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Basic Mathematics for Everyone

Maths Tips and Trick incl. Vedic Mathematics

This page describes some of the key shortcuts in maths primarily in algebra such as multiplication, square, quadratic equations and large exponents of a number. Do you know that 2, 3, 7 ad 8 cannot be at unit's place of a perfect square, that square of a number having 'n' digits will results in a number with 2n or 2n-1 digits, that the square of any odd number is sum to two consecutive numbers and so on. That is:

52 = 12 + 13

92 = 40 + 41

...

132 = 64 + 65

We have always been explained the standard method of finding the roots of a quadratic equation. Do you know that the first derivative of the equation = discriminant!

What is the time you will need to calculate 997×998? Using Vedic mathematics, you can simply write the product 997×998 = 995006!


Vedic mathematics contain fascinating hidden gems which are not only interesting but also help improve calculation speed. This is a complementary page where we endeavour to provide tips and tricks in mathematics. The questions dealing with standard aptitude tests are described such as finding remainder in divisions, efficiency method, total effort method, mathematical induction and so on.

Maths - NCERT Class IX Maths - NCERT Class X
Maths - NCERT Class XI Maths - NCERT Class XII-A
Maths - NCERT Class XII-B -
Trigonometry by S. L. Loney Trigonometry by Hall & Frink
Maths book on Coordinate Geometry by S. L. Loney
Division Rules

  • Divisibility by 2: A number is divisible by 2 if digit at unit place is even.
  • Divisibility by 3: A number is divisible by 3 if sum of all digits is also divisible by 3. E.g. 1323 is divisible by 3 because (1 + 3 + 2 + 3) = 9 which is divisible by 3.
  • Divisibility by 4: A number is divisible by 4 if number comprising of digits at tens and units is divisible by 4. E.g. 123456 is divisible by 4 because 56 is divisible by 4.
  • Divisibility by 5: A number is divisible by 5 if digit at unit place is either 0 or 5.
  • Divisibility by 6: A number is divisible by 6 (2 x 3) if [a] digit at unit place is an even number - divisibility by 2 and [b] sum of all digits is divisible by 3 - divisibility by 3.
  • Divisibility by 8: A number is divisible by 8 if number comprising of digits at hundreds, tens and units is divisible by 4. E.g. 123456 is divisible by 8 because 456 is divisible by 8.
  • Divisibility by 9: A number is divisible by 9 if sum of all digits is also divisible by 9. E.g. 1323 is divisible by 9 because (1 + 3 + 2 + 3) = 9 which is divisible by 9. The numerator is repeated, decimal place = number of digits less by 1. Example:
    Step-1: 513/9 = 513 513 513 513 repeat the numerator.
    Step-2: Location of decimal place = 2 (no of digits in 513) - 1. Thus, 513/9 = 51.3513
  • Divisibility by 11: A number is divisible by the difference in sum of digits at alternate place is divisible by 11. E.g. 7238561 is divisible by 11 because [1+5+3+7] - [6+8+2] = 0 which is divisible by 11.
  • Divisibility by 12: A number is divisible by 12 (3 x 4) if [a] sum of all digits is also divisible by 3 and [b] number comprising of digits at tens and units is divisible by 4.


Multiplication by 9

Multiplication by 11


Number of digits in products

Product of integers with N and M digits
N = 2, M = 3: 10 * 100   =   1000,   D = 4 = [N + M - 1]
N = 2, M = 3: 99 * 100   =   9900,   D = 4 = [N + M - 1]
N = 2, M = 3: 10 * 999   =   9990,   D = 4 = [N + M - 1]
N = 2, M = 3: 99 * 999   = 98901,   D = 5 = [N + M]

N = 2, M = 4: 10 * 1000 =   10000, D = 5 = [N + M - 1]
N = 2, M = 4: 99 * 1000 =   99000, D = 5 = [N + M - 1]
N = 2, M = 4: 10 * 9999 =   99990, D = 5 = [N + M - 1]
N = 2, M = 4: 99 * 9999 = 989901, D = 6 = [N + M]
Thus: the answer is [N + M - 1] and [N + M].


