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πŸ”’ Number Systems
βž— Fractions & Indices
πŸ“Š Ratios & Averages
πŸ’° Applied Arithmetic

Arithmetic β€” The Foundation of All Math

Arithmetic is the oldest and most practical branch of mathematics. Every competitive exam β€” SSC, Banking, Railways, Defence, CTET, CLAT, UPSC CSAT β€” tests your arithmetic fluency. This lesson builds deep intuition, not just formulas.

Part 1

πŸ”’ Number Systems & Divisibility

The Human Need to Count
Why Numbers Were Invented
Thousands of years ago, a shepherd needed to know if all his sheep were back. He'd put a pebble in a pouch for each sheep that left the pen. At sunset, he matched pebbles to returning sheep. That matching is the origin of counting β€” one-to-one correspondence. From that simple need, humanity built the entire number system: Natural numbers (the pebbles), then Integers (adding debt), Rational numbers (sharing food), and Real numbers (measuring everything).
Every number type was invented to solve a practical human problem.

The Number Zoo

Here is how numbers are classified. Each box contains the previous:

Real Numbers ℝ Rational β„š Integers β„€ Whole Numbers β„•β‚€ Natural Numbers β„• Counting: 1, 2, 3, ... + 0 β†’ Whole + negatives β†’ Integers + fractions β†’ Rational Irrational: Ο€, √2, e β€” non-repeating, non-terminating decimals Irrationals
The hierarchy of number systems. Each layer adds new numbers that solve new problems.

Divisibility Rules β€” Shortcuts That Save Minutes

Quick checks to see if one number divides another without actually dividing:

By 2
Last digit even (0,2,4,6,8)
By 3
Sum of digits Γ· 3
By 4
Last 2 digits Γ· 4
By 5
Last digit 0 or 5
By 6
Divisible by 2 AND 3
By 8
Last 3 digits Γ· 8
By 9
Sum of digits Γ· 9
By 11
(Sum odd βˆ’ sum even) Γ· 11
Example
Is 7,284 divisible by 11? Sum of odd-position digits: 7 + 8 = 15. Sum of even-position digits: 2 + 4 = 6. Difference: 15 βˆ’ 6 = 9. Since 9 is not a multiple of 11, 7,284 is not divisible by 11.

LCM & HCF β€” The Building Blocks

Every number can be broken into prime factors. HCF (GCD) is the biggest common factor. LCM is the smallest common multiple.

Key Insight
For any two numbers: LCM Γ— HCF = Product of numbers
Always true. Use it to find one when you know the other.
Real-World LCM
The Traffic Light Problem
Two traffic lights blink together. One blinks every 24 seconds, the other every 36 seconds. When will they blink together again? This is an LCM problem: find LCM(24, 36). Prime factors: 24 = 2Β³ Γ— 3, 36 = 2Β² Γ— 3Β². Take highest powers: 2Β³ Γ— 3Β² = 8 Γ— 9 = 72 seconds.
LCM = "least common multiple" β€” the next time two cycles align.
1
Find HCF of 144 and 198 β–Ά
Method 1 (Prime factorization):
144 = 2⁴ Γ— 3Β²
198 = 2 Γ— 3Β² Γ— 11
Common factors: 2 Γ— 3Β² = 2 Γ— 9 = 18

Method 2 (Division/Euclid's algorithm):
198 Γ· 144 = 1 remainder 54
144 Γ· 54 = 2 remainder 36
54 Γ· 36 = 1 remainder 18
36 Γ· 18 = 2 remainder 0
HCF = 18
Part 2

βž— Fractions, Decimals, Surds & Indices

Breaking Things into Parts
The Pizza That Taught Fractions
A pizza cut into 8 slices. You eat 3. You've eaten Β³β„β‚ˆ of the pizza. That's a fraction β€” the numerator (3) counts the parts you have, and the denominator (8) tells how many equal parts make the whole. Fractions are just division that hasn't been performed yet.
A fraction is a quotient written vertically. Β³β„β‚ˆ = 3 Γ· 8.

Types of Fractions

  • Proper: numerator < denominator (e.g., ²⁄₅) β€” value < 1
  • Improper: numerator β‰₯ denominator (e.g., ⁷⁄₃) β€” value β‰₯ 1
  • Mixed: whole + proper fraction (e.g., 2β…“ = ⁷⁄₃)
  • Decimal: fraction in base-10 (e.g., Β³β„β‚ˆ = 0.375)
Addition
ᡃ⁄ᡦ + αΆœβ„β‚” = (ad + bc) / bd
Multiplication
ᡃ⁄ᡦ Γ— αΆœβ„β‚” = ac / bd
Division
ᡃ⁄ᡦ Γ· αΆœβ„β‚” = a/b Γ— d/c
Comparison
ᡃ⁄ᡦ > αΆœβ„β‚” iff ad > bc

Decimal Conversions

Every fraction can be written as a decimal β€” either terminating (denominator's prime factors are only 2 and 5) or repeating.

