Formulae and Shortcuts - making life easier!

[rezwansk]

Guys, great job ... this thread is exactly what i was looking for and so far have found only here. many thanks to you all.

 

Got it from this blog: Divisibility Rules « TutorTeddy BLog

 

 

Let us take up an example. Consider the number 150. Let us find out if it is divisible by the other numbers:

 

150 is divisible by 2 because the last digit is 0.

150 is divisible by 3 because when we add the digits (1+5+0), we get 6 which is divisible by 3.

150 is NOT divisible by 4, since the last 2 numbers i.e., 5 and 0 are not divisible by 0.

150 is divisible by5 because the last digit is 0

150 is divisible by 6 because it is divisible by both 2 and 3

150 is not divisible by 7 because when we double 0 we get o again. If we subtract that from 15, we get 15, which is not divisible by 7.

150 is not divisible by 8 because the three digits 1, 5 and 0are not divisible by 8.

150 is NOT divisible by 9 because the sum of the digits 1+5+0=6, which is not divisible by 9.

150 is divisible by 10 because the last digit is 0.

 

 

Further Tips on Divisibility:

(i) A number divisible by 2 or 5

Any number ending in 0 or an even number is divisible by 2 e.g. 12, 256, 328, 2060. If the last digit of a number is 0 or 5, that number is divisible by 5 e.g. 150, 2025, 3175.

(ii) A number divisible by 4 or 25

Any number is divisible by 4 if the last two digits are divisible by 4, e.g. 132, 5276, 208. Similarly if the last two digits of a number are divisible by 25, that number is divisible by 25, e.g. 1375, 2500.

(iii) A number divisible by 8 or 125

For a number to be divisible by 8, its last three digits must be divisible by 8, e.g. 864, 1248, 3000. Similarly numbers ending with the last three digits divisible by 125 are divisible by 125, e.g. 4250, 12375, 12000.

(iv) A number divisible by 16 or 625

A number having its last four digits divisible by 16, will be divisible by 16, e.g. 31776, 28528. Similarly numbers having their last four digits divisible by 625 are divisible by 625, e.g. 83125, 125000.

(v) A number divisible by 3 or 9

If a number is divisible by 3, the sum of the digits is divisible by 3, e.g. 38451, 285612.

In the above numbers, sum of digits 3+8+4+5+1 = 21 and 2+8+5+6+1+2 = 24, which are all divisible by 3. Hence, the numbers are divisible by 3.

In a similar manner, a number is divisible by 9, if the sum of all its digits is divisible by 9, e.g. 1548, 653229.

In the above numbers, the sum of digits = 1+5+4+8 = 18, 6+5+3+2+2+9 = 27. Hence, the numbers are divisible by 9

(vi) A number divisible by 11

A number is divisible by 11 if the difference between the sum of the digits in the even places and the sum of the digits in the odd places is either 0 or a number divisible by 11, e.g. 65274, 538472.

In 65274, difference in the sum of numbers in the odd places and the sum of numbers in the even places = (6+2+4) – (5+7) = 0

In 538472, difference in the sum of numbers in the odd places and the sum of numbers in the even places = (5+8+7) – (3+4+2) = 20 – 9 = 11, a multiple of 11.

Hence the two numbers are divisible by 11

 

Thanks

 

:D:D:D:D

[lastnightilie]

Hey, would it be possible for a moderator to put all of the formulae in the first post so everyone doesn't have to go through so many pages? Alternatively, you could put in direct links to each post as a sort of table of contents.

[sexymax15]

thanks

[Mash]

Thanks a lot for these very useful formulae

[nandhur]

very usefull thread

[giggle]

enjoy... happy studying

 

Hey, would it be possible for a moderator to put all of the formulae in the first post so everyone doesn't have to go through so many pages? Alternatively, you could put in direct links to each post as a sort of table of contents.

[holymanbinny]

here are some which u find helpful i will post more

 

 

1. If an equation (i.e. f(x) = 0) contains all positive co-efficients of any powers of x, it has no positive roots.

Eg: x³+3x²+2x+6=0 has no positive roots

 

2. For an equation, if all the even powers of x have same sign coefficients and all the odd powers of x have the opposite sign coefficients, then it has no negative roots.