Square of a number

a2 = (a + b)*(a - b) + b2. For example: 43 * 43 = (43+3)*(43-3) + (3*3) = (46*40)+9 = 1849.

A great improvisation of this formula as per Vedic mathematics is as follows: "Whatever the extent of its deficiency, lessen it still further to that very extent; and also set-up the square of that deficiency". The calculations steps are then defined as:
  • Take the power of (10, 100, 1000 ...) nearest to the number being squared. E.g. n = 84 has base number b = 100.
  • Calculate deficiency, d = b - n = 100 - 84 = 16
  • Calculate square of the deficiency: d2. Write the right two digits, xx and carry the third digit if any say c. Here, d2 → 162 → xx = 56, c = 2
  • Subtract twice of the deficiency from base and add the carry c. Thus: aa = b - 2 * d + c. Here: a = 100 - 2 * 16 + 2 = 70.
  • Thus: 842 = aa|xx = 70|56 = 7056
  • Note that this method is best suitable if the deficiency from suitable base (10, 100, 1000 ...) is in single digit. Some adaptation will be presented later.
  • Followings are the specific applications of this rule:
    • Square of any number ending is 5: eg. x52 = x*(x+1)|25. Thus: 752 = 7*8|25 = 5625, 1152 = 11*12|25 = 13225
    • Square of numbers, when the sum of digits at unit's place is 10 and other digits are same, can be calculated using similar method. That is xy * xq where y + q = 10. xy * xq = x * (x+1)|p*q. For example: 67 * 63 = 6*(6+1)|7*3 = 4221.

      Product when sum of units place is 10


a2 - b2 = (a + b) × (a - b): This equation has many applications in solving maths problems quickly. For example:
  • The sum of two numbers is 45 and their difference is 10. The difference of their squares = 45 × 10 = 450.
  • The difference between the squares of two consecutive numbers is 25. Thus: a + b = 25 and a - b = 1 and hence a = 12, b = 13.
  • The difference in squares of two consecutive number 50 and 51, that is 502 - 512 = (50 + 51) × (51 - 50) = 101.
  • a2 + b2 = (a - b)2 + 2ab. Thus, if the product of two consecutive numbers is known, the sum of their squares = twice the product + 1.
  • a2 + b2 = (a + b)2 - 2ab. Example, the sum of two numbers is 23 and their product is 126. Find the sum of squares of these numbers. a2 + b2= 232 - 2×126 = 277.

a3 - b3 = (a - b)3 +3ab(a - b): This equation has many applications in solving maths problems quickly. For example:
  • The difference between the cubes of two consecutive numbers = 3ab + 1. Thus, if the product of two consecutive number is 380, find the difference of their cubes. The answer would be 3×380+1 = 1141.
  • The difference in cubes of two numbers is 14716 and difference between the two number is 4. Find the product of the numbers. a3 - b3 = 14716, a - b = 4, ab = [14716 - 43]/3/4 = 1221.

a4 - b4 = (a - b) (a + b) (a2 + b2): For two consecutive numbers: a2 + b2 = 2ab + 1, and hence a4 - b4 = (a + b) × (2ab + 1). Example: The sum of two consecutive numbers is 15 and their product is 56. Find the difference in their 4th exponent. a4 - b4 = 15 × (2×56 + 1) = 15 × 113 = 1695.
Similarly, a4 + b4 = (a2 + b2)2 - 2a2b2: For two consecutive numbers: a2 + b2 = 2ab + 1, and hence a4 + b4 = (2ab + 1)2 - 2a2b2. Note that the sum of fourth exponent of any two consecutive number now can be evaluated from the product of these numbers.

Note: 24 raised to an even power always ends with 76, 24 raised to an odd power always ends with 24, 76 raised to any power ends with 76.

What is digit at unit's place of the number 792351681/5? The answer is 8. This is due to the fact that any number raised to its fifth power will have its last digit (that at unit's place) unchanged!


