Add by Compensation
Shift one addend to a friendly number, then compensate by shifting the other amount back.
Example: 398 + 247 ā 400 + 247 ā 2 = 645.
Read guide āš§ Smart Math ā Tips, Tricks & Better Ways to Think
Learn faster methods without losing the mathematics underneath them. Smart Math combines mental-math shortcuts, number sense, worked examples, problem-solving habits, study strategies and deeper math ideas in one searchable learning library.
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Understand firstMethods explain why they work, not just what buttons to press.
Represent itUse numbers, diagrams, tables, equations and units to make relationships visible.
Check itEstimate, reverse-check, test units and compare representations before trusting an answer.
Practice smartRetrieval, spacing, mixed practice and error analysis build durable skill.
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Open todayās guide āEstimate before you calculate
A rough answer gives your exact work a target and catches many errors.
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Subtract by Compensation
Add the same amount to both numbers so the subtrahend becomes friendly.
Example: 503 ā 198 ā 505 ā 200 = 305.
Read guide āDouble One Factor, Halve the Other
For multiplication, doubling one factor and halving the other keeps the product unchanged.
Example: 16 Ć 35 ā 8 Ć 70 ā 4 Ć 140 = 560.
Read guide āMultiply by 5 Fast
Multiplying by 5 is the same as multiplying by 10 and dividing by 2.
Example: 68 Ć 5 = 680 Ć· 2 = 340.
Read guide āMultiply by 25 Fast
Because 25 is one quarter of 100, multiply by 100 and divide by 4.
Example: 48 Ć 25 = 4800 Ć· 4 = 1200.
Read guide āMultiply by 99
Use 99 = 100 ā 1 so the multiplication becomes a place-value shift followed by subtraction.
Example: 73 Ć 99 = 7300 ā 73 = 7227.
Read guide āSquare Numbers Ending in 5
For a two-digit number ending in 5, multiply the leading digit by the next integer and append 25.
Example: 65Ā²: 6Ć7 = 42, then append 25 ā 4225.
Read guide āMultiply Near 10, 100, or 1000
Numbers close to a power of ten can be multiplied using their small offsets from the base.
Example: 97Ć96 = (100ā3)(100ā4) = 10000ā700+12 = 9312.
Read guide āEstimate Before Exact Work
A quick estimate gives you a target range and catches many calculator or arithmetic errors.
Example: 49.8 Ć 19.7 ā 50 Ć 20 = 1000; exact work should land near 1000.
Read guide āDecompose by Place Value
Break numbers into hundreds, tens, ones, tenths, and so on to expose structure.
Example: 347 + 286 = (300+200)+(40+80)+(7+6)=500+120+13=633.
Read guide āUse Benchmark Numbers
Anchor thinking to familiar values such as 0, 1/2, 1, 10, 100, and 1000.
Example: 0.49 is just under 0.5, so 0.49Ć80 should be just under 40.
Read guide āUse Divisibility Tests
Simple digit tests can reveal factors without long division.
Example: 378 has digit sum 18, so it is divisible by 9; 378Ć·9=42.
Read guide āPrime Factorization as a Toolkit
Writing a number as a product of primes makes gcd, lcm, simplifying fractions, and divisibility easier.
Example: 360 = 2Ā³Ć3Ā²Ć5.
Read guide āConnect GCD and LCM
Prime factors or the Euclidean algorithm can reveal the greatest common divisor and least common multiple.
Example: 18=2Ć3Ā² and 24=2Ā³Ć3, so GCD=6 and LCM=72.
Read guide āUse Odd/Even Structure
Parity can predict whether a result is odd or even before you compute it.
Example: Odd + odd = even; 37 + 59 must be even, and 96 is.
Read guide āThink in Orders of Magnitude
Scientific notation helps compare very large or very small quantities by separating scale from detail.
Example: 3.2Ć10ā¶ is about ten times 3.2Ć10āµ.
Read guide āCompare Fractions with Benchmarks
Compare a fraction with 1/2, 1, or another familiar fraction before finding decimals.
Example: 7/15 is less than 1/2 because 7 < 7.5.
Read guide āAdd Fractions with Meaning
A common denominator creates equal-sized parts before numerators can be combined.
