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Elementary Number Sense (Grades 4–6)

2026–2027 Number Sense Problem Sequencing Chart — Check the skills you want to drill, or practice a full test mix.

Contest Rule: Problems marked with (*) require integral answers within ±5% of the exact value. All other problems require exact mental answers!

Problems 1–20

Problems 1 – 20 (Foundations & Shortcuts)

E1.1 Addition of Whole Numbers 27 + 48 = 75 E1.2 Subtraction of Whole Numbers 425 − 332 = 93 E1.3 Multiplication of Whole Numbers 26 × 6 = 156 E1.4 Division of Whole Numbers 312 ÷ 3 = 104 | 3434 ÷ 34 = 101 E1.5 Recognizing Place Value Digit in hundredths place of 34917.20568 → 0 E1.6 Rounding Off Whole Numbers & Decimals 38670.478 rounded to hundreds → 38700 E1.7 Multiplying by 11 52 × 11 = 572 | 178 × 11 = 1958 E1.8 Multiplying by 12 26 × 12 = 312 | 36 × 12 = 432 E1.9 Multiplying by 15 19 × 15 = 285 | 24 × 15 = 360 E1.10 Multiplying by 25 16 × 25 = 400 | 27 × 25 = 675 E1.11 Multiplying by 50 37 × 50 = 1850 | 16 × 50 = 800 E1.12 Multiplying by 75 28 × 75 = 2100 | 37 × 75 = 2775 E1.13 Remainders When Dividing by 2 983 ÷ 2 has a remainder of 1 E1.14 Remainders When Dividing by 3 832 ÷ 3 → 8+3+2=13 → remainder 1 E1.15 Remainders When Dividing by 4 382 ÷ 4 → 82 ÷ 4 has remainder 2 E1.16 Remainders When Dividing by 5 20583 ÷ 5 has a remainder of 3 E1.17 Remainders When Dividing by 8 5327 ÷ 8 → 327 ÷ 8 has remainder 7 E1.18 Remainders When Dividing by 9 60318 ÷ 9 → 6+0+3+1+8=18 → remainder 0 E1.19 Remainders When Dividing by 10 8712 ÷ 10 has a remainder of 2 E1.20 Remainders When Dividing by 11 7329 ÷ 11 → 9−2+3−7 = 3 E1.21 Smallest / Largest Even or Odd Number Problems Smallest even number greater than 17 → 18 E1.22 Counting Odd Whole Numbers Between Two Numbers Odd whole numbers between 11 and 48 → 18 E1.23 Counting Even Whole Numbers Between Two Numbers Even whole numbers between 13 and 25 → 6 E1.24 Expanded Notation (Whole Numbers & Powers of 10) 1×1000 + 9×100 + 7×10 + 4×1 = 1974 E1.25 Sums of Whole Numbers with Regrouping & Evenly Spaced Series 12 + 16 + 20 + 24 = 72 E1.26 Differences of Whole Numbers & Reversed 3-Digit Subtraction 721 − 127 = 594 E1.27 Roman Numerals / Arabic Numbers Conversion MCMXXV = 1925 | DLX = 560

Problems 21–40

Problems 21 – 40 (Fractions, Conversions & Number Theory)

E2.1 Addition & Subtraction of Fractions with Common Denominators 7/24 + 13/24 = 5/6 E2.2 Multiplication of Fractions 4/5 × 5/6 = 2/3 E2.3 Division of Fractions (9/10) ÷ (3/1000) = 300 E2.4 Comparing Fractions & Decimals Which is larger: 9/13 or 3/4? → 3/4 E2.5 Converting Fractions & Decimals (Either Way) .88 = 22/25 | 3/8 = .375 E2.6 Converting Percents & Fractions (Either Way) 38% = 19/50 | 83 1/3% = 5/6 E2.7 Converting Percents & Decimals (Either Way) 37% = .37 | 1.8 = 180% E2.8 Order of Operations (PEMDAS) 12 + 8 × 2 = 28 | 16 − 8 ÷ 2 = 12 E2.9 Multiplying Two Numbers: Same Tens Digit, Ones Digits Sum to 10 23 × 27 = 621 | 34 × 36 = 1224 E2.10 Multiplying Two Numbers Centered Around a Third Number 19 × 21 = 399 | 18 × 22 = 396 E2.11 Multiplying Two Numbers Close to and Less Than 100 99 × 98 = 9702 | 91 × 95 = 8645 E2.12 Multiplying Two Numbers Close to and Greater Than 100 102 × 103 = 10506 | 104 × 105 = 10920 E2.13 Multiplying Two Numbers Ending in 5 45 × 15 = 675 | 35 × 85 = 2975 E2.14 Distributive Property Shortcuts: a×b ± a×c 17 × 3 + 17 × 7 = 170 | 57 × 15 − 57 × 5 = 570 E2.15 Ratio & Proportion Problems 10 is to 8 as 15 is to n → n = 12 E2.16 Consumer Problems: Coins & Money Conversions 11 quarters + 7 nickels = 310¢ E2.17 Consumer Problems: Unit Pricing & Proportional Cost If 12 apples cost $2.40, then 8 apples cost $1.60 E2.18 Prime Numbers Less Than 100 Smallest prime more than 20 → 23 E2.19 Greatest Common Divisor (GCD) GCD of 24 and 32 = 8 E2.20 Least Common Multiple (LCM) LCM of 18 and 24 = 72 E2.21 Conversions: Length & Distance (Inches, Feet, Yards, Miles, Metric) 108 inches = 9 feet E2.22 Conversions: Capacity & Weight (Ounces, Cups, Pints, Quarts, Gallons, Pounds) 64 ounces = 2 quarts E2.23 Conversions: Time (Seconds, Minutes, Hours, Days, Months, Years) 15 years = 180 months

