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Mastering Mental Math: Best Tricks and Techniques for Quick Calculations and Advanced Problem Solving.

Math tricks can be a fun way to speed up calculations and impress your friends. Here are some of the best math tricks:

Multiplying by 11:
When multiplying a two-digit number by 11, separate the two digits and add them together. Then place the sum between the original two digits.
Example: \(35 \cdot 11=3 (3+5) 5=385 \)

Squaring numbers ending in 5:
Take the first digit, multiply it by the next higher digit, and then append 25 to the result.
Example: \( 75^2=(7 \cdot 8)25=5625 \)

Multiplying by 5:
When multiplying a number by 5, you can multiply by 10 and then divide by 2.
Example: \( 48 \cdot 5=\frac{48 \cdot 10}{2} = 240 \)

Multiplying by 9:
To multiply a one-digit number by 9, subtract 1 from the number, and then subtract the result from 9 to get the second digit.
Example: \( 7\cdot 9=63 (7-1=6,9-6=3) \)

Quick percentage calculation:
To find the percentage of a number, you can move the decimal point two places to the left and multiply by the percentage.
Example: \( 45 % of 200= 0.45 \cdot 200=90 \)

Adding large numbers:
When adding large numbers, it's often easier to round them to the nearest 10, 100, or 1000 and then subtract the difference.
Example: \( \small 568+379=(570–2)+(380–1)=950–3=947 \)

Divisibility rules:
A number is divisible by 2 if its last digit is even.
A number is divisible by 3 if the sum of its digits is divisible by 3.
A number is divisible by 4 if the last two digits are divisible by 4.
A number is divisible by 5 if it ends in 0 or 5.
A number is divisible by 6 if it's divisible by both 2 and 3.
A number is divisible by 9 if the sum of its digits is divisible by 9.
A number is divisible by 10 if it ends in 0.

Doubling and halving:
When multiplying two numbers, you can double one number and halve the other to make the calculation easier. Repeat the process if necessary.
Example: \( 14 \cdot 24=(7 \cdot 48)=(3.5 \cdot 96)=336 \)

Multiplying by 15:
To multiply a number by 15, you can multiply the number by 10, then add half of the product to the result.
Example: \( 15 \cdot 8=(8 \cdot 10)+(8 \cdot 5)=80+40=120 \)

Subtracting from 1,000:
To subtract a three-digit number from 1,000, subtract each digit from 9, except for the last digit, which you subtract from 10.
Example: \( 1,000 –634=(9-6)(9-3)(10-4)=366 \)

Squaring numbers close to 100:
If a number is close to 100, you can square it by finding the difference from 100, adding/subtracting that difference, and then multiplying the difference by itself.
Example: \( 97^2=(100–3)^2=(97–3) 3^2=94 09=9409 \)

Finding the average of two numbers:
To find the average of two numbers, add them together and divide by 2, or you can find the difference between the numbers, divide it by 2, and then add the result to the smaller number.
Example: Average of 45 and 65 equal \( \frac{45+65}{2} = \frac{110}{2} = 55 \)

Fast exponentiation using squaring:
To raise a number to a power, square the number and then multiply by itself the required number of times. This is particularly useful for calculating exponents with even powers.
Example: \( 3^4= (3^2 )^2= 9^2= 81 \)

Converting between Fahrenheit and Celsius:
To convert from Fahrenheit to Celsius, subtract 32 from the Fahrenheit temperature, then multiply the result by \( \frac{5}{9} \). To convert from Celsius to Fahrenheit, multiply the Celsius temperature by \( \frac{9}{5} \) and add 32.
Example: 68 F to Celsius \( (68-32) \cdot \frac{5}{9} = 36 \cdot \frac{5}{9} \approx 20^\circ \)

Finding the sum of integers from 1 to \(n\):
To find the sum of all integers from 1 to n, use the formula: \( \frac{n(n + 1)}{2} \)
Example, Sum of integers from 1 to 100:
\(1 \to 100=\frac{100 \cdot (100 + 1)}{2}=5050 \)

Finding the area of an equilateral triangle:
Given the side length of an equilateral triangle, find the area using the formula:
\( \frac{\text{side length}^2 \cdot \sqrt{3}}{4} \)
Example, Area of an equilateral triangle with a side length of 6:
\( \frac{6^2 \cdot \sqrt{3}}{4} = 9 \sqrt{3} \)

Estimating square roots:
To estimate the square root of a number close to a perfect square, find the closest perfect squares and use them as a reference.
Example: \( \sqrt{82} \) is between \( \sqrt{81} \) \((9^2 )\) and \( \sqrt{100} \) \((10^2 )\), so the square root is slightly larger than 9.

Fast multiplication by 12:
To multiply a number by 12, first multiply it by 10 and then add twice the original number.
Example: \( 12 \cdot 7=(10 \cdot 7)+(2 \cdot 7)=70+14=84 \)

Fast division by 5:
To divide a number by 5, you can multiply the number by 2 and then divide by 10.
Example: \( 48 \div 5 = \frac{48 \cdot 2}{10} = \frac{96}{10} = 9.6 \)

Fast multiplication with a number near 100:
To multiply a number by a number close to 100, find the difference from 100, add/subtract the difference, and then multiply the differences.
Example: \( \small 97 \cdot 103=(100–3)(100+3)=10000–9=9991 \)

Multiplying numbers between 10 and 20:
To multiply two numbers between 10 and 20, first find the sum of the units digits, then add the sum to 20 and multiply the result by 10. Finally, add the product of the units digits.
Example: \( \small 17 \cdot 14=((7+4)+20) \cdot 10+(7 \cdot 4)=310+28=338 \)

Adding/subtracting mixed numbers:
To add or subtract mixed numbers, add or subtract the whole numbers and fractions separately, and then simplify the result.
Example: \( \small 4 \frac{1}{4} - 2 \frac{3}{4}=(4-2)+(\frac{1}{4} - \frac{3}{4})=2 - \frac{2}{4}= 1 \frac{1}{2} \)

Finding the mode in a data set:
To find the mode (the most frequently occurring value) in a data set, count the number of times each value appears, and then identify the value with the highest count.

