✕ Multiplication
2-Digit × 2-Digit (FOIL)
Core FOIL technique — the foundation of every NS test.
Multiply by 11
Digit-sandwich trick — the most common NS shortcut.
Multiply by 101
Gap-digit trick — paste then adjust middle digits.
Multiply by 25
÷4 then ×100 — instant results for even numbers.
Multiply by 75
¾ × 100 trick — ÷4, ×3, ×100.
Double & Half Trick
Double one, halve the other — great for ×15, ×35, etc.
Multiply Near 100
Use distance from 100 to split the problem instantly.
Square Numbers Ending in 5
Last two digits are always 25. Rest = n(n+1).
Squares 41 – 59
Use distance from 50 to square any number in this range.
Squaring 2-Digit Numbers
(a+b)² expansion and difference-of-squares methods.
Equidistant Multiplication
(m−d)(m+d) = m² − d² — faster than full FOIL.
Multiplying Reverse-Digit Numbers
ab × ba formula using digits and their squares.
Difference of Squares
a² − b² = (a+b)(a−b) — skip computing both squares.
Two Numbers Ending in 5
Special formula when both numbers end in 5.
Mixed Number Multiplication
Convert to improper fractions, multiply, simplify back.
Factoring Numerical Expressions
Factor out common terms to simplify messy problems.
÷ Division & Remainders
Remainder Dividing by 4
Only the last two digits matter for divisibility by 4.
Remainder Dividing by 9
Digit-sum rule — add all digits, find the remainder.
Remainder Dividing by 11
Alternating digit-sum rule (right to left).
Dividing by 9 (Running Sum)
Running cumulative digit sum produces the quotient digits.
Percentage of a Number
Anchor on 10%, then scale — fast mental percentages.
Discriminant & Roots
Sum, product, discriminant, and smallest root of quadratics.
Vertex of a Parabola
Find the x and y coordinates of the vertex instantly.
＋ Addition & Subtraction
Multi-Number Addition
Pair numbers to build round sums — saves carries.
Order of Operations (PEMDAS)
Multiply first, then add/subtract left to right.
Subtracting Reverse Digits
Middle digit cancels — result is always a multiple of 99.
a/b + b/a Fraction Trick
= (a²+b²)/(ab) — one formula, instant answer.
Mixed Number Subtraction
Convert, find LCD, subtract, simplify — with fraction answer.
Fibonacci Numbers
Find terms, identify positions, and sum first-n using the formula.
Infinite Geometric Series
a₁/(1−r) — converging series with fraction answers.
Sum of Squares
Factor out the smaller square to simplify a²+b² instantly.
Consecutive Integers
Sum of first n integers, odds, evens, and word problems.