Mathematical Shortcut Techniques

Master mathematical shortcut techniques for RRB exam preparation with comprehensive methods to solve problems quickly and accurately.

Basic Arithmetic Shortcuts

Addition Techniques

Mental Addition Methods

• Column Method: Add from right to left
• Rounding and Compensating:
  • Example: 48 + 39 = (48 + 40) - 1 = 87
  • Example: 67 + 28 = (67 + 30) - 2 = 95
• Breaking Numbers:
  • Example: 234 + 189 = 234 + 200 - 11 = 423
  • Example: 456 + 278 = 456 + 300 - 22 = 734

Speed Addition Tricks

• Adding 9: Add 10, subtract 1
• Adding 11: Add 10, add 1
• Adding 99: Add 100, subtract 1
• Adding 101: Add 100, add 1

Subtraction Techniques

Mental Subtraction Methods

• Borrow and Pay Back:
  • Example: 432 - 267 = 432 - 300 + 33 = 165
  • Example: 567 - 189 = 567 - 200 + 11 = 378
• Complement Method:
  • Example: 1000 - 467 = 999 - 467 + 1 = 533
  • Example: 10000 - 3456 = 9999 - 3456 + 1 = 6544

Quick Subtraction

• Subtracting 9: Subtract 10, add 1
• Subtracting 11: Subtract 10, subtract 1
• Subtracting 99: Subtract 100, add 1
• Subtracting from 100: Use complements (9-7=2 for 100-67)

Multiplication Shortcuts

Multiplication by Special Numbers

Multiplication by 11

• Single digit: 11 × 7 = 77
• Two digits: 11 × 34 = 374 (3+7=10, carry over)
• General Method: Write sum of digits between original digits

Multiplication by 9

• Method: Multiply by 10, subtract original number
• Example: 9 × 47 = 470 - 47 = 423
• Example: 9 × 234 = 2340 - 234 = 2106

Multiplication by 5

• Even numbers: Halve and add zero
  • Example: 5 × 84 = 84 ÷ 2 = 42, then add zero = 420
• Odd numbers: Subtract 1, halve, then add 5
  • Example: 5 × 67 = (67-1) ÷ 2 = 33, then add 5 = 335

Multiplication by 25

• Method: Multiply by 100, divide by 4
• Example: 25 × 64 = 6400 ÷ 4 = 1600
• Example: 25 × 68 = 6800 ÷ 4 = 1700

Quick Multiplication Techniques

Base Method

• Same Base: Use 10, 100, 1000 as base
• Example: 97 × 96 = (100-3) × (100-4) = 10000 - 700 + 12 = 9312
• Example: 103 × 104 = (100+3) × (100+4) = 10000 + 700 + 12 = 10712

Split and Multiply

• Example: 43 × 27 = 43 × (30 - 3) = 1290 - 129 = 1161
• Example: 67 × 34 = 67 × (40 - 6) = 2680 - 402 = 2278

Division Shortcuts

Division by Special Numbers

Division by 5

• Method: Multiply by 2, divide by 10
• Example: 345 ÷ 5 = (345 × 2) ÷ 10 = 690 ÷ 10 = 69
• Example: 823 ÷ 5 = (823 × 2) ÷ 10 = 1646 ÷ 10 = 164.6

Division by 25

• Method: Multiply by 4, divide by 100
• Example: 450 ÷ 25 = (450 × 4) ÷ 100 = 1800 ÷ 100 = 18
• Example: 675 ÷ 25 = (675 × 4) ÷ 100 = 2700 ÷ 100 = 27

Division by 125

• Method: Multiply by 8, divide by 1000
• Example: 1000 ÷ 125 = (1000 × 8) ÷ 1000 = 8
• Example: 875 ÷ 125 = (875 × 8) ÷ 1000 = 7000 ÷ 1000 = 7

Estimation Techniques

• Rounding: Round to nearest convenient number
• Example: 487 ÷ 23 ≈ 500 ÷ 25 = 20
• Example: 894 ÷ 31 ≈ 900 ÷ 30 = 30

Percentage Shortcuts

Quick Percentage Calculations

Common Percentages

• 10%: Move decimal point one place left
  • Example: 10% of 450 = 45
• 1%: Move decimal point two places left
  • Example: 1% of 3750 = 37.5
• 5%: Half of 10%
  • Example: 5% of 840 = 84 ÷ 2 = 42
• 50%: Half of the number
  • Example: 50% of 680 = 340