Repeating Decimal Trick
To convert 0.Μ…142857 to a fraction: Let x = 0.142857142857... Multiply by 1,000,000 (6 digits repeat): 1,000,000x = 142,857.142857... Subtract x: 999,999x = 142,857. So x = 142,857/999,999 = 1/7. That's why 1/7 = 0.Μ…142857.

Surds β€” Irrational Square Roots

Surds are roots that can't be simplified to integers (√2, √3, √5). The key skill is rationalization β€” removing the root from the denominator.

Rationalization Formula
1 / (√a + √b) = (√a βˆ’ √b) / (a βˆ’ b)
Multiply numerator and denominator by the conjugate.
2
Simplify: 1/(√5 + √2) β–Ά
Multiply top and bottom by the conjugate (√5 βˆ’ √2):
= (√5 βˆ’ √2) / ((√5)Β² βˆ’ (√2)Β²)
= (√5 βˆ’ √2) / (5 βˆ’ 2)
= (√5 βˆ’ √2) / 3

Laws of Indices (Exponents)

Product
aᡐ Γ— aⁿ = aᡐ⁺ⁿ
Quotient
aᡐ ÷ aⁿ = aᡐ⁻ⁿ
Power of Power
(aᡐ)ⁿ = aᡐⁿ
Zero Exponent
a⁰ = 1 (a β‰  0)
Negative
a⁻ⁿ = 1/aⁿ
Fractional
aᡐ/ⁿ = ⁿ√(aᡐ)
Part 3

πŸ“Š Ratios, Percentages & Averages

Comparing Quantities
The Lemonade Stand
You make lemonade with 2 cups of water and 1 cup of lemon juice. The ratio of water to juice is 2:1. If you scale to 10 cups of water, you need 5 cups of juice to keep the same taste. That's proportion β€” ratios stay constant when you multiply both parts by the same number. In exams, ratios tell you how to share, mix, and compare.
A ratio compares two quantities. A proportion says two ratios are equal.

Ratio & Proportion

If a:b = c:d, then ad = bc (cross-multiplication). Key applications:

  • Sharing: Divide a total in a given ratio
  • Mixtures: Find ratio of ingredients, or combine two mixtures
  • Direct proportion: y = kx (more work β†’ more pay)
  • Inverse proportion: y = k/x (more people β†’ less time)
Mixture Problem
Two vessels contain milk and water in ratios 3:1 and 5:3. Mix them in equal quantity. What's the new ratio?
First vessel: milk = ΒΎ, water = ΒΌ. Second: milk = ⅝, water = β…œ.
Total milk = ΒΎ + ⅝ = 6/8 + 5/8 = 11/8. Total water = ΒΌ + β…œ = 2/8 + 3/8 = 5/8.
Ratio milk:water = 11:5.

Percentage β€” "Per Hundred"

Percentages are ratios with denominator 100. They are the most tested topic in every exam.

Percentage Change
(New βˆ’ Old) / Old Γ— 100
Successive %
a + b + ab/100
% to Fraction
a% = a/100
% of a Number
a% of b = (a Γ— b) / 100
Fraction ↔ Percentage Shortcuts
¹⁄₂ = 50%, ¹⁄₃ = 33β…“%, ¹⁄₄ = 25%, ¹⁄₅ = 20%,
¹⁄₆ = 16β…”%, ¹⁄₇ β‰ˆ 14.28%, ΒΉβ„β‚ˆ = 12.5%, ¹⁄₉ β‰ˆ 11.11%, ¹⁄₁₀ = 10%
Memorize these β€” they save 5-10 seconds per question.
3
Salary increased by 20%, then decreased by 20%. What's net change? β–Ά
Use successive formula: a + b + ab/100 = 20 + (βˆ’20) + (20 Γ— βˆ’20)/100
= 0 + (βˆ’400)/100 = βˆ’4% (net 4% decrease)

Intuition: 20% increase multiplies by 1.2, then 20% decrease multiplies by 0.8.
Net: 1.2 Γ— 0.8 = 0.96 = 4% decrease. Always a net loss when you increase then decrease by the same percentage.

Averages & Mixtures

Average = Sum Γ· Count. But exams rarely test the simple average β€” they test weighted averages and mixtures (alligation).

