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Important Math Formula for quick solving of questions
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22 Apr 2017, 08:23
Hi, I have consolidated few important math formula for quick solving of questions. Topics included are: PERCENTAGES PROFIT AND LOSS TIME AND DISTANCE BOATS AND STREAMS PROGRESSIONS INEQUALITIES AND MODULUS PERMUTATIONS AND COMBINATIONS QUADRATIC EQUATIONS GEOMETRY AND MENSURATIONPlease give kudos if you find these useful. PERCENTAGES PROFIT AND LOSS1)If the value of an item goes up/down by x%,the percentage reduction/increment to be now made to bring it back to the original level is \(\frac{100x}{(100+x)}\)%/ \(\frac{100x}{(100x)}\)% respectively.
2) If A is x% more/less than B, then B is \(\frac{100x}{(100+x)}\)% / \(\frac{100x}{(100x)}\)% less/more than A respectively.
3)If the price of an item goes up/down by x% then the quantity consumed should be reduced/increased by \(\frac{100x}{(100+x)}\) % / \(\frac{100x}{(100x)}\)% respectively so that the total expenditure remains the same.
4) S.P. = C.P * \(\frac{(100+P)}{100}\) , where S.P.= Selling Price, C.P.= Cost Price, P=Profit percentage
5) S.P. = C.P. * \(\frac{(100L)}{100}\) , where S.P.= Selling Price, C.P.= Cost Price, L=Loss percentage
6) C.P. = S.P* \(\frac{100}{(100+P)}\) ,where S.P.= Selling Price, C.P.= Cost Price, P=Profit percentage
7)C.P. = S.P. * \(\frac{100}{(100L)}\) ,where S.P.= Selling Price, C.P.= Cost Price, L=Loss percentage
8) When two articles are sold at the same price such that there is a profit P% on one article and a loss of P% on the other, the net result of the transaction is LOSS. Loss % = \(\frac{〖(Common profit Or loss)〗^2}{100}\)
= \(\frac{P^2}{100}\)
9)If A makes a profit of x% by selling an item to B and B makes a profit of y% by selling it to C, then the resultant profit% is (x + y + \(\frac{xy}{100}\))%
10) Selling Price = Marked Price – Discount
11) Discount% = \(\frac{(Marked Price Selling Price)}{(Marked Price)}\)* 100
TIME AND DISTANCE 1) In general, if a person travelling between two points reaches ‘p’ hours late travelling at a speed of ‘u’ kmph and reaches ‘q’ hours early travelling at ‘v’ kmph, the distance between two points is given by \(\frac{(vu)(p+q)}{(vu)}\)
2) If a body travels from point A to point B with a speed of ‘p’ and back to point A (from B) with a speed of ‘q’, then the average speed of the body can be calculated as \(\frac{2pq}{(p+q)}\) . This does not depend upon the distance between A and B.
3) If a body covers part of the journey at speed ‘p’ and the remaining part of the journey at speed ‘q’ and the distance of the two parts of the journey are in the ratio m:n, then the average speed for the entire journey is: \(\frac{(m+n)pq}{(mq+np)}\)
4) Time taken by a train to overtake another train = \(\frac{( Length of faster train + Length of slower train)}{(Their relative speed)}\)
5) When 2 people are running around a circular track: Time taken to meet for the first time ever in same direction: \(\frac{L}{(ab)}\) Time taken to meet for the first time ever in opposite direction: \(\frac{L}{(a+b)}\) Time taken to meet for the first time at the starting time in same direction: LCM ( \(\frac{L}{a}\) , \(\frac{L}{b}\) ) Time taken to meet for the first time at the starting time in opposite direction: LCM ( \(\frac{L}{a}\) , \(\frac{L}{b}\) )
6) When 3 people are running around a circular track: Time taken to meet for the first time ever in same direction: LCM { \(\frac{L}{(ab)}\) , \(\frac{L}{(bc)}\) } Time taken to meet for the first time ever in opposite direction: LCM {\(\frac{L}{(ab)}\) , \(\frac{L}{(bc)}\) } Time taken to meet for the first time at the starting time in same direction: LCM (\(\frac{L}{a}\) , \(\frac{L}{b}\) , \(\frac{L}{c}\)) Time taken to meet for the first time at the starting time in opposite direction: LCM (\(\frac{L}{a}\) , \(\frac{L}{b}\) , \(\frac{L}{c}\)) BOATS AND STREAMS1) Speed of the boat against stream= Stream of the boat in still water – Speed of the stream 2) Speed of the boat with stream= Stream of the boat in still water +Speed of the stream 3) Speed of the boat in still water = \(\frac{(u+v)}{2}\) where, u = Speed of the boat upstream, v = Speed of the boat downstream. 4) Speed of the water current = \(\frac{(vu)}{2}\) where, u = Speed of the boat upstream, v = Speed of the boat downstream. PROGRESSIONS1) If both progressions are AP’s, then the common difference is the L.CM. of the common difference of the 2 progressions.
Nth common term of the 2 progression = First common term of the 2 progressions + (N1)* L.C.M (\(d_{1}\),\(d_{2}\)) Where, \(d_{1}\) = common difference of the first arithmetic progression And \(d_{2}\) = common difference of the second arithmetic progression
2) Sum of the terms of an arithmetic progression = (number of terms)*(the middle term of the A.P.), if number of terms is odd. 3) Difference between sum of odd numbered terms and even numbered terms is \(\frac{(n*d)}{2}\) if n i.e., number of terms is even.