Example: 2/3 + 1/4 = 8/12 + 3/12 = 11/12.
Read guide āUnderstand Fraction Division
Dividing by a fraction asks how many of that fractional-sized group fit into the original quantity.
Example: 3 Ć· 3/4 = 3Ć4/3 = 4.
Read guide āFind a Percent by Decomposing
Break awkward percentages into easy pieces such as 10%, 5%, 1%, 25%, and 50%.
Example: 17% of 240 = 10% (24) + 5% (12) + 2% (4.8) = 40.8.
Read guide āWork Backward from a Percent
When a final amount represents a known percent of the original, divide by the decimal multiplier.
Example: After a 20% discount, $64 is 80% of original: 64Ć·0.8=$80.
Read guide āPercent Change Uses the Original
Percent change compares the difference to the starting value, not the ending value.
Example: Price rises from 50 to 65: change=15; 15/50=30%.
Read guide āSuccessive Percent Changes Multiply
Repeated percentage changes apply to the updated amount, so they do not simply add.
Example: +20% then ā20% gives 1.2Ć0.8=0.96, a net 4% decrease.
Read guide āMove Between Decimals and Fractions
Terminating decimals can be written over powers of ten and reduced.
Example: 0.375 = 375/1000 = 3/8.
Read guide āTreat an Equation Like a Balance
Whatever operation you perform on one side of an equation must preserve equality on the other side.
Example: 3x+5=20 ā 3x=15 ā x=5.
Read guide āUse the Distributive Property Both Ways
Distribution expands products; factoring reverses the same structure.
Example: 7Ć48 = 7(50ā2)=350ā14=336.
Read guide āSpot a Difference of Squares
The identity aĀ²ābĀ²=(aāb)(a+b) turns some hard products and factorizations into easy ones.
Example: 101Ā²ā99Ā²=(101ā99)(101+99)=2Ć200=400.
Read guide āSolve Proportions with Structure
A proportion states that two ratios are equal; scaling or cross-products can solve the missing value.
Example: 3/5 = x/20; denominator Ć4, so numerator Ć4 ā x=12.
Read guide āThink of Slope as Rate of Change
Slope is vertical change divided by horizontal change and carries units.
Example: From (2,3) to (6,11), slope=(11ā3)/(6ā2)=8/4=2.
Read guide āUse Elimination for Systems
Add or subtract equations so one variable cancels.
Example: x+y=10 and xāy=4 ā 2x=14 ā x=7, y=3.
Read guide āFactor Simple Quadratic Trinomials
For xĀ²+bx+c, look for two numbers whose sum is b and product is c.
Example: xĀ²+7x+12 ā 3 and 4 ā (x+3)(x+4).
Read guide āComplete the Square
Completing the square rewrites a quadratic expression in vertex form and clarifies its geometry.
Example: xĀ²+6x+2 = (x+3)Ā²ā7.
Read guide āFind Area by Decomposing Shapes
Break an irregular figure into rectangles, triangles, circles, or other familiar pieces.
Example: An L-shape can be a 10Ć8 rectangle minus a 4Ć3 cutout: 80ā12=68 square units.
Read guide āUse the Pythagorean Theorem
For a right triangle, the square of the hypotenuse equals the sum of the squares of the legs.
Example: Legs 6 and 8: c=ā(36+64)=10.
Read guide āUse Similar Triangles for Indirect Measurement
Similar triangles have equal corresponding angles and proportional corresponding sides.
Example: A 2 m stick casts a 3 m shadow; a tree casts 12 m. Height=2Ć12/3=8 m.
Read guide āEstimate Circles Before Using Ļ
Use Ļā3.14 or Ļā22/7 when suitable, but first judge whether radius or diameter is being used.
Example: r=5: circumferenceā31.4, areaā78.5.
Read guide āUse Angle Sums as Constraints
Known angle totals can turn geometry into simple subtraction.
Example: Triangle angles 48Ā° and 67Ā° leave 180ā115=65Ā°.
Read guide āDistance on the Coordinate Plane
The distance formula is the Pythagorean theorem applied to horizontal and vertical changes.
Example: (1,2) to (4,6): ā(3Ā²+4Ā²)=5.