 

3. For an equation f(x)=0 , the maximum number of positive roots it can have is the number of sign changes in f(x) ; and the maximum number of negative roots it can have is the number of sign changes in f(-x)

 

4. Complex roots occur in pairs, hence if one of the roots of an equation is 2+3i, another has to be 2-3i and if there are three possible roots of the equation, we can conclude that the last root is real. This real root could be found out by finding the sum of the roots of the equation and subtracting (2+3i)+(2-3i)=4 from that sum.

 

5.

ü For a cubic equation ax³+bx²+cx+d=o

· Sum of the roots = - b/a

· Sum of the product of the roots taken two at a time = c/a

· Product of the roots = -d/a

 

ü For a bi-quadratic equation ax⁴+bx³+cx²+dx+e = 0

· Sum of the roots = - b/a

· Sum of the product of the roots taken three at a time = c/a

· Sum of the product of the roots taken two at a time = -d/a

· Product of the roots = e/a

 

6. If an equation f(x)= 0 has only odd powers of x and all these have the same sign coefficients or if f(x) = 0 has only odd powers of x and all these have the same sign coefficients, then the equation has no real roots in each case (except for x=0 in the second case)

 

7. Consider the two equations

 

a₁x+b₁y=c₁

a₂x+b₂y=c₂

 

Then,

ü If a₁/a₂ = b₁/b₂ = c₁/c₂, then we have infinite solutions for these equations.

ü If a₁/a₂ = b₁/b₂ c₁/c₂, then we have no solution.

ü If a₁/a₂ b₁/b₂, then we have a unique solution.

 

8. Roots of x² + x + 1=0 are 1, w, w² where 1 + w + w²=0 and w³=1

 

9. |a| + |b| = |a + b| if a*b>=0

else, |a| + |b| >= |a + b|

 

10. The equation ax²+bx+c=0 will have max. value when a0. The max. or min. value is given by (4ac-b²)/4a and will occur at x = -b/2a

 

11.

ü If for two numbers x + y=k (a constant), then their PRODUCT is MAXIMUM if x=y (=k/2). The maximum product is then (k²)/4.

 

ü If for two numbers x*y=k (a constant), then their SUM is MINIMUM if

x=y (=root(k)). The minimum sum is then 2*root (k).

 

12. Product of any two numbers = Product of their HCF and LCM. Hence product of two numbers = LCM of the numbers if they are prime to each other.

 

13. For any 2 numbers a, b where a>b

 

ü a>AM>GM>HM>b (where AM, GM ,HM stand for arithmetic, geometric , harmonic means respectively)

 

ü (GM)^2 = AM * HM

 

14. For three positive numbers a, b, c

 

ü (a + b + c) * (1/a + 1/b + 1/c)>=9

 

15. For any positive integer n

 

ü 2

 

16. a² + b² + c² >= ab + bc + ca

If a=b=c, then the case of equality holds good.

 

17. a⁴ + b⁴ + c⁴ + d⁴ >= 4abcd (Equality arises when a=b=c=d=1)

 

18. (n!)² > nⁿ

 

19. If a + b + c + d=constant, then the product a^p * b^q * c^r * d^s will be maximum if a/p = b/q = c/r = d/s

 

20. If n is even, n(n+1)(n+2) is divisible by 24

 

21. x^n -a^n = (x-a)(x^(n-1) + x^(n-2) + .......+ a^(n-1) ) ......Very useful for finding multiples. For example (17-14=3 will be a multiple of 17^3 - 14^3)

 

22. e^x = 1 + (x)/1! + (x^2)/2! + (x^3)/3! + ........to infinity

Note: 2

 

23. log(1+x) = x - (x^2)/2 + (x^3)/3 - (x^4)/4 .........to infinity [Note the alternating sign . .Also note that the logarithm is with respect to base e]

 

24. (m + n)! is divisible by m! * n!

 

25. When a three digit number is reversed and the difference of these two numbers is taken, the middle number is always 9 and the sum of the other two numbers is always 9.

