🧮 Mental Math Toolkit

Master essential mental math techniques to perform calculations quickly and accurately during the contest. Speed and accuracy in basic arithmetic will save valuable time.

🎯 Core Mental Math Skills

Addition & Subtraction

Break down numbers: $47 + 38 = 47 + 30 + 8 = 77 + 8 = 85$
Use complements: $100 - 47 = 53$ (since $47 + 53 = 100$)
Round and adjust: $298 + 156 = 300 + 156 - 2 = 456 - 2 = 454$
Left to right: $234 + 567 = 200 + 500 + 30 + 60 + 4 + 7 = 700 + 90 + 11 = 801$

Multiplication

Break down factors: $24 \times 15 = 24 \times 10 + 24 \times 5 = 240 + 120 = 360$
Use distributive property: $13 \times 17 = (10 + 3)(10 + 7) = 100 + 70 + 30 + 21 = 221$
Round and adjust: $19 \times 23 = 20 \times 23 - 23 = 460 - 23 = 437$
Use known products: $25 \times 16 = 25 \times 4 \times 4 = 100 \times 4 = 400$

Division

Break down divisors: $144 \div 12 = 144 \div 4 \div 3 = 36 \div 3 = 12$
Use multiplication: $168 \div 14 = ?$ → $14 \times 12 = 168$, so $168 \div 14 = 12$
Round and adjust: $147 \div 7 = 140 \div 7 + 7 \div 7 = 20 + 1 = 21$
Use factors: $180 \div 15 = 180 \div 3 \div 5 = 60 \div 5 = 12$

🔢 Fraction Mastery

Converting Fractions

To decimals: $\frac{1}{8} = 0.125$, $\frac{3}{8} = 0.375$, $\frac{5}{8} = 0.625$, $\frac{7}{8} = 0.875$
To percents: $\frac{1}{4} = 25%$, $\frac{1}{3} \approx 33.3%$, $\frac{2}{3} \approx 66.7%$
Common fractions: $\frac{1}{2} = 0.5$, $\frac{1}{4} = 0.25$, $\frac{3}{4} = 0.75$

Adding & Subtracting Fractions

Same denominator: $\frac{3}{7} + \frac{2}{7} = \frac{5}{7}$
Different denominators: $\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}$
Mixed numbers: $2\frac{1}{3} + 1\frac{2}{3} = 3\frac{3}{3} = 4$

Multiplying & Dividing Fractions

Multiplication: $\frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}$
Division: $\frac{2}{3} \div \frac{3}{4} = \frac{2}{3} \times \frac{4}{3} = \frac{8}{9}$
Mixed numbers: $2\frac{1}{3} \times 1\frac{1}{2} = \frac{7}{3} \times \frac{3}{2} = \frac{21}{6} = 3\frac{1}{2}$

📊 Percent Calculations

Basic Percentages

10%: Move decimal one place left ($10%$ of $240 = 24$)
5%: Half of 10% ($5%$ of $240 = 12$)
1%: Move decimal two places left ($1%$ of $240 = 2.4$)
50%: Half the number ($50%$ of $240 = 120$)

Percent Applications

Percent increase: $120$ increased by $25% = 120 + 30 = 150$
Percent decrease: $200$ decreased by $15% = 200 - 30 = 170$
Percent of percent: $20%$ of $30% = 0.2 \times 0.3 = 0.06 = 6%$

Quick Percent Tricks

$12\frac{1}{2}% = \frac{1}{8}$: $12\frac{1}{2}%$ of $80 = 10$
$33\frac{1}{3}% = \frac{1}{3}$: $33\frac{1}{3}%$ of $90 = 30$
$66\frac{2}{3}% = \frac{2}{3}$: $66\frac{2}{3}%$ of $60 = 40$

🔢 Perfect Squares & Roots

Perfect Squares to 30

$1^2 = 1$, $2^2 = 4$, $3^2 = 9$, $4^2 = 16$, $5^2 = 25$
$6^2 = 36$, $7^2 = 49$, $8^2 = 64$, $9^2 = 81$, $10^2 = 100$
$11^2 = 121$, $12^2 = 144$, $13^2 = 169$, $14^2 = 196$, $15^2 = 225$
$16^2 = 256$, $17^2 = 289$, $18^2 = 324$, $19^2 = 361$, $20^2 = 400$

Square Root Approximations

$\sqrt{2} \approx 1.414$, $\sqrt{3} \approx 1.732$, $\sqrt{5} \approx 2.236$
$\sqrt{6} \approx 2.449$, $\sqrt{7} \approx 2.646$, $\sqrt{8} \approx 2.828$
$\sqrt{10} \approx 3.162$, $\sqrt{11} \approx 3.317$, $\sqrt{12} \approx 3.464$

Square Root Tricks

Perfect squares: $\sqrt{144} = 12$, $\sqrt{169} = 13$, $\sqrt{196} = 14$
Near perfect squares: $\sqrt{150} \approx \sqrt{144} = 12$ (actual: 12.25)
Estimation: $\sqrt{200} \approx \sqrt{196} = 14$ (actual: 14.14)