Large Exponent of a Large Number

10012 = 1002001.      Hence, 10012 - 1 is divisible by 1000 or multiple of 1000 as per divisibility rules of 2, 3, 4 ...

10013 = 1003003001. Hence, 10013 - 1 is divisible by 1000 or multiple of 1000 as per divisibility rules of 2, 3, 4 ...

Special cases
  • 1111...12 [where '1' is repeated 'n' times] = 123..n(n-1)...1. e.g. 111112 = 123454321 where n = 5. This is valid till n ≤ 9.

    For n ≥ 10: 111...1 2 = 12...79012..U..2098...1 where U = unit place of (n+1). For example, 1 repeated 12 times: 1111111111112 = 12345679012320987654321, here U = 3 in (12+1 = 13).

    1 repeated 13 times: 11111111111112 = 1234567901234320987654321, here U = 4 in (13+1 = 14).

    Exception: 1 repeated 10 times: 111111111112 = 1234567900987654321.

  • 3333...32 [where '3' is repeated 'n' times] = 11..1088..89 where '1' and '8' are repeated 'n-1' times. e.g. 3333332 = 111110888889 where n = 6.
  • 666...62 [where '6' is repeated 'n' times] = 44..4355..56 where '4' and '5' are repeated 'n-1' times. e.g. 66666662 = 44444435555556 where n = 7.
  • 999...92 [where '9' is repeated 'n' times] = 99..9800..01 where '9' and '0' are repeated 'n-1' times. e.g. 99992 = 9999800001 where n = 4.
  • Can you find a pattern in following square numbers formed by repetition of 2, 4, 5 and 8?

    square of larger numbers


Approximate Square Root of a number

Square Root

Infinite sequence of square root can be calculated as:

Infinite sequence Square Root

The addition and subtraction of conjugate squares:

Conjugate square root sum


Last digit in power (exponent) of 3, 4, 7 ...
  • 34 = 81, 35 = 243, 36 = 729, 37 = 2187, 38 = 6561
    Thus:
    3n = 1 if remainder(n/4) = 0
      = 3 if remainder(n/4) = 1
      = 9 if remainder(n/4) = 2
      = 7 if remainder(n/4) = 3
  • Similarly, last digit of 4n is 6 is n is an even number and 4 if n is an odd number.
  • Last digit of 5n is always 5.
  • Last digit of 6n is always 6.
  • Last digit of 7n:

    7 power n

    7n = 1 if remainder(n/4) = 0
      = 7 if remainder(n/4) = 1
      = 9 if remainder(n/4) = 2
      = 3 if remainder(n/4) = 3
  • 8n = 6 if remainder(n/4) = 0
      = 8 if remainder(n/4) = 1
      = 4 if remainder(n/4) = 2
      = 2 if remainder(n/4) = 3
  • Using the remainders described above, remainder of expressions like 334/5 can be easily obtained as explained below.

    If Q = A × n + R, then remainder of Qm/n = remainder of Rm/n.

    If Q = (A × n + R) and P = (B × n + S), then remainder of (Q * P)m/n = remainder of (R * S)/n.

    Thus: remainder of 334/n = remainder[ { remainder of (34)8/5 × remainder(32/5) } / 5 ]

    Remainder of large exponent

    The trick is to break the larger exponent in smaller exponent such that remainder is 1.

The cube of a number will have no change in digit at unit's place if the digits at the unit place are 4, 6 or 9. In case of any other digit at unit's place, the digit in its cube would be [10 - that digit]. E.g. 133 = 2197, 173 = 4913.

Similarly, the square of square of a number (4th power of a number) has either 1, 5 or 6 at units place. That is if unit's place is an odd number (except 5), the 4th exponent of that number will always end in 1. Similarly, if unit's place is an even number , the 4th exponent of that number will always end in 6.