 

26. Any function of the type y=f(x)=(ax-b)/(bx-a) is always of the form x=f(y)

 

27. To Find Square of a 3-Digit Number

 

Let the number be XYZ

 

[TABLE]

[TR]

[TD]Step No.[/TD]

[TD]Operation to be Performed[/TD]

[/TR]

[TR]

[TD]1[/TD]

[TD]Last digit = Last digit of Sq(Z)[/TD]

[/TR]

[TR]

[TD]2[/TD]

[TD]Second last digit = 2*Y*Z + any carryover from STEP 1[/TD]

[/TR]

[TR]

[TD]3[/TD]

[TD]Third last digit 2*X*Z+ Sq(Y) + any carryover from STEP 2[/TD]

[/TR]

[TR]

[TD]4[/TD]

[TD]Fourth last digit is 2*X*Y + any carryover from STEP 3[/TD]

[/TR]

[TR]

[TD]5[/TD]

[TD]Beginning of result will be Sq(X) + any carryover from Step 4[/TD]

[/TR]

[/TABLE]

 

 

Eg) Let us find the square of 431

 

[TABLE]

[TR]

[TD]Step No.[/TD]

[TD]Operation to be Performed[/TD]

[/TR]

[TR]

[TD]1[/TD]

[TD]Last digit = Last digit of Sq(1) = 1[/TD]

[/TR]

[TR]

[TD]2[/TD]

[TD]Second last digit = 2*3*1 + any carryover from STEP 1=6+0=6[/TD]

[/TR]

[TR]

[TD]3[/TD]

[TD]Third last digit 2*4*1+ Sq(3) + any carryover from STEP 2 = 8+9+0 = 17 i.e. 7 with carry over of 1[/TD]

[/TR]

[TR]

[TD]4[/TD]

[TD]Fourth last digit is 2*4*3 + any carryover from STEP 3 = 24+1 = 25 i.e. 5 with carry over of 2[/TD]

[/TR]

[TR]

[TD]5[/TD]

[TD]Beginning of result will be Sq(4) + any carryover from Step 4 = 16+2 = 18[/TD]

[/TR]

[TR]

[TD=colspan: 2]THUS SQ(431) = 185761[/TD]

[/TR]

[/TABLE]

 

 

If the answer choices provided are such that the last two digits are different, then, we need to carry out only the first two steps only.

 

28.

ü The sum of first n natural numbers = n(n+1)/2

 

ü The sum of squares of first n natural numbers is n(n+1)(2n+1)/6

 

ü The sum of cubes of first n natural numbers is (n(n+1)/2)²/4

 

ü The sum of first n even numbers= n (n+1)

 

ü The sum of first n odd numbers= n²

 

29. If a number ‘N’ is represented as a^x * b^y * c^z… where {a, b, c, …} are prime numbers, then

 

ü the total number of factors is (x+1)(y+1)(z+1) ....

 

ü the total number of relatively prime numbers less than the number is

N * (1-1/a) * (1-1/b) * (1-1/c)....

 

ü the sum of relatively prime numbers less than the number is

N/2 * N * (1-1/a) * (1-1/b) * (1-1/c)....

 

ü the sum of factors of the number is {a^(x+1)} * {b^(y+1)} * ...../(x * y *...)

 

30.

ü Total no. of prime numbers between 1 and 50 is 15

 

ü Total no. of prime numbers between 51 and 100 is 10

 

ü Total no. of prime numbers between 101 and 200 is 21

 

31.

ü The number of squares in n*m board is given by m*(m+1)*(3n-m+1)/6

 

ü The number of rectangles in n*m board is given by ⁿ⁺¹C₂ * ^m+1C₂

 

32. If ‘r’ is a rational no. lying between 0 and 1, then, r^r can never be rational.

 

33. Certain nos. to be remembered

 

ü 2¹⁰ = 4⁵ = 32² = 1024

 

ü 3⁸ = 9⁴ = 81² = 6561

 

ü 7 * 11 * 13 = 1001

 

ü 11 * 13 * 17 = 2431

 

ü 13 * 17 * 19 = 4199

 

ü 19 * 21 * 23 = 9177

 

ü 19 * 23 * 29 = 12673

 

34. Where the digits of a no. are added and the resultant figure is 1 or 4 or 7 or 9, then, the no. could be a perfect square.

 

35. If a no. ‘N’ has got k factors and a^l is one of the factors such that l>=k/2, then, a is the only prime factor for that no.

