🧮 Mental Math Tricks

Master mental math techniques to solve problems faster and more accurately.

Basic Arithmetic Shortcuts

Addition Tricks

Adding 9: Add 10, then subtract 1

• $47 + 9 = 47 + 10 - 1 = 56$

Adding 11: Add 10, then add 1

• $47 + 11 = 47 + 10 + 1 = 58$

Adding near multiples of 10:

• $47 + 19 = 47 + 20 - 1 = 66$
• $47 + 21 = 47 + 20 + 1 = 68$

Adding two-digit numbers:

• $47 + 23 = (40 + 20) + (7 + 3) = 60 + 10 = 70$

Subtraction Tricks

Subtracting 9: Subtract 10, then add 1

• $47 - 9 = 47 - 10 + 1 = 38$

Subtracting 11: Subtract 10, then subtract 1

• $47 - 11 = 47 - 10 - 1 = 36$

Subtracting near multiples of 10:

• $47 - 19 = 47 - 20 + 1 = 28$
• $47 - 21 = 47 - 20 - 1 = 26$

Multiplication Tricks

Multiplying by 5: Multiply by 10, then divide by 2

• $47 \times 5 = 470 \div 2 = 235$

Multiplying by 25: Multiply by 100, then divide by 4

• $47 \times 25 = 4700 \div 4 = 1175$

Multiplying by 11: Add the digits and place in the middle

• $47 \times 11 = 4(4+7)7 = 517$

Multiplying by 12: Multiply by 10, then add double the number

• $47 \times 12 = 470 + 94 = 564$

Division Tricks

Dividing by 5: Multiply by 2, then divide by 10

• $47 \div 5 = 94 \div 10 = 9.4$

Dividing by 25: Multiply by 4, then divide by 100

• $47 \div 25 = 188 \div 100 = 1.88$

Dividing by 4: Divide by 2 twice

• $47 \div 4 = 23.5 \div 2 = 11.75$

Squares and Square Roots

Perfect Squares to Memorize

Square Calculation Tricks

Squares ending in 5:

• $25^2 = 2 \times 3 = 6$, then add 25 = 625
• $35^2 = 3 \times 4 = 12$, then add 25 = 1225

Squares near perfect squares:

• $47^2 = 50^2 - 2(50)(3) + 3^2 = 2500 - 300 + 9 = 2209$
• $53^2 = 50^2 + 2(50)(3) + 3^2 = 2500 + 300 + 9 = 2809$

Square Root Estimation

For perfect squares: Use memorized values For other numbers: Use approximation

• $\sqrt{50} \approx 7.07$ (between $7^2 = 49$ and $8^2 = 64$)
• $\sqrt{120} \approx 10.95$ (between $10^2 = 100$ and $11^2 = 121$)

Fraction and Decimal Shortcuts

Common Fraction-Decimal Pairs

Fraction Operations

Adding fractions with same denominator:

• $\frac{3}{7} + \frac{2}{7} = \frac{5}{7}$

Adding fractions with different denominators:

• $\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$

Multiplying fractions:

• $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$

Dividing fractions:

• $\frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9}$

Percent Calculations

Finding percentages:

• $20%$ of 150 = $0.2 \times 150 = 30$
• $15%$ of 200 = $0.15 \times 200 = 30$

Percentage increase/decrease:

• Increase 80 by 25%: $80 + 0.25 \times 80 = 80 + 20 = 100$
• Decrease 120 by 15%: $120 - 0.15 \times 120 = 120 - 18 = 102$

Special Number Patterns

Pythagorean Triples

Powers of 2

Powers of 10

Algebraic Shortcuts

Factoring Patterns

Difference of squares: $a^2 - b^2 = (a+b)(a-b)$

• $25 - 9 = 5^2 - 3^2 = (5+3)(5-3) = 8 \times 2 = 16$

Perfect square trinomials: $a^2 + 2ab + b^2 = (a+b)^2$

• $x^2 + 6x + 9 = (x+3)^2$

Sum/difference of cubes: $a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)$

• $8 - 1 = 2^3 - 1^3 = (2-1)(4+2+1) = 1 \times 7 = 7$

Quick Substitution

When $x = 2$:

• $x^2 = 4$
• $x^3 = 8$
• $2x = 4$
• $3x = 6$

When $x = 3$:

• $x^2 = 9$
• $x^3 = 27$
• $2x = 6$
• $3x = 9$

Practice Exercises

Basic Arithmetic

1. $47 + 29 = ?$
2. $73 - 28 = ?$
3. $24 \times 5 = ?$
4. $156 \div 4 = ?$

Squares and Roots

1. $17^2 = ?$
2. $23^2 = ?$
3. $\sqrt{144} = ?$
4. $\sqrt{225} = ?$

Fractions and Percents

1. $\frac{3}{8} + \frac{1}{4} = ?$
2. $\frac{2}{3} \times \frac{3}{4} = ?$
3. $25%$ of 80 = ?
4. $15%$ of 120 = ?

Algebraic Shortcuts

1. If $x = 4$, find $x^2 + 2x + 1$
2. If $x = 5$, find $x^2 - 4$
3. Factor $x^2 - 16$
4. Factor $x^2 + 8x + 16$

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