question explicitly says otherwise. | | Adding ratios directly when combining | You cannot add 2:3 and 1:2 as 3:5. Convert to fractions with common denominators or find actual quantities first. | | Confusing SI formula with CI formula | SI uses simple multiplication (P×R×T/100). CI uses power formula with (1 + R/100)^T. Don't interchange. | | Forgetting to convert time to years | If time is in months, divide by 12. For 6 months at 10% p.a., use T = 0.5, not 6. |

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Quick Reference

• **x% of N** = (x × N)/100 — always divide by 100.
• **Successive % change** (a%, then b%): Net = a + b + (ab/100).
• **Profit % = (Profit/CP) × 100** — never on SP.
• **SI = PTR/100** — linear growth; **CI uses (1 + R/100)^T** — exponential growth.
• **Ratio a:b** → parts are a/(a+b) and b/(a+b) of the total.
• **For 2-year CI−SI difference**: P × (R/100)².