 

36. To find out the sum of 3-digit nos. formed with a set of given digits

 

This is given by (sum of digits) * (no. of digits-1)! * 1111…1 (i.e. based on the no. of digits)

 

Eg) Find the sum of all 3-digit nos. formed using the digits 2, 3, 5, 7 & 8.

Sum = (2+3+5+7+8) * (5-1)! * 11111 (since 5 digits are there)

= 25 * 24 * 11111

=6666600

 

37. Consider the equation x^n + y^n = z^n

 

As per Fermat’s Last Theorem, the above equation will not have any solution whenever n>=3.

 

38. Further as per Fermat, where ‘p’ is a prime no. and ‘N’ is co-prime to p, then,

N^(p-1) – 1 is always divisible by p.

 

39. 145 is the 3-digit no. expressed as sum of factorials of the individual digits i.e.

 

145 = 1! + 4! + 5!

 

40.

ü Where a no. is of the form a^n – b^n, then,

· The no. is always divisible by a - b

· Further, the no. is divisible by a + b when n is even and not divisible by

a + b when n is odd

 

ü Where a no. is of the form a^n + b^n, then,

· The no. is usually not divisible by a - b

· However, the no. is divisible by a + b when n is odd and not divisible by

a + b when n is even

41. The relationship between base 10 and base ‘e’ in log is given by

log₁₀N = 0.434 log_eN

 

42. WINE and WATER formula

 

Let Q - volume of a vessel, q - qty of a mixture of water and wine be removed each time from a mixture, n - number of times this operation is done and A - final qty of wine in the mixture, then,

 

A/Q = (1-q / Q)^n

 

43. Pascal’s Triangle for computing Compound Interest (CI)

 

The traditional formula for computing CI is

CI = P*(1+R/100)^N – P

 

Using Pascal’s Triangle,

 

Number of Years (N)

-------------------

1 1

2 1 2 1

3 1 3 3 1

4 1 4 6 4 1

… 1 .... .... ... ... ..1

 

Eg: P = 1000, R=10 %, and N=3 years. What is CI & Amount?

 

Step 1:

Amount after 3 years = 1 * 1000 + 3 * 100 + 3 * 10 + 1 * 1 = Rs.1331

 

The coefficients - 1,3,3,1 are lifted from the Pascal's triangle above.

 

Step 2:

CI after 3 years = 3*100 + 3*10 + 3*1 = Rs.331 (leaving out first term in step 1)

 

If N =2, we would have had,

Amt = 1 * 1000 + 2 * 100 + 1 * 10 = Rs.1210

CI = 2 * 100 + 1* 10 = Rs.210

 

44. Suppose the price of a product is first increased by X% and then decreased by Y% , then, the final change % in the price is given by:

Final Difference% = X - Y - XY/100

 

Eg) The price of a T.V set is increased by 40 % of the cost price and then is decreased by 25% of the new price. On selling, the profit made by the dealer was Rs.1000. At what price was the T.V sold?

 

Applying the formula,

Final difference% = 40 – 25 - (40*25/100) = 5 %.

 

So if 5 % = 1,000

Then, 100 % = 20,000.

Hence, C.P = 20,000

& S.P = 20,000+ 1000= 21,000

 

45. Where the cost price of 2 articles is same and the mark up % is same, then, marked price and NOT cost price should be assumed as 100.

 

46.

ü Where ‘P’ represents principal and ‘R’ represents the rate of interest, then, the difference between 2 years’ simple interest and compound interest is given by P * (R/100)²

 

ü The difference between 3 years’ simple interest and compound interest is given by (P * R² *(300+R))/100³

 

47.

ü If A can finish a work in X time and B can finish the same work in Y time then both of them together can finish that work in (X*Y)/ (X+Y) time.

 

ü If A can finish a work in X time and A & B together can finish the same work in S time then B can finish that work in (XS)/(X-S) time.

 

ü If A can finish a work in X time and B in Y time and C in Z time then all of them working together will finish the work in (XYZ)/ (XY +YZ +XZ) time

 

ü If A can finish a work in X time and B in Y time and A, B & C together in S time then

· C can finish that work alone in (XYS)/ (XY-SX-SY)

· B+C can finish in (SX)/(X-S); and

· A+C can finish in (SY)/(Y-S)

 

48. In case ‘n’ faced die is thrown k times, then, probability of getting atleast one more than the previous throw = ⁿC₅/n⁵

 

49.

