Vedic mathematics, mental math shortcuts, speed calculation, algebra tricks, and everyday estimation methods to solve problems faster.
A Vedic sutra for multiplying numbers close to a base (10, 100, 1000, etc.). Works by finding how far each number is from the base.
A general Vedic multiplication method that works for any two numbers. Uses vertical and crosswise multiplication patterns.
Quick method to square any number near 50 using the identity (50±d)² = 2500 ± 100d + d².
Instantly square any number ending in 5. Multiply the tens digit by (tens digit + 1), then append 25.
Transform numbers with large digits (6-9) into "vinculum" form using complements from 10, making calculations easier.
Instantly multiply any two-digit number by 11 by splitting and adding the digits.
x% of y = y% of x. When one direction is hard, flip it! This works because (x/100)×y = (y/100)×x.
Multiply by 10 and divide by 2 (or vice versa) for instant results.
Subtract each digit from 9 except the last, which you subtract from 10. Works for any power of 10.
Quickly check if a number is divisible by 2 through 12 without dividing.
Use the identity (a+b)(a-b) + b² = a² to square numbers quickly by going to the nearest easy number.
Memorize common fraction-to-decimal conversions for instant calculation.
Split large multiplications into simpler parts using the distributive property.
A quick way to verify multiplication and addition results by checking digit sums modulo 9.
Multiplying any number by a string of nines is instant: subtract 1 from the number for the left part, then take the complement for the right part.
When numbers are near a convenient working base that is a multiple of a power of 10, use a proportional base to simplify the multiplication.
The Dwandwa (duplex) method squares any number by summing duplex values digit by digit — a general Vedic squaring technique.
Multiply by 10 and subtract the original number — much faster than multiplying by 9 directly.
Since 25 = 100 ÷ 4, multiply by 100 and divide by 4 for instant results.
Multiply by 10, then add half of that — because 15 = 10 + 5 and 5 is half of 10.
When one factor is even, halve it and double the other to reach an easier multiplication.
Add the big place values first (left to right) instead of the traditional right-to-left carrying — easier to hold in your head.
For numbers a little above 100, use the surplus from 100 to multiply almost instantly.
Any percentage can be built from 10%, 5%, and 1% — quick to find and easy to combine.
Use (a+1)² = a² + 2a + 1. Square the round number below, add twice it, add 1.
Multiplying by 4 is doubling twice; by 8 is doubling three times. No memorised tables needed.
Gauss's formula: the sum of the first n whole numbers is n(n+1)/2 — no adding required.
The sum of the first n odd numbers is always a perfect square, exactly n².
a² − b² = (a + b)(a − b). Turns a subtraction of squares into an easy product.
Multiply two binomials by combining First, Outer, Inner, Last terms.
Two numbers equally spaced around a middle value m multiply to m² − d², using the difference of squares.
Find 10% first, then build 15% or 20% from it in your head.
Estimate how many years it takes an investment to double: divide 72 by the annual interest rate.
A fast approximation: to go C→F, double and add 30. For exact values use ×9/5 + 32.
Multiply miles by 1.6 for km, or divide km by 1.6 for miles. Consecutive Fibonacci numbers are handy approximations.
To find the better deal, divide price by quantity to get the price per unit and compare.
Bracket the number between two perfect squares, then refine with a quick average step.
Round the total up to a friendly number, divide by the number of people, then adjust down slightly.
Instantly multiply two numbers that share the same tens digit and whose units digits add up to 10 (e.g. 43 × 47, 62 × 68).
Square any number close to 100 (or 1000) in seconds using how far it sits from the base.
Square any number ending in 9 by leaning on the next round ten, which is much easier to square.
Any two-digit number times 101 is simply the number written twice — no working needed.
Dividing by 5 is awkward; doubling then moving the decimal one place is far faster.
Multiplying by 50 is the same as multiplying by 100 and halving — both easy steps.
Multiply by 12 by using the easy ×10 and then adding twice the number — the same idea extends to 13 and 14.
Add up 2 + 4 + 6 + … up to the Nth even number instantly with a single multiplication.
Find 1² + 2² + 3² + … + N² with one tidy formula instead of adding each square.
Find the cube root of a perfect cube between 1 and 1,000,000 mentally using the last digit and the leading group.
Extend the ×11 trick to three-digit numbers by adding each neighbouring pair of digits.
Multiply any two numbers from 11 to 19 in your head with a quick three-step pattern.
Multiply two numbers close to 1000 using their deficits from the base, just like the near-100 trick.