A combination of this the above rules can be used to find digit at unit's place for any other exponent (such as 6th, 8th, 10th ...) of a number. Use the following table to check the rules described thus far.
Number X2 X3 X4 X5 X6
11 121 1331 14641 161051 1771561
12 144 1728 20736 248832 2985984
13 169 2197 28561 371293 4826809
14 196 2744 38416 537824 7529536
15 225 3375 50625 759375 11390625
16 256 4096 65536 1048576 16777216
17 289 4913 83521 1419857 24137569
18 324 5832 104976 1889568 34012224
19 361 6859 130321 2476099 47045881

Cube root of exact cubes
First digit of cubeFirst digit of cube root
11
28
37
44
55
66
73
82
99

The number of digits in a cube root = number of 3 digit groups in original cube including a single digit or double digit group if there is any.

The first (leftmost) digit of the cube root can be guessed easily from the first group (leftmost) of the cube. Thus: 17,233,489,287: there are 4 groups of 3 digits. Hence, the cube root of this number will have 4 digits. The digit at unit's place, as per the table above will be 3. The digit at thousand's place will be 2 as 23 < 17. Alternatively, except 4, 6 and 9, the digit at unit place in = 10 - the digit.


Simple Equations

Simple fraction equality

Quadratic equations in terms of reciprocals:

Reciprocal fraction equality

Same method applies when there is a minus sign between x and 1/x.

Reciprocal fraction equality with minus sign


fraction equality equation


Find tens place of any exponent of a number ending in 1

E.g. tens place of 3157. The unit place would be 1. The tens place would be unit place of product (3 * 7) and hence 1 will be at tens place of 3157. The number is calculated at product of digit at tens place in the base that is '3' here and the first digit (at units place) in the exponent which is '7' here.

So far easy for exponents. How to find digit at ten's place in multiplication of 3 or 4 digits? Refer to the method below for product 24 × 37 × 68 × 94.

Digit at tens place


Work and time - Use efficiency as short-cut

If A does the task in 20 days and B does it in 16 days. How long will they take it to complete the task together?

Efficiency Method A's efficiency = 100/20 = 5%, B's efficiency = 100/16 = 6.25%, (A+B)'s efficiency = 5+6.25 = 11.25 %. Thus: Time required = 100/11.25 = 8.89 days

Another variant of such problem is given as:
If a group-A of 4 persons each of equal efficiency can do a task in 12 days. How many day would be required to do the same task by group-B of 6 persons with efficiency 2 times that of group-A?

Equality of total effort Let x be the efficiency of persons in group A. Total effort required = 4 * x * 12 = 48x [mandays]. This is constant for that particular task.

Now, let N be the number of days required for persons in group-B to complete the task. Thus: N * 2x * 6 = 48x. Hence, N = 4 days.


Sum of Arithmetic progression of numbers and squares
Arithmetic Progression

Sum of Arithmetic Progression

Sum of A.P. of squares

Sum of Arithmetic Progression of Squares

Sum of A.P. of Cubes and Products

Sum of Arithmetic Progression of Cubes


Sum of exponents of a number and its inverse

Sum of exponents of a number and its inverse 01

Sum of exponents of a number and its inverse 02

Sum of exponents of a number and its inverse 03


Vectors in Coordinate Geometry Find length of BP, the projection of vector B over vector A.

vector Projection Length


Quadratic Equations
The method to find roots of a quadratic equation is well known. However, there are times when the standard textbook method can be shortened. For example: a·x2 + b·x + c = 0 can be re-written as

x2 + b/m · x + c/m = 0

If r1 and r2 are roots of the equation, (x - r1) × (x - r2) = x2 - (r1 + r2)x + r1×r2 = 0. Thus: r1 + r1 = -b/a and r1 × r2 = c/a.

Example:

3·x2 + 2·x - 48 = 0, the shortcut is to find two numbers which adds to -2 and whose multiplication = -48. 6 and -8 are the numbers and hence the roots of the quadratic equation are [6/3, -8/3].
Fractions: There are some general rules of the fractions which should be remembered to get answers quicker.

fraction addition to numerator and demoninator

A quick check during the aptitude test is to use: [a = 1, b = 5, c = 1] or [a = 1, b =2, c = 1].
Puzzles

There are two containers having equal volume of A and B. Now, amount x of liquid A from container is taken out and mixed with liquid B. Thereafter, same volume x is taken out from second container and mixed with liquid A. Question is: which of the containers have higher impurity? In other words, is the volume fraction of B in A in the first container is higher, lower or same as volume fraction of A in B in the second container?
Answer: The volume fraction of B in A in container 1 = volume fraction of A in B in container 2. Before the answer is demonstrated through complex calculation, the easiest way to get to the answer is by noticing the variable 'x'. It can be between 0 to 100. Hence, assume the case when x = 100 and the answer is straightforward.