Finding the median in a data set:
To find the median (the middle value) in a data set, first arrange the values in ascending order, and then locate the middle value. If there is an even number of values, find the average of the two middle values.

Fast exponentiation using the doubling and halving method:
To quickly raise a number to a power, you can repeatedly double the number and halve the exponent until the exponent is 1.
Example: \( \small 2^6 = 2 \cdot 2^5 = 4 \cdot 2^4 = 8 \cdot 2^3 = 16 \cdot 2^2 = 32 \cdot 2^1 = 64 \)

Pythagorean triples:
A Pythagorean triple consists of three positive integers \(a\), \(b\), and \(c\), such that \(a^2 + b^2 = c^2 \). One way to generate Pythagorean triples is by using Euclid's formula:
\( a = m^2 - n^2\), \(b=2mn\), and \(c=m^2 + n^2 \), where \(m\) and \(n\) are positive integers with \(m > n \).

Difference of squares:
The difference of two squares can be factored as \( (a^2 - b^2 )= (a+b)(a–b) \). This can be useful for simplifying expressions and solving equations.

Sum of cubes:
The sum of two cubes can be factored as \((a^3 +b^3 )=(a+b)(a^2 -ab+b^2 ) \)

Difference of cubes:
The difference of two cubes can be factored as \( (a^3 - b^3) =(a–b)(a^2 + ab + b^2) \)

Quadratic formula:
To solve a quadratic equation of the form \( ax^2+bx+c=0 \), use the quadratic formula:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Binomial theorem:
The binomial theorem states that for any non-negative integer \(n\) and any real numbers \(a\) and \(b\),
\((a + b)^n = \sum_{k=0}^{n} \binom{n}{k} b^k a^{n-k} \), where \(\binom{n}{k}\) are the binomial coefficients, also known as "n choose k" or the number of ways to choose k elements from a set of \(n\) elements.

Logarithm properties:

\( log_a (x \cdot y) =log_a (x) +log_a(y) \)

\( log_a ( \frac{x}{y} ) = log_a (x) -log_a (y) \)

\( log_a (x^n ) = n \cdot log_a (x) \)

Euler's identity:
Euler's identity is an elegant and profound equation that relates the most important constants in mathematics: \(e^ {i \pi}+1=0 \), where \(e\) is the base of the natural logarithm, \(i\) is the imaginary unit, and \( \pi \) is the ratio of a circle's circumference to its diameter.

Calculating derivatives:
The derivative of a function measures the rate of change of the function with respect to its input variable. Some basic rules for calculating derivatives include:

\( (cf(x))'= c \cdot f' (x) \)

\( (f(x) \pm g(x))'=f' (x) \pm g' (x) \)

\( (f(x) \cdot g(x))' =f' (x) \cdot g(x)+f(x) \cdot g' (x) \)

\( ( \frac{f(x)}{g(x)} )'=\frac{(f' (x) \cdot g(x) - f(x) \cdot g' (x))}{g(x)^2} \)

\( (f(g(x)))'= f' (g(x)) \cdot g' (x) \)

Calculating integrals:
Integration is the reverse process of differentiation, used to find the area under a curve or to solve differential equations. Some basic rules for calculating integrals include:

\( \int (c \cdot f(x))dx = c \cdot \int (f(x)) dx \)

\( \int (f(x) \pm g(x)) dx = \int (f(x)) dx \pm \int (g(x)) dx \)

\( u=g(x) \) and \( \frac{du}{dx} = g'(x) \), then \( \int f(g(x)) \cdot g'(x) dx = \int f(u) du \)

\( \int u dv = uv - \int v du \) , where \(u\) and \(v\) are functions.

Taylor series:
A Taylor series is a representation of a function as an infinite sum of terms, calculated from the values of its derivatives at a single point. For a function \(f(x)\) and a point \(a\), the Taylor series is:
\( f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x - a)^n \) where \( f^{(n) } a \) represents the nth derivative of \(f\) evaluated at \(a\)

Partial fraction decomposition:
Partial fraction decomposition is a technique used to break down a rational function (a fraction where the numerator and the denominator are both polynomials) into a sum of simpler fractions. This can be helpful for integration or solving differential equations.

Cramer's rule:
Cramer's rule is a method for solving a system of linear equations using determinants. For a system of \(n\) linear equations with \(n\) variables, if the determinant of the coefficient matrix is non-zero, the unique solution can be found by calculating the determinants of the matrices obtained by replacing one column with the constants and dividing each by the determinant of the coefficient matrix.

Diagonalization of matrices:
Diagonalization is a technique used to find a diagonal matrix that is similar to a given square matrix, if possible. Diagonalizing a matrix can simplify the process of raising the matrix to a power or solving differential equations.

Dot product and cross product:
The dot product of two vectors is a scalar value, and it can be calculated as the sum of the products of the corresponding components of the two vectors. The cross product of two vectors is a vector perpendicular to both input vectors, with a magnitude equal to the area of the parallelogram formed by the two input vectors.

Stokes' theorem:
Stokes' theorem relates the line integral of a vector field around a closed curve to the surface integral of the curl of the vector field over a surface bounded by the curve:
\(\oint_C F \cdot d r = \iint_S \nabla \times F \cdot d S\) , where \(F\) is a vector field, \(C\) is a closed curve, and \(S\) is a surface.