ü When an unbiased coin is tossed odd no. (n) of times, then, the no. of heads can never be equal to the no. of tails i.e. P (no. of heads=no. of tails) = 0

 

ü When an unbiased coin is tossed even no. (2n) of times, then,

P (no. of heads=no. of tails) = 1-(²ⁿC_n/2²ⁿ)

 

50. Where there are ‘n’ items and ‘m’ out of such items should follow a pattern, then, the probability is given by 1/m!

 

Eg)1. Suppose there are 10 girls dancing one after the other. What is the probability of A dancing before B dancing before C?

 

Here n=10, m=3 (i.e. A, B, C)

 

Hence, P (A>B>C) = 1/3!

= 1/6

 

Eg)2. Consider the word ‘METHODS’. What is the probability that the letter ‘M’ comes before ‘S’ when all the letters of the given word are used for forming words, with or without meaning?

 

P (M>S) = 1/2!

= 1/2

 

51. CALENDAR

 

ü Calendar repeats after every 400 years.

 

ü Leap year- it is always divisible by 4, but century years are not leap years unless they are divisible by 400.

 

ü Century has 5 odd days and leap century has 6 odd days.

 

ü In a normal year 1st January and 2nd July and 1st October fall on the same day. In a leap year 1st January 1st July and 30th September fall on the same day.

 

ü January 1, 1901 was a Tuesday.

 

52.

ü For any regular polygon, the sum of the exterior angles is equal to 360 degrees, hence measure of any external angle is equal to 360/n (where n is the number of sides)

 

ü For any regular polygon, the sum of interior angles =(n-2)*180 degrees

So measure of one angle is (n-2)/n *180

 

ü If any parallelogram can be inscribed in a circle, it must be a rectangle.

 

ü If a trapezium can be inscribed in a circle it must be an isosceles trapezium (i.e. oblique sides equal).

 

53. For an isosceles trapezium, sum of a pair of opposite sides is equal in length to the sum of the other pair of opposite sides (i.e. AB+CD = AD+BC, taken in order)

 

54.

ü For any quadrilateral whose diagonals intersect at right angles, the area of the quadrilateral is

0.5*d₁*d₂, where d₁, d₂ are the length of the diagonals.

 

ü For a cyclic quadrilateral, area = root((s-a) * (s-b) * (s-c) * (s-d)), where

s=(a + b + c + d)/2

Further, for a cyclic quadrilateral, the measure of an external angle is equal to the measure of the interior opposite angle.

 

ü Area of a Rhombus = Product of Diagonals/2

 

55. Given the coordinates (a, b); (c, d); (e, f); (g, h) of a parallelogram , the coordinates of the meeting point of the diagonals can be found out by solving for

[(a + e)/2, (b + f)/2] = [(c + g)/2, (d + h)/2]

 

56. Area of a triangle

 

ü 1/2*base*altitude

ü 1/2*a*b*sin C (or) 1/2*b*c*sin A (or) 1/2*c*a*sin B

ü root(s*(s-a)*(s-b)*(s-c)) where s=(a+b+c)/2

ü a*b*c/(4*R) where R is the circumradius of the triangle

ü r*s ,where r is the inradius of the triangle

 

57. In any triangle

ü a=b*cos C + c*cos B

ü b=c*cos A + a*cos C

ü c=a*cos B + b*cos A

ü a/sin A=b/sin B=c/sin C=2R, where R is the circumradius

ü cos C = (a^2 + b^2 - c^2)/2ab

ü sin 2A = 2 sin A * cos A

ü cos 2A = cos^2 (A) - sin^2 (A)

 

58. The ratio of the radii of the circumcircle and incircle of an equilateral triangle is 2:1

 

59. Appollonius Theorem

 

In a triangle ABC, if AD is the median to side BC, then

AB² + AC² = 2(AD² + BD²) or 2(AD² + DC²)

 

60.

ü In an isosceles triangle, the perpendicular from the vertex to the base or the angular bisector from vertex to base bisects the base.