The Alligation Shortcut
Mixing Tea at Different Prices
You mix β‚Ή200/kg tea with β‚Ή300/kg tea. You want β‚Ή240/kg mixture. How much of each?
The alligation method: Draw a cross. Difference from mean: 300βˆ’240 = 60, 240βˆ’200 = 40. Ratio = 60:40 = 3:2. So mix 3 parts of β‚Ή200 tea with 2 parts of β‚Ή300 tea.
This works for any mixture: prices, percentages, concentrations, speeds.
Alligation = weighted average in reverse. Given the average, find the ratio.
Part 4

πŸ’° Applied Arithmetic β€” Profit, Interest, Time & Work

Profit & Loss

The language of business. Every transaction has Cost Price (CP) and Selling Price (SP).

Profit
SP βˆ’ CP
Loss
CP βˆ’ SP
Profit %
(Profit / CP) Γ— 100
Discount
MP βˆ’ SP
Key Insight
Profit percentage is ALWAYS on CP (unless stated otherwise). Discount percentage is ALWAYS on Marked Price (MP).
4
A shopkeeper marks goods 30% above cost and gives 10% discount. Find profit %. β–Ά
Let CP = β‚Ή100. MP = 100 + 30% of 100 = β‚Ή130.
Discount = 10% of MP = 10% of 130 = β‚Ή13.
SP = MP βˆ’ Discount = 130 βˆ’ 13 = β‚Ή117.
Profit = SP βˆ’ CP = 117 βˆ’ 100 = β‚Ή17.
Profit % = (17/100) Γ— 100 = 17%.

Quick formula: Profit % = (Markup% βˆ’ Discount% βˆ’ (Markup% Γ— Discount%)/100)
= 30 βˆ’ 10 βˆ’ (30 Γ— 10)/100 = 30 βˆ’ 10 βˆ’ 3 = 17%.

Simple & Compound Interest

Money grows over time. Simple Interest (SI) grows linearly. Compound Interest (CI) grows exponentially.

Simple Interest
SI = P Γ— R Γ— T / 100
Amount (SI)
A = P + SI
Compound Interest
A = P(1 + R/100)α΅€
CI = A βˆ’ P
The Power of Compounding
β‚Ή1 That Became β‚Ή16
You invest β‚Ή1 at 100% annual interest. After 1 year: β‚Ή2. After 2: β‚Ή4. After 3: β‚Ή8. After 4: β‚Ή16. That's doubling every year β€” exponential growth. In SI, β‚Ή1 at 100% would be β‚Ή1 + β‚Ή1Γ—4 = β‚Ή5 after 4 years. CI gives β‚Ή16 β€” more than 3Γ— higher! This is why Einstein called compounding "the eighth wonder of the world."
CI grows faster because you earn interest on interest. Time magnifies the difference massively.
CI vs SI Difference (2 years)
CI βˆ’ SI = P Γ— (R/100)Β²
For 3 years: CI βˆ’ SI = P Γ— (R/100)Β² Γ— (3 + R/100)

Time & Work

The fundamental relationship: Work = Rate Γ— Time. The trick is to invert: a person who takes T days to finish does ¹⁄ᡀ of the work per day.

Classic Problem
A can finish a job in 12 days, B in 18 days. How many days together?
A's daily work = ¹⁄₁₂, B's daily work = ΒΉβ„β‚β‚ˆ.
Combined = ¹⁄₁₂ + ΒΉβ„β‚β‚ˆ = (3+2)/36 = ⁡⁄₃₆.
Time = 1 Γ· (⁡⁄₃₆) = ³⁢⁄₅ = 7.2 days.

Speed, Distance & Time

Speed
Speed = Distance / Time
Average Speed
Total Dist / Total Time
Rel Speed (same direction)
v₁ βˆ’ vβ‚‚
Rel Speed (opposite)
v₁ + vβ‚‚
Average Speed Trap
If you travel at v₁ for half the DISTANCE and vβ‚‚ for the other half, average speed = 2v₁vβ‚‚ / (v₁ + vβ‚‚).
Not (v₁ + vβ‚‚)/2!
Similarly, if you travel at v₁ for half the TIME, then average = (v₁ + vβ‚‚)/2.
5
A train 300m long passes a pole in 15s and a platform in 45s. Find platform length. β–Ά
Speed of train = 300m / 15s = 20 m/s.
To pass the platform, train must cover its own length + platform length.
Distance = Speed Γ— Time = 20 Γ— 45 = 900m.
Platform length = 900 βˆ’ 300 = 600m.

🎯 Practice β€” 12 Questions

Test your understanding. Click an option to see if you're right.

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