Now mathematics is as follows:
Mixing of Liquids

What is the shortest path between two diagonally opposite vertices of a cube when the travel path has to be all along the walls, ground and ceiling of the cube? Two such set of vertices have been shown by circles and diamonds in the following figure.

Cube Shortest Path


Multiplications of large numbers: can you find a pattern or data is insufficient?

mutliplicationOnesFivesEights

More symmetry in maths
1 x 8 + 1 = 91 x 9 + 2 = 11
12 x 8 + 2 = 9812 x 9 + 3 = 111
123 x 8 + 3 = 987123 x 9 + 4 = 1111
1234 x 8 + 4 = 98761234 x 9 + 5 = 11111
12345 x 8 + 5 = 987 6512345 x 9 + 6 = 111111
123456 x 8 + 6 = 987654123456 x 9 + 7 = 1111111
1234567 x 8 + 7 = 98765431234567 x 9 + 8 = 11111111
12345678 x 8 + 8 = 9876543212345678 x 9 + 9 = 111111111
123456789 x 8 + 9 = 987654321123456789 x 9 +10 = 1111111111
1 x 1 = 1
9 x 9 + 7 = 8811 x 11 = 121
98 x 9 + 6 = 888111 x 111 = 12321
987 x 9 + 5 = 88881111 x 1111 = 1234321
9876 x 9 + 4 = 8888811111 x 11111 = 123454321
98765 x 9 + 3 = 888888111111 x 111111 = 12345654321
987654 x 9 + 2 = 88888881111111 x 1111111 = 1234567654321
9876543 x 9 + 1 = 8888888811111111 x 11111111 = 123456787654321
98765432 x 9 + 0 = 888888888 111111111 x 111111111 = 12345678987654321

Trivia

Kaprekar’s Constant - Take any four-digit number except an integral multiple of 1111 (i.e. don’t take one of the nine numbers with four identical digits). Rearrange the digits of your number to form the largest and smallest strings possible. That is, write down the largest permutation of the number, the smallest permutation (allowing initial zeros as digits), and subtract. Apply this same process to the difference just obtained. Within the total of seven steps, you always reach 6174. At that point, further iteration with 6174 is pointless: 7641–1467 = 6174. Example: Start with 8028. The largest permutation is 8820, the smallest is 0288, and the difference is 8532. Repeat with 8532 to calculate 8532–2358 = 6174. Your own example may take more steps, but you will always reach 6174.

