Guide — strategy & formula sheet
How Number Sense scoring works
80 questions, 10 minutes, no calculator. +5 for a correct answer, −4 for a wrong answer or a missed question, 0 for questions left blank after the last answered question. A blank before the last answered question counts as missed (−4) — so never leave a mid-test blank.
Estimate questions accept any answer within ±5% of the exact value, as on the real test. Equivalent answers count: 1/2, 0.5, and 2/4 are all the same answer.
Test-day strategy
- Pace checkpoints: reach Q20 by 2:30, Q40 by 5:00, Q60 by 7:30, and Q80 by 10:00. If you are behind at a checkpoint, move faster and come back only if time remains.
- Never leave a mid-test blank. A blank before the last answered question counts as missed (−4), so always guess — a guess can only help.
- Blanks after the last answered question cost nothing (0). If time is nearly up, skip the hard ones and answer the cheap ones.
- Sweep the cheap late items first: units digits, squares, GCD/LCM, and quick fractions are worth the same 5 points as the hard ones.
- On estimates, check the digit count before you trust an answer. 285,714 × 13 is about 3.7 million, not 39 million.
- Write the answer on the line even when you did the work in the margin. A correct answer in the margin with a blank line scores zero.
- One answer only per line. If you change an answer, cross the old one out cleanly and write the new one — overwritten answers are unreadable and marked wrong.
Formula sheet — all 40 types
Early paper
| Type | What it is | Rule / trick |
|---|---|---|
| Digits | Place values, digit products and digit sums. | Thousands, hundreds, tens, ones — read the place, not the digit. |
| Order of operations | Mixed + − × ÷ in one line. | × and ÷ before + and −. |
| Quick add & subtract | Two- and three-term addition and subtraction. | Add the hundreds first, then the rest. |
| Quick multiply & divide | Multiplication, division, and triple products. | Break numbers apart: 17 × 8 = 10 × 8 + 7 × 8. |
| Remainders | Division with a remainder. | Dividend = divisor × quotient + remainder. |
| Roman numerals | Convert Roman numerals to Arabic. | I=1 V=5 X=10 L=50 C=100 D=500 M=1000. A smaller value before a larger one subtracts. |
| Squares & cubes | Squares (early) and cubes (mid) of small bases. | n² and n³ — memorize squares to 25² and cubes to 10³. |
Middle paper
| Type | What it is | Rule / trick |
|---|---|---|
| Arithmetic sequences | nth term, term count, and sums. | aₙ = a₁ + (n−1)d; sum = n × (first + last) ÷ 2. |
| Base conversions | Decimal, binary, and octal conversions. | Place values are powers of the base; binary 11111₂ = 16 + 8 + 4 + 2 + 1 = 31. |
| Counting multiples | How many multiples of k lie in a range. | Count = ⌊last ÷ k⌋ − ⌊(first − 1) ÷ k⌋. |
| Decimals | Decimal multiply, divide, and powers of ten. | × 10⁻² moves the point two places left; ÷ 0.1 is the same as × 10. |
| Decimals to fractions | Terminating and repeating decimals as fractions. | 0.0̄8 = 8/90 = 4/45; a two-digit repeat ab̄ = ab/99. |
| Estimation (±5%) | Estimate a product, quotient, or sum within 5%. | Round to friendly numbers first; check the digit count before you trust an answer. |
| Exact roots | Square roots, cube roots, and root products. | √(a²) = a; ³√(a³) = a; √a × √b = √(ab). |
| Fractions: add & subtract | Add and subtract fractions, incl. three unit fractions. | Common denominator first, then add the tops. |
| Fractions: compare & reduce | Which fraction is greater; reduce to lowest terms. | Cross-multiply to compare; divide top and bottom by the GCD to reduce. |
| Greatest common divisor | GCD of a pair. | Largest number that divides both — factor both numbers first. |
| Least common multiple | LCM of a pair. | LCM = a × b ÷ GCD. |
| Mean, median & mode | Mean, median, or mode of a short list. | Mean = sum ÷ count; median = the middle value when sorted. |
| Mixed-number multiplication | Mixed numbers times fractions. | Convert to improper fractions first, then cancel before multiplying. |
| Multiplication tricks | Difference of squares and near-round-number products. | (a − b)(a + b) = a² − b²; 998 × 992 = (1000 − 2)(1000 − 8). |
| Percent as a fraction | Fractional percents, e.g. 16⅔% of a number. | 16⅔% = 1/6; 12½% = 1/8; 33⅓% = 1/3; 9 1/11% = 1/11. |
| Percent of a number | x% of N. | x% of N = (x ÷ 100) × N. |
| Rectangles & squares | Area and perimeter, incl. squares from perimeter. | Area = length × width; perimeter = 2(length + width); square side = perimeter ÷ 4. |
| Unit conversions | Yards, hours, ounces, days, feet. | 1 yard = 3 ft; 1 hour = 60 min; 1 day = 24 hr = 1440 min. |
| Units digit | Units digit of a power or a product. | Only the units digits matter; powers cycle every 4. |
Late paper
| Type | What it is | Rule / trick |
|---|---|---|
| Advanced exponents | Fractional exponents, exponent equations, exponent juggling. | a^(m/n) = ⁿ√(a^m); to solve a^x = b, write both sides with the same base. |
| Circles | Area and radius, incl. π-cancelling cases. | Area = πr²; circumference = 2πr; if r = k/√π, the π cancels and the area is k². |
| Combinations | Choose k from n. | C(n,2) = n(n−1) ÷ 2. |
| Counting outcomes | Coin flips and subsets. | n coins → 2ⁿ outcomes; proper subsets of an n-letter set = 2ⁿ − 1. |
| Dice probability | Probability of a sum, product, or event on two dice. | 36 outcomes in all; P = favourable ÷ 36. Product is odd only when both dice are odd. |
| Divisors | How many divisors; sum of distinct prime divisors. | From the prime factorization, add 1 to each exponent and multiply. Sum each distinct prime once. |
| Geometric sequences | nth term and infinite sums. | aₙ = a₁rⁿ⁻¹; infinite sum = a ÷ (1 − r) when |r| < 1. |
| Polygon angles | Interior sums, exterior angles, side from perimeter. | Interior sum = (n − 2) × 180°; exterior angle = 360° ÷ n; side = perimeter ÷ n; parallelogram consecutive angles add to 180°. |
| Primes | Next prime, previous prime, sums of primes. | 2 is the only even prime. Test divisibility by primes up to √n. |
| Root estimation | Estimate square and cube roots, incl. roots of products. | √n sits near the square root of the closest perfect square; ³√n near the closest perfect cube. |
| Solids | Prism and cube surface area; sphere radius and diameter. | Prism SA = 2(lw + lh + wh); cube SA = 6s², V = s³; sphere V = 4/3·π·r³, SA = 4·π·r². |
| Solve, then evaluate | Solve for x, then evaluate an expression in x. | Solve first, then plug the value in — don't try to do it in one step. |
| Trapezoids & rhombuses | Trapezoid area; rhombus perimeter from diagonals. | Trapezoid area = (base₁ + base₂) ÷ 2 × height; rhombus side = ½√(d₁² + d₂²), perimeter = 4 × side. |
| Triangles | Area, perimeter from leg + area, isosceles-right triangles. | Area = ½ × base × height; a² + b² = c²; isosceles-right leg = √(2 × Area). |
Practice for Math League-style Number Sense. This trainer is not affiliated with or endorsed by Math League. Every question here is original — written fresh from the question type, never copied from a competition test.