Problems 41–60

Problems 41 – 60 (Algebra, Geometry & Base Systems)

E3.1 Addition & Subtraction of Mixed Numbers 6 3/8 − 2 1/2 = 3 7/8 E3.2 Multiplication & Division of Mixed Numbers 5 1/5 × 5 4/5 = 30 4/25 E3.3 Substitution & Function Evaluation f(x) If f(x) = 3x − 5, then f(4) = 7 E3.4 Perimeter & Area of Squares and Rectangles Perimeter of square with area 169 cm² → 52 cm E3.5 Perimeter & Area of Triangles Area of triangle with base 8 and height 4½ → 18 E3.6 Radius, Diameter, Circumference & Area of a Circle Diameter of circle with area 100π → 20 E3.7 Powers & Exponents of Numbers 2⁴ × 3⁰ = 16 | 2⁵ = 32 E3.8 Square Roots & Cube Roots of Numbers √529 = 23 | ∛64 = 4 E3.9 Solving One-Variable Equations If 7x = 2x + 20, then x = 4 E3.10 Sequences & Pattern Recognition Find k in 3, 12, 27, k, 75, 108 → 48 E3.11 Sets (Union, Intersection & Subsets) Elements in {2,4,6} ∪ {1,2,3,4} → 5 E3.12 Word Problems 12 grapes per oz → 1 lb equals 192 grapes E3.13 Volume & Surface Area of Cube / Rectangular Box Volume of box 6.5 cm × 11 cm × 4 cm = 286 cm³ E3.14 Right Triangle Problems (Pythagorean Triples & Area) Area of right triangle with legs 4¼ and 16 → 34 E3.15 Multiplying by 101 101 × 468 = 47268 E3.16 Multiplying by 111 111 × 579 = 64269 E3.17 Multiplying by 143 (Multiples of 7) 143 × 28 = 4004 E3.18 Base Systems Conversions 241 (base 6) = 97 (base 10)

Problems 61–80

Problems 61 – 80 (Integers, Advanced Geometry & Probability)

E4.1 Addition, Subtraction, Multiplication & Division of Integers (-10) + (-3) × (5) = -25 E4.2 Additive Inverses Additive inverse of -2/3 → 2/3 E4.3 Multiplicative Inverses (Reciprocals) Multiplicative inverse of -3 3/8 (-27/8) → -8/27 E4.4 Basic Geometry Facts (Triangle Angles & Regular Polygons) Triangle angles 24° and 46° → third angle is 110° E4.5 Area of a Square from Diagonal & Nested Squares Area of square with diagonal 8 in → 32 in² E4.6 Squaring Numbers Ending in 5 65² = 4225 | 85² = 7225 E4.7 Squaring Two-Digit Numbers Near 50 (41–59) 57² = 3249 | 47² = 2209 E4.8 Squaring General Two-Digit Numbers & Sum of Squares a² + (2a)² 71² = 5041 | 24² + 6² = 612 E4.9 Multiplying by 125, 375, and 625 375 × 80 = 30000 | 125 × 24 = 3000 E4.10 Difference of Two Squares: a² − b² 27² − 23² = 50 × 4 = 200 E4.11 More Powers, Roots & Squared Series (1 + 2 + 3 + ... + 10)² = 3025 E4.12 Consumer Problems: Simple Interest & Sales Tax Simple interest on $500 at 4% for 3 yrs = $60 E4.13 Solving Inequalities If 4x − 11 < 53, then x < 16 E4.14 Probability (Dice, Cards & Coins) Probability of rolling sum of 9 on 2 dice → 1/9 E4.15 Area of a Parallelogram Parallelogram with base 14 and height 3 → 42 E4.16 Area of a Rhombus Rhombus with diagonals 10 and 12 → 60 E4.17 Area of a Trapezoid Trapezoid bases 5, 7 and altitude 12 → 72 E4.18 Distances and Midpoints on the Number Line Distance between -19 and 12 on number line → 31 E4.19 More Percent Type Problems 18% of what is 36% of 54? → 108 E4.20 Two or More Operations with Fractions (a/b + b/a) 3/4 + 4/3 = 2 1/12