  • Other than the trivial examples of 0 and 1, the only natural numbers that equal the sum of the cubes of their digits are 153, 370, 371, and 407.
  • Aside from 1, first positive number which is square and cube of a number is 64.
  • 365 = 102 + 112 + 122 = 132 + 142
  • 1 googol = 1100 - first used in 1940 by nine-year old Mitton Sirotta
  • 1 + 4 + 6 + 7 = 18 = 2 + 3 + 5 + 8 and 12 + 42 + 62 + 72 = 102 = 23 + 52 + 82 + 12
  • 1729 = 13 + 123 = 93 + 103- is known as Ramanujan-Hardy number. It is the smallest 'taxicab' number.
  • Squares having odd number of divisors are called "perfect squares".
  • Two consecutive prime numbers which differ by 2 are called Twin Primes.
  • Interesting squares, exponents and summation from web:
    • 71 + 73 + 71 + 77 + 77 + 73 + 78 = 13177388.
    • 132 = 169, 312 = 961
    • 206156734 = 26824404 + 153656394 + 187967604 - integer solution for the equation of the form x4 + y4 + z4 = a4 found by Noam Elkies.
    • 2042 = 233 + 243 + 253
    • 163 + 503 + 333 = 165033
  • Lagrange’s Four Square Theorem: every natural number can be written as sum of squares of four non-negative integers (including 0) where numbers can be repeated. The summation for few Fibonacchi numbers are:
    • 21 = 02 + 12 + 22 + 42 = 22 + 22 + 22 + 32
    • 34 = 02 + 02 + 32 + 52 = 02 + 32 + 32 + 42 = 12 + 12 + 42 + 42 = 12 + 22 + 22 + 52
    • 55 = 12 + 12 + 22 + 72 = 12 + 22 + 52 + 52 = 12 + 32 + 32 + 62
    • 89 = 02 + 02 + 52 + 82 = 02 + 22 + 22 + 92 = 02 + 22 + 62 + 72 = 02 + 32 + 42 + 82 = 12 + 42 + 62 + 62
  • Euler’s Equation: x3 + y3 + z3 = u3   No general solution exists though some specific solutions are expressed as following parametric form for arbitrary constants m and p.
    • x = m7 - 3m4 (1+p) + m(2 + 6p + 3p2)
    • y = 2m6 - 3m3(1+2p) + (1+3p+3p2)
    • z = m6 - (1+3p+3p2)
    • u = m4(m3-3p) + m(3p2-1)
    • Exception: 13 + 63 + 83 = 93
    • Another form of the parametric solution of Euler's equation is:

      (6a2 - 4ab + 4b2)3 = (3a2 + 5ab - 5b2)3 + (4a2 - 4ab + 6b2)3 + (5a2 - 5ab - a2)3

    • Yet another form of specific solution is given below when a2 + ab + b2 = 3mn2

      (a+m2n)3 + (mb + n)3 = (ma + n)3 + (b + m2n)3


Integral of Trigonometric Squares

The following integration formula have wide applications in applications such as Fourier transforms, calculation of lift coefficient of airfoils...

Integration of cosine functions

Integration of sine functions

Integration of higher exponent of sin-cos


Multiply your Money

If you invest money which compounds annually:
  • (Approx.) number of years required to double the money = 72/ROI where ROI = annual rate of interest. E.g. if ROI = 4%, YTD (years to double) = 72/4 = 18 years. YTD varies as 70/ROI (ROI = 2%) to 75/ROI (ROI = 17%). Exact value = ln(2) / ln(1 + ROI/100).
  • (Approx.) number of years required to triple the money = 115/ROI. E.g. if ROI = 4%, YTT (years to triple) = 115/4 = 24 years. YTT varies as 111/ROI (ROI = 2%) to 120/ROI (ROI = 19%). Exact value = ln(3) / ln(1 + ROI/100)

Lateral Thinking
This is an activity where children are encouraged to find a link between arrangement of words and numbers into standard phrases, something similar to clues given in a treasure hunt. Refer to few examples below:

Examples of lateral thinking

In the example above, ECNALG can be interpreted as Reverse Glance. Similarly, PRO / MISE can be used to refer "Broken Promise".

What is the next two letters in this sequence: B C D G J ... ...? The answers are: O and P. Why and how?

How many straight lines are required to connect 16 dots arranged in a 4 × 4 square, the lines must not cross each other?

connecting 16 squares by straight lines

One of the quickest answer is: 7 as shown below. Can you find a better way?

connecting 16 squares Answer


Can we have two right angles in a triangle? The classical textbook thinking will lead to answer in negative. That is because we always think triangles or rectangles to be formed by connecting ends of straight lines! What if the lines are not straight? Refer the image below and revisit your answer.

Triangle with two right angles


Area of a circular section cut by a chord

Area of a circular section cut by a chord

Note that the formula is applicable to both the situation shown in the image. In case chord location is above R, the expression (R-H) will negative and the two areas A1 and A2 will add up. Similarly, the angle θ will be calculated to be > 90°.
Integration: exponential-harmonic function

exponential Sin Function Integration

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