Percentage Increase/Decrease

• Quick Method: Use multiplication factors
• Example: 20% increase = × 1.2
• Example: 15% decrease = × 0.85
• Example: 25% of 480 = 480 × 0.25 = 120

Successive Percentage Changes

• Formula: A + B + (A×B)/100
• Example: 20% increase followed by 10% increase
  • Total = 20 + 10 + (20×10)/100 = 30%
• Example: 10% decrease followed by 5% increase
  • Total = -10 + 5 + (-10×5)/100 = -5.5%

Time and Work Shortcuts

Work and Efficiency Problems

Basic Formula

• Work = Rate × Time
• Efficiency: Work done per unit time
• Combined Work: Sum of individual rates

Quick Methods

• LCM Method: Use LCM of days as total work

  • Example: A can do work in 15 days, B in 20 days
  • LCM = 60 units, A = 4 units/day, B = 3 units/day
  • Together = 7 units/day, Time = 60 ÷ 7 = 8.57 days

• Fraction Method: Use fractions for efficiency

  • Example: A = 1/15, B = 1/20, Together = 1/15 + 1/20 = 7/60
  • Time = 60/7 days

Pipe and Cistern Problems

Filling and Emptying

• Filling Rate: Positive, Emptying Rate**: Negative
• Net Rate: Sum of all rates
• Formula: Time = Total Capacity ÷ Net Rate

Quick Examples

• Example: Pipe A fills in 6 hours, Pipe B in 8 hours, Pipe C empties in 24 hours
  • Rates: A = 1/6, B = 1/8, C = -1/24
  • Together: 1/6 + 1/8 - 1/24 = (4+3-1)/24 = 6/24 = 1/4
  • Time = 4 hours

Time and Distance Shortcuts

Speed, Distance, Time

Basic Formula

• Speed = Distance ÷ Time
• Distance = Speed × Time
• Time = Distance ÷ Speed

Unit Conversion Tricks

• km/h to m/s: Multiply by 5/18
  • Example: 54 km/h = 54 × 5/18 = 15 m/s
• m/s to km/h: Multiply by 18/5
  • Example: 20 m/s = 20 × 18/5 = 72 km/h

Relative Speed

Same Direction

• Relative Speed: Difference of speeds
• Formula: Time = Distance ÷ (Speed₁ - Speed₂)

Opposite Direction

• Relative Speed: Sum of speeds
• Formula: Time = Distance ÷ (Speed₁ + Speed₂)

Quick Examples

• Example: Train A 60 km/h, Train B 80 km/h, same direction

  • Relative speed = 80 - 60 = 20 km/h
  • Time to catch up over 100 km = 100 ÷ 20 = 5 hours

• Example: Train A 60 km/h, Train B 80 km/h, opposite direction

  • Relative speed = 60 + 80 = 140 km/h
  • Time to meet over 210 km = 210 ÷ 140 = 1.5 hours

Average Shortcuts

Quick Average Calculations

Simple Average

• Direct Formula: Sum of all values ÷ Number of values
• Mental Method: Add numbers, count and divide

Combined Average

• Formula: (n₁ × a₁ + n₂ × a₂) ÷ (n₁ + n₂)
• Example: Class of 30 students average 80, class of 40 students average 90
  • Combined average = (30×80 + 40×90) ÷ (30+40) = (2400+3600) ÷ 70 = 86

Average Speed

• Equal Distances: 2xy/(x+y)
  • Example: Speeds 60 and 80, equal distances
  • Average = (2×60×80) ÷ (60+80) = 9600 ÷ 140 = 68.57 km/h
• Equal Times: (x+y)/2
  • Example: Speeds 60 and 80, equal times
  • Average = (60+80) ÷ 2 = 70 km/h

Ratio and Proportion Shortcuts

Ratio Simplification

Basic Rules

• Divide by GCF: Simplify by greatest common factor
• Equivalent Ratios: Multiply or divide by same number

Proportion Problems

• Cross Multiplication: a/b = c/d, then a×d = b×c
• Direct Proportion: Increase/decrease together
• Inverse Proportion: One increases, other decreases

Quick Examples

• Example: If 8:x = 12:18, find x

  • 8/x = 12/18, so 8×18 = 12×x
  • 144 = 12x, x = 12

• Example: If 5 workers can complete work in 12 days, how many workers for 8 days?