Read guide āFind a Midpoint by Averaging Coordinates
A midpoint lies halfway in both horizontal and vertical directions.
Example: Between (2,8) and (10,4): midpoint=(6,6).
Read guide āScale Factors Affect Area and Volume Differently
If lengths scale by k, areas scale by kĀ² and volumes by kĀ³.
Example: Doubling every dimension makes area 4Ć and volume 8Ć.
Read guide āUse UnderstandāPlanāSolveāCheck
A repeatable problem-solving cycle reduces rushed mistakes and makes reasoning visible.
Example: For a trip-cost problem, list miles, mpg, and fuel price before choosing formulas.
Read guide āDraw a Diagram
A labeled sketch can reveal relationships that are hidden in words.
Example: A ladder problem becomes a right triangle once wall, ground, and ladder are drawn.
Read guide āMake a Table
Tables organize repeated cases, patterns, and input-output relationships.
Example: For y=3x+2, values x=0,1,2 give y=2,5,8 and reveal a constant increase of 3.
Read guide āWork Backward
When the final result is known and operations are reversible, undo them in reverse order.
Example: A number is doubled then 7 added to get 25: 25ā7=18, then Ć·2=9.
Read guide āLet Units Guide the Formula
Units can show which operations make sense and expose mismatched formulas.
Example: miles Ć· miles/hour = hours, so time = distance Ć· speed.
Read guide āTest Simple or Extreme Cases
A formula or conjecture is easier to understand by checking 0, 1, symmetric cases, or large/small limits.
Example: A shipping formula should usually give zero variable cost when quantity is zero.
Read guide āLook for Invariants and Patterns
Repeated structure can suggest a general rule, but the rule still needs justification.
Example: Odd numbers 1+3+5+7=16 suggests the sum of first n odd numbers is nĀ² for n=4.
Read guide āSwitch Representations
A problem may become easier when moved among words, equations, tables, graphs, diagrams, and number lines.
Example: A constant rate can be seen as a ratio, a table, a straight graph, or y=mx.
Read guide āChoose Mean or Median Thoughtfully
The mean uses every value; the median is more resistant to extreme values.
Example: Data 10,11,12,13,100 has median 12 but mean 29.2.
Read guide āUse Weighted Averages
When values contribute unequally, multiply each value by its weight before averaging.
Example: Tests 80 (40%) and 95 (60%) give 80Ć0.4+95Ć0.6=89.
Read guide āUse the Complement Rule
Sometimes it is easier to find the probability that an event does not happen.
Example: If P(rain)=0.3, P(no rain)=0.7.
Read guide āAdd Probabilities Without Double Counting
For events A and B, P(A or B)=P(A)+P(B)āP(A and B).
Example: If P(A)=0.5, P(B)=0.4, overlap=0.2, then union=0.7.
Read guide āMultiply Independent Probabilities
For independent events, the probability that both occur is the product of their probabilities.
Example: Two fair coin heads: 1/2Ć1/2=1/4.
Read guide āThink in Expected Value
Expected value is a probability-weighted average of possible outcomes, useful for repeated decisions.
Example: 50% chance of $10 and 50% chance of $0 gives EV=$5.
Read guide āPermutation or Combination?
Use permutations when order matters and combinations when order does not.
Example: Choosing president and vice president from 5 people: 5P2=20; choosing any 2-person committee: 5C2=10.
Read guide āSeparate Correlation from Causation
A statistical association does not by itself prove that one variable causes the other.
Example: Ice cream sales and sunburns may rise together because hot sunny weather affects both.
Read guide āCompare Unit Prices
Divide total price by a common unit so differently sized packages can be compared fairly.
Example: $6 for 24 oz = $0.25/oz; $5.50 for 20 oz = $0.275/oz.
Read guide āUnderstand Stacked Discounts
Successive discounts multiply; a second discount applies to the already-discounted price.
Example: 20% off then 10% off: 0.8Ć0.9=0.72, so total discount is 28%.
Read guide āEstimate Tips Quickly
Use 10% as an anchor and combine it to get common tip rates.
Example: $46 bill: 20%ā$9.20; 15%ā$6.90.
