Arithmetic

Number System, Percentage, Ratio, Profit-Loss, Time-Work

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Overview

Arithmetic forms the quantitative backbone of the HTET reasoning section across all three levels (PRT, TGT, PGT). Questions typically test speed and accuracy rather than deep mathematical insight—examiners want to see if you can handle classroom-level calculations confidently.

This topic carries 5–8 questions on average. Mastery here directly boosts your score because these are often the most straightforward questions in the paper, provided you know the shortcuts. The concepts also overlap with the Mathematics subject paper for Level 1 and Level 2 candidates.

Focus areas: quick mental math for percentages, recognising standard ratio patterns, applying profit-loss formulas without confusion over cost vs selling price, and choosing the right approach (LCM method or efficiency method) for time-work problems.

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Key Concepts

• **Number System Basics**: Natural numbers (1, 2, 3…), Whole numbers (0, 1, 2…), Integers (…−2, −1, 0, 1, 2…), Rational numbers (p/q form), Irrational numbers (non-terminating, non-repeating decimals like √2).

• **Divisibility Rules**: A number is divisible by 3 if sum of digits is divisible by 3; by 4 if last two digits form a number divisible by 4; by 9 if digit sum divisible by 9; by 11 if difference of sums of alternate digits is 0 or divisible by 11.

• **Percentage as "Per Hundred"**: x% of Y = (x × Y)/100. Converting fractions to percentages: multiply by 100. Key equivalents: 1/2 = 50%, 1/4 = 25%, 1/5 = 20%, 1/8 = 12.5%.

• **Ratio Expresses Relative Magnitude**: If A : B = 3 : 5, then A = 3k and B = 5k for some constant k. Ratios can be combined using LCM of common terms.

• **Profit-Loss Reference Point**: Profit or loss is always calculated on Cost Price (CP), not Selling Price (SP). Profit% = (Profit/CP) × 100.

• **Time-Work Relationship**: If A completes work in x days, A's one-day work = 1/x. Total work is often assumed as LCM of individual times for easier calculation.

• **Pipes and Cisterns Follow Same Logic**: Inlet adds, outlet subtracts; combined rate = algebraic sum of individual rates.

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Formulas / Key Facts

| Topic | Formula / Fact | |-------|----------------| | HCF-LCM Relation | HCF × LCM = Product of two numbers | | Percentage Increase | New Value = Original × (1 + r/100) | | Percentage Decrease | New Value = Original × (1 − r/100) | | Successive % Change | Net effect on 100: (a + b + ab/100)% for changes a% and b% | | Ratio Proportion | If a : b = c : d, then ad = bc (cross-multiplication) | | Profit | SP − CP (when SP > CP) | | Loss | CP − SP (when CP > SP) | | SP when Profit% given | SP = CP × (100 + P%)/100 | | SP when Loss% given | SP = CP × (100 − L%)/100 | | Time-Work (A + B together) | 1/A + 1/B = 1/T, where T = combined time | | Work-Wage Rule | Wages distributed in ratio of work done (or efficiency) |

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Worked Examples

### Example 1: Percentage **Problem**: A shopkeeper increases the price of an item by 20% and then offers a 10% discount. What is the net percentage change in price?

**Solution**: Using successive change formula: a = +20, b = −10 Net change = 20 + (−10) + (20 × −10)/100 = 20 − 10 − 2 = **8% increase**

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### Example 2: Profit and Loss **Problem**: An article is sold for ₹540 at a profit of 8%. Find the cost price.

**Solution**: SP = CP × (100 + Profit%)/100 540 = CP × 108/100 CP = 540 × 100/108 = 54000/108 = **₹500**

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### Example 3: Time and Work **Problem**: A can finish a job in 12 days, B can finish it in 18 days. How many days will they take working together?

**Solution**: A's one-day work = 1/12 B's one-day work = 1/18 Combined one-day work = 1/12 + 1/18 = (3 + 2)/36 = 5/36 Time together = 36/5 = **7.2 days** (or 7 days 4.8 hours)

*Shortcut*: LCM of 12 and 18 = 36 units (total work). A does 3 units/day, B does 2 units/day. Together = 5 units/day. Time = 36/5 days.

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### Example 4: Ratio **Problem**: The ratio of milk to water in a mixture is 5 : 3. If 8 litres of water is added, the ratio becomes 5 : 5. Find the original quantity of milk.

**Solution**: Let milk = 5k, water = 3k After adding 8 litres water: 5k : (3k + 8) = 5 : 5 = 1 : 1 So 5k = 3k + 8 → 2k = 8 → k = 4 Original milk = 5 × 4 = **20 litres**

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Common Mistakes

| Wrong Thinking | Correct Fix | |----------------|-------------| | Calculating profit% on SP instead of CP | Always use CP as base: Profit% = (Profit/CP) × 100 | | Adding successive percentages directly (20% + 10% = 30%) | Use formula: a + b + ab/100 to account for compounding | | Confusing ratio with actual values | Ratio 3:5 means 3k and 5k; find k using given total or difference | | Subtracting work rates when both are working together | Add rates: 1/A + 1/B. Subtract only for opposing actions (pipes: inlet vs outlet) | | Forgetting to convert time units | If A works in days and B in hours, convert both to same unit before combining | | Applying percentage decrease formula for increase | Increase: multiply by (1 + r/100); Decrease: multiply by (1 − r/100) |

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Quick Reference

• **Percentage ↔ Fraction**: 25% = 1/4, 33.33% ≈ 1/3, 12.5% = 1/8, 16.67% ≈ 1/6

• **Profit% shortcut**: If SP is x% more than CP, profit = x%

• **Two successive changes a% and b%**: Net = a + b + ab/100

• **Combined work time**: T = (A × B)/(A + B) when A and B are individual times

• **HCF for dividing equally, LCM for simultaneous occurrence**

• **Ratio a : b : c with total T**: Each part = (respective ratio term / sum of ratios) × T