  • Inverse proportion: 5×12 = 8×workers
  • Workers = (5×12) ÷ 8 = 7.5 ≈ 8 workers

Interest and Profit-Loss Shortcuts

Simple Interest

Quick Formula

• SI = P × R × T ÷ 100
• P = SI × 100 ÷ (R × T)
• R = SI × 100 ÷ (P × T)
• T = SI × 100 ÷ (P × R)

Mental Calculation

• Example: P = 5000, R = 8%, T = 3 years
  • SI = 5000 × 8 × 3 ÷ 100 = 1200
  • Amount = 5000 + 1200 = 6200

Compound Interest

Approximation Method

• For 2 years: CI = P × (2R + R²/100) ÷ 100
• Example: P = 10000, R = 10%, T = 2 years
  • CI = 10000 × (20 + 1) ÷ 100 = 2100

Rule of 72

• Doubling Time: 72 ÷ Interest Rate
• Example: At 12% interest, money doubles in 72 ÷ 12 = 6 years

Profit and Loss

Quick Formulas

• Profit% = (Profit ÷ CP) × 100
• Loss% = (Loss ÷ CP) × 100
• SP = CP × (1 + Profit%/100)
• SP = CP × (1 - Loss%/100)

Mental Tricks

• Example: CP = 500, Profit% = 25%
  • SP = 500 × 1.25 = 625
• Example: SP = 600, Profit% = 20%
  • CP = 600 ÷ 1.2 = 500

Number System Shortcuts

Divisibility Rules

Quick Tests

• 2: Last digit even
• 3: Sum of digits divisible by 3
• 4: Last two digits divisible by 4
• 5: Last digit 0 or 5
• 6: Divisible by both 2 and 3
• 8: Last three digits divisible by 8
• 9: Sum of digits divisible by 9
• 11: Difference between sum of digits at odd and even places

Prime Numbers

• Quick Test: Check divisibility up to √n
• Common Primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31

Square and Cube Shortcuts

Perfect Squares (1-30)

• Pattern: Last digits repeat in cycles
• Quick Recognition: Look for perfect square patterns
• Mental Squares: Use (a+b)² = a² + 2ab + b²

Square Roots

• Estimation: Find nearest perfect squares
• Example: √85 ≈ 9 (since 9²=81, 10²=100)

Cubes (1-20)

• Last Digit Pattern: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
• Quick Recognition: Look for cube patterns

Practice Questions

Question 1

Calculate 48 × 52 using base method.

Question 2

Find 15% of 240 using shortcut method.

Question 3

If 3 workers can complete a work in 12 days, how many days will 6 workers take?

Question 4

A train travels 240 km at 60 km/h and returns at 80 km/h. Find average speed.

Question 5

Find the average of first 20 odd numbers.

Question 6

Calculate 875 ÷ 25 using division shortcut.

Question 7

If the ratio of A:B is 3:4 and B:C is 5:6, find A:B:C.

Question 8

Find simple interest on ₹5000 at 8% for 3 years.

Question 9

Calculate 97 × 94 using base method.

Question 10

Find the time taken by a train 200m long to cross a platform 300m long at 60 km/h.

Time-Saving Tips

Mental Math Strategies

1. Break Complex Problems: Split into simpler parts
2. Use Estimation: Round to convenient numbers
3. Memorize Tables: Learn multiplication tables up to 30
4. Practice Daily: Regular practice improves speed
5. Use Patterns: Recognize number patterns

Exam Strategy

1. Time Management: Allocate specific time per question
2. Skip Difficult Questions: Return later if time permits
3. Use Rough Work: Organized calculations reduce errors
4. Double-Check: Verify answers if time allows
5. Practice Mock Tests: Simulate exam conditions

Common Mistakes to Avoid

1. Calculation Errors: Careful with basic operations
2. Unit Conversion: Ensure consistent units
3. Formula Application: Use correct formulas
4. Reading Questions: Understand what’s asked
5. Answer Choices: Check all options before selecting

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