Read guide āUse the DistanceāRateāTime Triangle
Distance, rate, and time are linked by d=rt.
Example: 150 miles at 50 mph takes 150/50=3 hours.
Read guide āScale Recipes with Ratios
Multiply every ingredient by the same scale factor to preserve proportions.
Example: Recipe serves 4; need 10: scale factor 10/4=2.5. Two cups becomes five cups.
Read guide āSeparate Simple from Compound Interest
Simple interest grows on principal only; compound interest grows on principal plus accumulated interest.
Example: $1000 at 5% simple interest for 3 years: I=1000Ć0.05Ć3=$150.
Read guide āUse the Rule of 72 as an Estimate
For moderate percentage growth rates, 72 divided by the annual rate gives a rough doubling time.
Example: At 8% growth, doubling time ā72/8=9 years.
Read guide āTurn a Budget into Percentages
Percent-of-income views make differently sized budgets easier to compare.
Example: $600 housing from $2400 take-home pay = 25%.
Read guide āPractice Retrieval, Not Just Rereading
Trying to recall a method from memory strengthens access better than only looking at worked solutions.
Example: After reviewing fraction addition, solve 3/4+2/5 without notes.
Read guide āSpace Practice Across Days
Several shorter sessions spread over time usually build more durable memory than one long cram session.
Example: Practice ratios Monday, revisit Wednesday, then again on the weekend.
Read guide āInterleave Problem Types
Mixing related problem types forces you to choose the method rather than merely repeat it.
Example: Mix area, perimeter, and volume questions instead of doing 20 of one type in a row.
Read guide āKeep an Error Log
Classify mistakes so the same pattern becomes visible and fixable.
Example: Mistake: used new value in percent-change denominator. Fix: divide change by original.
Read guide āExplain Each Step Aloud
Explaining why a step is valid exposes gaps that can hide during silent symbol pushing.
Example: For 2(x+3)=14, say: divide both sides by 2, then subtract 3.
Read guide āFade Worked Examples
Move gradually from fully worked examples to partially completed ones to independent problems.
Example: First see a full quadratic factorization, then complete the last two steps of the next one, then solve one alone.
Read guide āPair Visual and Symbolic Representations
Connecting pictures, number lines, graphs, and symbols builds meaning behind procedures.
Example: 3/4 can be shown as three of four equal regions and as 0.75 on a number line.
Read guide āUse Productive Struggle
Give yourself time to reason before reaching for a solution, while still using hints when truly stuck.
Example: On a hard ratio problem, draw a table before looking up the formula.
Read guide āUse Exponent Laws Consistently
Exponent rules compress repeated multiplication and make algebraic simplification predictable.
Example: xĀ³Ā·xāµ=xāø; (xĀ³)Ā²=xā¶; xā»Ā²=1/xĀ² for xā 0.
Read guide āTranslate Logarithms into Exponents
A logarithm answers: what exponent on the base produces this number?
Example: logā32=5 because 2āµ=32.
Read guide āMultiply in Scientific Notation
Multiply coefficients and add powers of ten, then normalize.
Example: (3Ć10ā“)(2Ć10ā»Ā³)=6Ć10Ā¹=60.
Read guide āRecognize Arithmetic Sequences
An arithmetic sequence changes by a constant difference.
Example: 5,8,11,... has d=3, so a_10=5+9Ć3=32.
Read guide āRecognize Geometric Sequences
A geometric sequence changes by a constant ratio.
Example: 3,6,12,... has r=2, so a_6=3Ć2āµ=96.
Read guide āRead a Quadratic in Vertex Form
In y=a(xāh)Ā²+k, the vertex is (h,k) and a controls opening and vertical scale.
Example: y=2(xā3)Ā²ā5 has vertex (3,ā5).
Read guide āThink of Function Composition as a Pipeline
In f(g(x)), the output of g becomes the input of f.
Example: f(x)=x+1, g(x)=2x: f(g(3))=f(6)=7.
Read guide āUse Dimensional Analysis for Conversions
Multiply by conversion factors equal to 1 so unwanted units cancel.
Example: 60 mi/h Ć 5280 ft/mi Ć 1 h/3600 s = 88 ft/s.
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