±5% Estimation

Starred (*) Estimation Problems (#10, #20, #30, #40, #50, #60, #70, #80)

E*.1 Starred (*) 5% Integral Estimation Problems *(10) 249 × 398 ≈ 250 × 400 = 100000 (94147–104057)

Junior High Number Sense (Grades 7–8)

2026–2027 Number Sense Problem Sequencing Chart — Check the skills you want to drill, or practice a full test mix.

Contest Rule: Problems marked with (*) require integral answers within ±5% of the exact value. All other problems require exact mental answers!

Problems 1–20

Problems 1 – 20 (Core Speed Arithmetic & Statistics)

J1.1 Addition, Subtraction, Multiplication & Division of Whole Numbers 1101 ÷ 3 = 367 | 411 − 109 = 302 J1.2 Addition, Subtraction, Multiplication & Division of Fractions 7/12 − 4/9 = 5/36 J1.3 Addition, Subtraction, Multiplication & Division of Decimals 24 ÷ 0.25 = 96 | 25 × 0.12 = 3 J1.4 Order of Operations (PEMDAS) 24 − 16 ÷ 4 + 2 = 22 J1.5 Use of the Distributive Property: a×b ± a×c 21 × 18 + 9 × 18 = 540 J1.6 Comparison of Fractions & Decimals Which is smaller: 9/11 or 11/13? → 9/11 J1.7 Multiplying by 11 38 × 11 = 418 J1.8 Multiplying by 25, 50, and 75 25 × 16 = 400 | 75 × 23 = 1725 J1.9 Multiplying Two Numbers: Same Tens Digit, Ones Sum to 10 73 × 77 = 5621 | 43 × 47 = 2021 J1.10 Multiplying Two Numbers: Tens Digits Sum to 10, Same Ones Digit 38 × 78 = 2964 J1.11 Multiplying Two Numbers Centered Around a Third Number 19 × 21 = 399 | 12 × 16 = 192 J1.12 Squaring Numbers (1 to 40) 17² = 289 | 27² = 729 J1.13 Squaring Numbers Near 50 (41 to 59) 56² = 3136 | 47² = 2209 J1.14 Squaring Numbers Ending in 5 75² = 5625 | 115² = 13225 J1.15 Roman Numerals / Arabic Numbers Conversion MDXL = 1540 | MMXXV = 2025 J1.16 Finding the Mean (Average) Average of 18, 26, and 28 = 24 J1.17 Finding the Median, Mode & Range Median of 18, 20, 15, 12 → 16.5 J1.18 Sums of Whole Numbers of Multiple Terms of a Finite Series 17 + 20 + 23 + 26 + 29 = 115

Problems 21–40

Problems 21 – 40 (Mixed Numbers, Number Theory & Plane Geometry)

J2.1 Addition, Subtraction, Multiplication & Division of Mixed Numbers 3 1/3 × 3 2/3 = 12 2/9 J2.2 Addition, Subtraction, Multiplication & Division of Integers (-12) − (-24) ÷ (-3) = -20 J2.3 Multiplying Two Numbers Close to and Greater Than 100 102 × 104 = 10608 J2.4 Multiplying Two Numbers Close to and Less Than 100 95 × 98 = 9310 | 99 × 96 = 9504 J2.5 Percent Problems & Fractional Percents 225% of 444 = 999 J2.6 Conversions: English / Metric & Rate Conversions (m/s ↔ km/hr) 20 meters/sec = 72 km/hr J2.7 Conversions: Length, Area, Capacity & Time 25 1/3 days = 608 hours J2.8 Consumer Type Problems (Sales Tax, Simple Interest & Discounts) 12% sales tax on $16 is $1.92 J2.9 Substitution Problems & Perfect Square Trinomials If f(x) = x² − 8x + 16, then f(9) = 25 J2.10 Solving One-Variable Equations If 2x/3 + 13 = 21, then x = 12 J2.11 Square Roots & Cube Roots ∛343 = 7 | √(28 × 63) = 42 J2.12 Greatest Common Divisor (GCD) GCD of 45 and 81 = 9 J2.13 Least Common Multiple (LCM) LCM of 36 and 24 = 72 J2.14 Prime Numbers & Counting Positive Integral Divisors Positive integral divisors of 45 → 6 J2.15 Perimeter & Area of Squares and Rectangles Area of square with diagonal 12 → 72 J2.16 Circumference & Area of Circles Radius of circle with area 16π → 4 J2.17 Ratio & Proportion 9 is to 12 as x is to 28 → x = 21 J2.18 Additive Inverses Additive inverse of -7.5 → 7.5 J2.19 Multiplicative Inverses (Reciprocals) Multiplicative inverse of 6 3/7 → 7/45 J2.20 Multiplication by 101 101 × 468 = 47268 | 101 × 935 = 94435 J2.21 Multiplication by 111 111 × 678 = 75258 | 494 × 111 = 54834