ü In any triangle the angular bisector of an angle bisects the base in the ratio of the other two sides.

 

61. The quadrilateral formed by joining the angular bisectors of another quadrilateral is always a rectangle.

 

62. Let W be any point inside a rectangle ABCD, then,

WD² + WB² = WC² + WA²

 

63. Let a be the side of an equilateral triangle, then, if three circles are drawn inside this triangle such that they touch each other, then each circle’s radius is given by a/(2*(root(3)+1))

 

64.

ü Distance between a point (x₁, y₁) and a line represented by the equation

ax + by + c=0 is given by |ax₁+by₁+c|/Sq(a²+b²)

ü Distance between 2 points (x₁, y₁) and (x₂, y₂) is given by

Sq((x₁-x₂)²+ (y₁-y₂)²)

 

65. Where a rectangle is inscribed in an isosceles right angled triangle, then, the length of the rectangle is twice its breadth and the ratio of area of rectangle to area of triangle is 1:2.

[aishavadhariya]

More formulae:

 

PROGRESSION:

 

Sum of first n natural numbers: 1 +2 +3 + .... + n = [n(n+1)]/2

Sum of first n odd numbers: 1 + 3 + 5 + .... upto n terms = n^2

Sum of first n even numbers: 2 + 4 + 6 + ... upto n terms = n(n+1)

ARITHMETIC PROGRESSION

 

nth term of an Arithmetic progression = a + (n-1)d

Sum of n terms in an AP = s = n/2 [2a + (n-1)d]

where, a is the first term and d is the common differnce.

 

If a, b and c are any three consequtive terms in an AP, then 2b = a + c

GEOMETRIC PROGRESSION

 

nth term of a GP is = a[r^(n-1)]

sum of n terms of a GP:

s = a [(r^n - 1)/(r-1)] if r > 1

s = a [(1 - r^n)/(r-1)] if r

 

sum of an infinite number of terms of a GP is

s(approx.) = a/ (1-r) if r

 

If a, b and c are any three consequtive terms in a GP, then b^2 = a + c

 

PS: Everybody!!! please contribute or this thread will never grow.

 

sum of squares of first n integers.= n(n+1)(2n+1)/6

 

 

sum of cube of first n integers=[n(n+1)/2]^2

[rbgandhi]

More formulae:

 

PROGRESSION:

 

Sum of first n natural numbers: 1 +2 +3 + .... + n = [n(n+1)]/2

Sum of first n odd numbers: 1 + 3 + 5 + .... upto n terms = n^2

Sum of first n even numbers: 2 + 4 + 6 + ... upto n terms = n(n+1)

ARITHMETIC PROGRESSION

 

nth term of an Arithmetic progression = a + (n-1)d

Sum of n terms in an AP = s = n/2 [2a + (n-1)d]

where, a is the first term and d is the common differnce.

 

If a, b and c are any three consequtive terms in an AP, then 2b = a + c

GEOMETRIC PROGRESSION

 

nth term of a GP is = a[r^(n-1)]

sum of n terms of a GP:

s = a [(r^n - 1)/(r-1)] if r > 1

s = a [(1 - r^n)/(r-1)] if r

 

sum of an infinite number of terms of a GP is

s(approx.) = a/ (1-r) if r

 

If a, b and c are any three consequtive terms in a GP, then b^2 = a + c

 

PS: Everybody!!! please contribute or this thread will never grow.

 

This formula is incorrect

Sum of first n odd numbers: 1 + 3 + 5 + .... upto n terms = n^2

 

It should be

Sum of first n odd numbers: 1 + 3 + 5 + .... upto (2n-1) terms = n^2

 

sorry if someone has already replied or corrected.

[usa4ever]

thanksss so much

[rajechP]

A short Cut...

 

Q. If the price of fuel rises by 10%, by what % should one reduce its consumption so that total expenses on fuel remain the same?

 

Step 1: Convert 10% into fraction, which is 1/10. Call it 1/a.

 

Step 2: Add '1' to the denominator of 1/a. It becomes 1/(1+a). For the problem, it is 1/11.

 

Step 3: Convert 1/(1+a)=1/11 fraction into %, it is 9.09%.

 

The consumption should be reduced by 9.09%.