Reverse the digits of any two-digit number and subtract — the result is always 9 times the difference of the digits.
For any whole number, the product of the number before it, itself and the one after, plus the number, equals its cube.
Since 125 = 1000 ÷ 8, multiplying by 125 is just adding three zeros and dividing by 8.
Since 75 = 3 × 25, multiply by 3 and then by 25 (which is ×100 ÷ 4).
Multiplying by 1.5 is simply adding half of the number to itself — handy for recipes and time-and-a-half pay.
A foolproof way for anyone to get any 9 times table answer (9×1 to 9×10) using just their ten fingers.
Builders use multiples of the 3-4-5 triangle to mark a perfect right angle — no protractor needed.
Multiply any number by a repunit of 9s (99, 999, 9999…). Split the answer into two halves: left half = number − 1, right half = complement of left half to all 9s.
The general Vedic sutra for multiplying any two numbers. Work digit by digit: units × units, then cross-multiply, then tens × tens — carrying as needed.
A Vedic method for dividing by numbers close to a power of 10. Use the complement as a "flag" and perform synthetic-style division mentally.
Verify any multiplication by checking digital roots. The digital root of the product must equal the digital root of (digital root of factor 1 × digital root of factor 2).
Solve simultaneous linear equations by simply adding or subtracting them to eliminate one variable instantly.
The Vedic completion method — solve quadratics by completing the square. Add and subtract (half the coefficient of x)² to create a perfect square.
Factor quadratics instantly: for x² − (a+b)x + ab = 0, find two numbers whose sum equals the middle coefficient and whose product equals the constant term.
Divide large numbers by multi-digit divisors using a flag (the non-leading digits of the divisor). Process one digit at a time, subtracting flag × quotient digit as you go.
When adding a number close to a round number, add the round number first, then subtract the small difference. Much easier mentally.
When two numbers are equidistant from a round "anchor" number, use the difference-of-squares identity: (anchor + d)(anchor − d) = anchor² − d².
Compare two fractions without finding a common denominator. Cross-multiply: if a/b vs c/d, compare a×d with c×b.
Multiply by rounding one factor to the nearest easy number, then adjusting. For example, 48 × 7 = 50 × 7 − 2 × 7.
Subtract from a round number by using the complement trick: subtract from (round number − 1) instead, then add 1. All digit subtractions become from 9.
Square any number by rounding to the nearest multiple of 10: n² = (n+d)(n−d) + d², where d is the distance to that multiple.
The average of any evenly spaced (arithmetic) sequence is simply (first + last) ÷ 2. No need to add all terms.
Convert fractions to percentages by using known benchmarks. For example, 1/8 = 12.5%, so 7/8 = 100% − 12.5% = 87.5%.
Divide by 9 using repeated subtraction of digit sums. Each step: bring down the next digit, the quotient digit equals the running remainder.
Multiply any number by 99 using: n × 99 = n × 100 − n. Just append two zeros and subtract the original number.
Multiply a two-digit number by 111: n × 111 = n × 100 + n × 11. Use the multiply-by-11 trick for the second part.
Since 37 × 3 = 111, multiply by 37 by first dividing the other number by 3, then multiplying by 111. Works perfectly when the number is divisible by 3.
Dividing by 25 = multiplying by 4 then dividing by 100. Much simpler for mental math since multiplying by 4 (double twice) is easy.
Divide by 8 by halving the number three times in a row. Since 8 = 2³, three successive halvings give the exact answer.
To square any number ending in 5: multiply the tens digit by (tens digit + 1), then append 25. Works instantly for any size.
When both numbers are close to 100, use their deficits: cross-subtract to get the hundreds, multiply deficits for the units.
Multiple successive discounts are NOT additive. Multiply the remaining fractions: 20% off then 10% off = 0.8 × 0.9 = 0.72, meaning 28% total discount (not 30%).
Cube a number near 10 using the binomial expansion: (10 + a)³ = 1000 + 300a + 30a² + a³. Compute each piece and add.
a³ + b³ = (a + b)(a² − ab + b²). Useful for factoring expressions or simplifying sums of perfect cubes.
a³ − b³ = (a − b)(a² + ab + b²). The companion formula to sum of cubes, with sign changes in specific positions.
Expand (a + b)ⁿ using Pascal triangle coefficients. Row n gives the coefficients: e.g. row 3 = 1, 3, 3, 1 for (a+b)³.
Sum = n × (first + last) / 2, where n is the number of terms. Find n using: n = (last − first) / common difference + 1.