Problems 41–60

Problems 41 – 60 (Sets, Pythagoras, Bases & Solid Geometry I)

J3.1 Sets (Union, Intersection & Power Sets) Elements in {1,2,3,5} ∪ {2,4,6,8} → 7 J3.2 Word Problems Three times a number added to 5 is 38 → 11 J3.3 Pythagorean Theorem (Right Triangles & Rectangle Diagonals) Perimeter of rectangle with diagonal 13 and length 12 → 34 J3.4 Sequences (Missing Terms & nth Term) In 3, n, 11, q, 19, 23: n + q = 22 J3.5 Volume & Surface Area of Rectangular Solid / Cube Edge of cube with surface area 150 → 5 J3.6 Base Systems: Conversions Between Bases 31 (base 4) = 1101 (base 2) J3.7 Base Systems: Addition, Subtraction & Multiplication 4 (base 5) × 41 (base 5) = 314 (base 5) J3.8 Area of Parallelogram, Rhombus & Trapezoid Rhombus area 60 with diagonal 12 → other diagonal = 10 J3.9 Solving Inequalities If 13 − 2x < 27, then x > -7 J3.10 Basic Geometry Facts (Polygon Interior & Exterior Angles) Interior angle of regular hexagon → 120° J3.11 Remainder Problems with Operations (12² + 8 × 6) ÷ 5 remainder → 2

Problems 61–80

Problems 61 – 80 (Advanced Number Theory, Solids, Coordinates & Series)

J4.1 Repeating Decimals to Common Fractions & Operations 0.4666... = 7/15 | 22 × 0.8181... = 18 J4.2 Number Theory: Sum of Positive Integral Divisors Sum of whole number divisors of 12 → 28 J4.3 Number Theory: Relatively Prime Integers (Euler Totient) Positive integers ≤ 24 relatively prime to 24 → 8 J4.4 Number Theory: Triangular & Polygonal Figurate Numbers 10th triangular number → 55 J4.5 Powers of Numbers & Exponent Laws 5³ ÷ 25 = 5 | 4⁴ ÷ 16 = 16 J4.6 Volume & Surface Area of a Circular Cylinder Cylinder radius 10 and length 25 has V = kπ → k = 2500 J4.7 Volume & Surface Area of a Cone and Sphere Surface area of sphere with diameter 8 is kπ → k = 64 J4.8 Arithmetic Series (Consecutive Whole, Odd & Even Sums) 20 + 19 + ... + 0 = 210 | 21 + 19 + ... + 1 = 121 J4.9 Factorials (0! to 8! and Factorial Expressions) 3! × 4! = 144 | (5! + 3!) ÷ 3! = 21 J4.10 Distances, Midpoints & Slope Using Coordinate Pairs Distance between (6,5) and (0,13) → 10 J4.11 Probability & Odds Probability of red queen from 52 cards → 1/26 J4.12 More Percent Type Problems: Advanced 18% of n equals 9% of 20 → n = 10 J4.13 More Remainder Type Problems (High Powers) 10⁵ ÷ 6 remainder → 4 J4.14 Multiplying by 143 and 286 (Multiples of 7) 286 × 49 = 14014 | 286 × 56 = 16016 J4.15 Sum of Squares Shortcut: a² + (2a)² and a² + (3a)² 15² + 30² = 5 × 225 = 1125 J4.16 Multiplication Using Like Bases & Factor Redistribution 48 × 24 = 1152 | 125 × 24 = 3000

±5% Estimation

Starred (*) Estimation Problems (#10, #20, #30, #40, #50, #60, #70, #80)

J*.1 Starred (*) 5% Integral Estimation Problems (142857, Roots, Powers of π) *(30) 28 ÷ 499 × 719 + 59 ≈ 99 (95–104)

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E1.4 • Multiplication Shortcuts ★ 5% Estimation Problem

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52 × 11 =

Write exact answer (use a/b for fractions or w a/b for mixed numbers)

✓ Correct! Answer: 572

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