⚡ Approximation Techniques

Rounding Strategies

Round to nearest 10: $47 + 38 \approx 50 + 40 = 90$ (actual: 85)
Round to nearest 100: $298 + 156 \approx 300 + 200 = 500$ (actual: 454)
Round to nearest 1000: $1,234 + 2,567 \approx 1,000 + 3,000 = 4,000$ (actual: 3,801)

Order of Magnitude

Powers of 10: $10^0 = 1$, $10^1 = 10$, $10^2 = 100$, $10^3 = 1,000$
Scientific notation: $2.5 \times 10^3 = 2,500$
Quick estimates: $47 \times 23 \approx 50 \times 20 = 1,000$ (actual: 1,081)

Approximation Examples

$19 \times 21$: $20 \times 20 = 400$ (actual: 399)
$98 \times 102$: $100 \times 100 = 10,000$ (actual: 9,996)
$47 \times 53$: $50 \times 50 = 2,500$ (actual: 2,491)

🎯 Practice Drills

5-Minute Sprint: Basic Arithmetic

Target: 20 problems in 5 minutes (90%+ accuracy)

$47 + 38 = ?$
$156 - 89 = ?$
$24 \times 15 = ?$
$168 \div 12 = ?$
$13 \times 17 = ?$
$144 \div 9 = ?$
$29 + 47 = ?$
$203 - 87 = ?$
$18 \times 25 = ?$
$225 \div 15 = ?$

5-Minute Sprint: Fractions & Percents

Target: 15 problems in 5 minutes (90%+ accuracy)

$\frac{3}{8} + \frac{1}{4} = ?$
$\frac{5}{6} - \frac{1}{3} = ?$
$\frac{2}{3} \times \frac{3}{4} = ?$
$\frac{3}{5} \div \frac{2}{5} = ?$
$25%$ of $80 = ?$
$15%$ of $120 = ?$
$\frac{1}{8} = ?%$
$\frac{2}{3} = ?%$
$12\frac{1}{2}%$ of $40 = ?$
$33\frac{1}{3}%$ of $90 = ?$

5-Minute Sprint: Perfect Squares

Target: 20 problems in 5 minutes (95%+ accuracy)

$11^2 = ?$
$12^2 = ?$
$13^2 = ?$
$14^2 = ?$
$15^2 = ?$
$16^2 = ?$
$17^2 = ?$
$18^2 = ?$
$19^2 = ?$
$20^2 = ?$

📊 Progress Tracking

Accuracy Targets

Basic arithmetic: 95%+ accuracy
Fractions & percents: 90%+ accuracy
Perfect squares: 95%+ accuracy
Approximations: 85%+ accuracy

Speed Targets

Basic arithmetic: 4 problems per minute
Fractions & percents: 3 problems per minute
Perfect squares: 4 problems per minute
Approximations: 3 problems per minute

Weekly Goals

Week 1: Focus on accuracy, don’t worry about speed
Week 2: Maintain accuracy, increase speed by 20%
Week 3: Maintain accuracy, increase speed by 40%
Week 4: Maintain accuracy, increase speed by 60%

⚡ Quick Reference

Essential Techniques:

Break down numbers: Use distributive property
Round and adjust: Round to easy numbers, then adjust
Use complements: Find numbers that add to round numbers
Left to right: Add from left to right for mental math
Use known products: Apply multiplication facts

Common Values:

$\frac{1}{8} = 0.125$, $\frac{3}{8} = 0.375$, $\frac{5}{8} = 0.625$, $\frac{7}{8} = 0.875$
$12\frac{1}{2}% = \frac{1}{8}$, $33\frac{1}{3}% = \frac{1}{3}$, $66\frac{2}{3}% = \frac{2}{3}$
$\sqrt{2} \approx 1.414$, $\sqrt{3} \approx 1.732$, $\sqrt{5} \approx 2.236$

Practice Schedule:

Daily: 15 minutes of mental math practice
Focus areas: Work on your weakest skills
Progressive difficulty: Increase complexity over time
Time pressure: Practice under time constraints
Accuracy first: Speed comes with accuracy

Next: Modular Arithmetic Sprints | Back to: Strategy Guide

🧮 Mental Math Toolkit#

🎯 Core Mental Math Skills#

Addition & Subtraction#

Multiplication#

Division#

🔢 Fraction Mastery#

Converting Fractions#

Adding & Subtracting Fractions#

Multiplying & Dividing Fractions#

📊 Percent Calculations#

Basic Percentages#

Percent Applications#

Quick Percent Tricks#

🔢 Perfect Squares & Roots#

Perfect Squares to 30#

Square Root Approximations#

Square Root Tricks#

⚡ Approximation Techniques#

Rounding Strategies#

Order of Magnitude#

Approximation Examples#

🎯 Practice Drills#

5-Minute Sprint: Basic Arithmetic#

5-Minute Sprint: Fractions & Percents#

5-Minute Sprint: Perfect Squares#

📊 Progress Tracking#

Accuracy Targets#

Speed Targets#

Weekly Goals#

⚡ Quick Reference#

Essential Techniques:#

Common Values:#

Practice Schedule:#