Sum = a(rⁿ − 1) / (r − 1) where a is the first term, r is the common ratio, and n is the number of terms.
Use log properties to compute logs mentally: log(5) = log(10/2) = 1 − log(2). Memorize log(2) ≈ 0.301 and log(3) ≈ 0.477 to derive everything else.
When raising an exponent to another power, multiply the exponents: (aᵐ)ⁿ = aᵐˣⁿ. Simplifies nested exponents instantly.
If you know the roots r₁ and r₂, the quadratic is x² − (r₁+r₂)x + r₁×r₂ = 0. Sum gives the middle term, product gives the constant.
Apply successive discounts by multiplying the "keep" fractions, not by adding the percentages. The result is always less than the sum of discounts.
To find how many years it takes for money to TRIPLE at a given interest rate, divide 114 by the rate. Like the Rule of 72 but for tripling.
BMI = weight (kg) ÷ height (m)². Square the height first, then divide. A healthy BMI is 18.5–24.9.
Fuel cost = (distance ÷ fuel efficiency) × price per litre. Know your car does and estimate trips in seconds.
If a price increased by p% to become X, the original = X ÷ (1 + p/100). People often mistakenly subtract the percentage instead.
Scale any recipe by multiplying each ingredient by (desired servings ÷ original servings). Works for any ratio.
Distance = Speed × Time. When time is in minutes, convert to hours by dividing by 60: e.g. 40 minutes = 40/60 = 2/3 hour.
Multiply any number by 6 by first multiplying by 3 (easy), then doubling the result. Two simple steps instead of one hard one.
Multiply by 3 by doubling the number and adding the original once more.
Multiply by 7 by multiplying by 8 (triple-double) and subtracting the original: 7n = 8n − n.
Split 13 into 10 + 3: multiply by 10, then add three times the number.
Split 14 into 10 + 4: multiply by 10, then add four times the number.
Since 16 = 2⁴, multiply by 16 by doubling the number four times in a row.
Use 35 = 70 ÷ 2: multiply by 70, then halve.
Use 45 = 90 ÷ 2: multiply by 90, then halve.
Use 55 = 110 ÷ 2: multiply by 110, then halve.
Use 65 = 130 ÷ 2: multiply by 130, then halve.
Use 85 = 170 ÷ 2: multiply by 170, then halve.
Use 95 = 100 − 5: multiply by 100, then subtract five times the number.
Use 105 = 100 + 5: multiply by 100, then add five times the number.
Use 110 = 100 + 10: multiply by 100, then add ten times the number.
Any 3-digit number × 1001 just repeats itself: abc × 1001 = abcabc.
Use 250 = 1000 ÷ 4: multiply by 1000, then divide by 4.
Use 500 = 1000 ÷ 2: multiply by 1000, then halve.
Use 2.5 = 10 ÷ 4: multiply by 10, then divide by 4.
Use 7.5 = 15 ÷ 2 = 30 ÷ 4: multiply by 30, then divide by 4.
Multiplying by 0.5 is the same as halving the number.
Multiplying by 0.25 is the same as dividing by 4 (halve twice).
Build a square from the one before it: n² = (n−1)² + (2n − 1). Great for a running sequence.
For n near 200: n² = 200² + 400×d + d², where d = n − 200 (d can be negative).
Split as (a + b)² = a² + 2ab + b², taking a as the round hundreds part and b the rest.
Divide by 4 by halving the number twice.
Divide by 6 by dividing by 2, then by 3.
Divide by 12 by dividing by 4 (halve twice), then by 3.
Use 15 = 30 ÷ 2: multiply by 2, then divide by 30 — or divide by 3 then by 5.
Use 50 = 100 ÷ 2: multiply by 2, then divide by 100.
Use 125 = 1000 ÷ 8: multiply by 8, then divide by 1000.
Use 250 = 1000 ÷ 4: multiply by 4, then divide by 1000.
Subtract starting from the biggest place value, adjusting as you go — how people naturally think about size.
Instead of taking away, count up from the smaller number to the larger one — easier for making change.
The average of two numbers is their midpoint: add them and halve. Or take the lower and add half the gap.
Round each number to a friendly value, compute, then note the direction of error for a quick sanity check.
Judge whether a fraction is above or below 1/2 by comparing the numerator to half the denominator.
Find 1% by moving the decimal two places, then multiply up to any percentage you need.
Use the distributive law: split one factor into easy parts, multiply each, then add.
Add a big number in friendly chunks instead of all at once, keeping a running total.
Add or subtract a nearby round number, then correct by the small difference.
The complement of a 2-digit number to 100: subtract the tens digit from 9 and the units from 10.
Round the divisor and dividend to compatible numbers that divide evenly for a fast estimate.
Find 10% by moving the decimal one place, then step to 5%, 20%, 15%, etc.
Get 5% instantly: find 10% and halve it.
Estimate 1 ÷ n by finding how many times n fits into a convenient power of 10.
When multiplying, double one factor and halve the other to reach an easier pair — the product stays the same.
Round each price to the nearest dollar as you add, then adjust for the cents at the end.
To find a fraction of a number, divide by the denominator, then multiply by the numerator.
Round both numbers to multiples of ten, multiply, then subtract the cross-correction for a close estimate.
When numbers are close, multiply one by the count and adjust by the small differences from it.
When two numbers are close, just find the gap directly instead of full subtraction.
a² − b² = (a + b)(a − b). Spot two perfect squares with a minus between them.
a² + 2ab + b² = (a + b)². If the first and last terms are squares and the middle is twice their roots’ product, it factors cleanly.
For ax² + bx + c = 0, x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant b²−4ac tells you the number of real roots.
Slope = rise over run = (y₂ − y₁) ÷ (x₂ − x₁).
Distance = √((x₂ − x₁)² + (y₂ − y₁)²) — the Pythagorean theorem on the coordinate grid.
Midpoint = ((x₁ + x₂)÷2, (y₁ + y₂)÷2) — average the x’s and the y’s.
(a + b)² = a² + 2ab + b². The middle term is the one people forget.
aₙ = a₁ + (n − 1)d, where d is the common difference.
aₙ = a₁ × r^(n−1), where r is the common ratio.
n! = n × (n−1)!. Build from a known factorial instead of starting over.
C(n, r) = n! ÷ (r! × (n−r)!). Order does not matter.
P(n, r) = n! ÷ (n−r)!. Order matters here.
If a/b = c/x, cross-multiply to get a×x = b×c, then solve for x.
Percent change = (new − old) ÷ old × 100.
Isolate x by undoing operations in reverse order: move constants first, then divide by the coefficient.
When two fractions are equal, cross-multiply to clear denominators in one step.
For two positive numbers, (a + b)/2 ≥ √(ab). Equality only when a = b.
1³ + 2³ + ... + n³ = (n(n+1)/2)² — the square of the nth triangular number.
The remainder when polynomial P(x) is divided by (x − a) is simply P(a).
Solve one equation for a variable, substitute into the other, then back-substitute.
Multiply kilograms by 2.2 to get pounds (close enough for daily use).
Divide pounds by 2.2 to get kilograms.
Multiply inches by 2.54 to get centimetres.
Divide centimetres by 2.54 to get inches (or multiply by 0.39).
Multiply metres by 3.28 to get feet.
Divide litres by 3.785 to get US gallons (roughly divide by 3.8).
One US cup ≈ 240 ml. Multiply cups by 240.
Interest = Principal × Rate × Time (rate as a decimal, time in years).
Each year multiply by (1 + rate). Over a few years, chain the multipliers.
Find the tax with the 1% or 10% method, then add it to the price.
Change = (new − old) ÷ old × 100. Negative means a decrease.
Selling price = cost × (1 + markup%). A 40% markup means multiply cost by 1.4.
Quick rule: double the hourly rate and add “000” (assumes ~2000 work hours/year).
Divide the annual salary by 12 to get gross monthly pay.
Miles per gallon = miles driven ÷ gallons used.
Area = length × width. Multiply the two sides for square footage/metres.
Divide the floor area by the area of one tile, then add ~10% for cuts and breakage.
Litres ≈ wall area ÷ coverage per litre (about 10 m²/L for one coat).
Average speed over equal distances is the harmonic mean: 2ab ÷ (a + b), not the plain average.
Probability = favourable outcomes ÷ total outcomes.
Odds of a:b convert to probability a ÷ (a + b).
Add the place values (1, 2, 4, 8, ...) wherever there is a 1.
Repeatedly divide by 2 and record remainders bottom-up.
Add values left to right; when a smaller numeral precedes a larger one, subtract it.
Count up to the next hour, then add the remaining whole hours and minutes.
Divide the total by the quantity to get the cost (or amount) per single unit.
Convert common discounts to easy fractions: 25% = 1/4 off, 20% = 1/5 off, 50% = half off.
For a 20% tip: find 10% (move the decimal) and double it.
Pace = total minutes ÷ distance in km, giving minutes per kilometre.
Rough rule: calories ≈ your weight in kg × kilometres walked.