4) Sum of n terms of an A.P. = \(\frac{(a+l)}{2}\)*n 5) The Arithmetic Mean (AM) between two numbers a and b = \(\frac{(a+b)}{2}\)
6) To solve most of the problems related to AP, the terms can be conveniently taken as 3 terms: (a – d), a, (a +d) 4 terms: (a – 3d), (a – d), (a + d), (a +3d) 5 terms: (a – 2d), (a – d), a, (a + d), (a +2d)
7) Tn = Sn  Sn1
8) The Harmonic Mean(HM) between two numbers a and b = \(\frac{2ab}{(a+b)}\)
9) If a, b, c are in HP, \(\frac{2}{b}\) = \(\frac{1}{a}\) + \(\frac{1}{c}\)
10) To solve most of the problems related to GP, the terms of the GP can be conveniently taken as 3 terms: \(\frac{a}{r}\), a, ar 5 terms: \(\frac{a}{r^2}\) , \(\frac{a}{r}\) , a, ar, \(ar^2\)
INEQUALITIES AND MODULUS1) A.M ≥ G.M. ≥ HM. The equality occurs only when the number are all equal.
2) If the sum of two positive quantities is given, their product is the greatest when they are equal. If the product of two positive quantities is given, their sum is the least when they are equal.
3) When the expression (ax+by) is constant, the maximum value of \(x^m y^n\) is realized when \(\frac{ax}{m}\) = \(\frac{by}{n}\)
4) When the expression \(x^m y^n\) is constant, the minimum value of (ax+by) is realized when \(\frac{ax}{m}\) = \(\frac{by}{n}\)
5) If the product of several numbers is constant, their sum is minimum when they are all equal.
6) \((n!)^2\) ≥ \(n^n\) PERMUTATIONS AND COMBINATIONS1) A group of 4n distinct items can be divided equally: Among 4 boys in \(\frac{(4n)!}{(n!)^4}\) ways. Into 4 parcels in \(\frac{(4n)!}{4! (n!)^4}\) ways.
2) Number of rectangles that can be formed in a n*n chess board = ∑\(i^3\) = \(\frac{[n(n+1)]}{4} ^2\)
3) Number of squares that can be formed in a n*n chess board = ∑\(i^2\) = \(\frac{n(n+1)(2n+1)}{6}\)
4) If p things are alike of one kind, q things are alike of second kind and r things are alike of third kind, then one or more things can be selected in (p+1) (q+1) (r+1) – 1 ways.
5) Number of ways of selecting one or more items from n given items is \(2^n\)  1.
6) Number of ways of dividing (p+q) items into two groups of p and q items respectively is = \(\frac{(p+q)!)}{p!q!}\)
7) Number of ways of dividing 2p items into 2 equal groups of p each is = \(\frac{(2p)!}{(p!)^2}\) , where 2 groups have distinct identity.
8) Number of ways of dividing 2p items into 2 equal groups of p each is = \(\frac{(2p)!}{((2!)(p!)^2 )}\) , where 2 groups do not have distinct identity.
9) Number of ways in which (p+q+r) things can be divided into 3 groups containing p,q and r things respectively is : \(\frac{(p+q+r)!}{p!q!r!}\)
10) Number of circular arrangements of n distinct items is: (n1)! , if there is a difference between clockwise and anticlockwise arrangements. \(\frac{(n1)!}{2}\) , if there is NO difference between clockwise and anticlockwise arrangements.
11) If there are m objects that are to be equally distributed among n people such that each person gets p objects, total number of ways in which this is done is : \(\frac{m!}{( (p!)^m )}\)
12) If all the possible n – digit numbers using n distinct digits are formed, the sum of all the numbers so formed = (n1)! * ( Sum of the n digits) * {1111…. n times}
QUADRATIC EQUATIONS1) \(ax^2\)+ bx +c = 0 Sum of the roots = (b)/a Product of the roots = c/a
2) When: Discriminant = \(b^2\) – 4ac < 0, roots are complex and unequal Discriminant = \(b^2\) – 4ac = 0, roots are real and equal Discriminant = \(b^2\) – 4ac > 0, roots are real and unequal
3) For a polynomial of degree 3 of the form, \(ax^3\) + \(bx^2\) + cx + d = 0, Product of the roots = \(\frac{d}{a}\) Sum of the roots = \(\frac{b}{a}\) Sum of the product of roots taken two at a time = \(\frac{c}{a}\)
4) If p is the sum of the roots of the quadratic equation and q is the product of the roots of the quadratic equation,then the equation can be written as \(x^2\) –px+q=0.
5) The quadratic expression \(ax^2\) + bx + c: Has a minimum value whenever a> 0 , the minimum value of the quadratic expression is \(\frac{(4acb^2)}{4a}\)and it occurs at x = \(\frac{ b}{2a}\). Has a maximum value whenever a < 0, the maximum value of the quadratic expression is \(\frac{(4acb^2)}{4a}\) and it occurs at x=\(\frac{b}{2a}\).
6) Number of roots = degree
7) Equation whose roots are reciprocals of the roots of the equation \(ax^2\) + bx + c = 0 is given by : \(cx^2\) + bx + a = 0.
8) To determine a 1st degree expression, 2 conditions are needed, for a quadratic expression, 3 degrees are needed and for a cubic expression, 4 conditions are needed. The difference of the roots of \(ax^2\) + bx +c = 0 is \(\frac{\sqrt{b^24ac}}{a}\)
GEOMETRY AND MENSURATION1) Circumradius of a right angled triangle = \(\frac{Hypotenuse}{2}\)
2) Inradius of a triangle must always be less than \(\frac{1}{2}\) (Smallest altitude in it).
3) Inradius of an equilateral triangle = \(\frac{1}{(2√3)}\) (Side of the triangle).
4) Circumradius of an equilateral triangle = \(\frac{1}{√3}\) (Side of the triangle).
5) Area of an isosceles triangle = \(\frac{(b√(4a^2b^2 ))}{4}\), where b is the third side of the isosceles triangle.
6) Sum of the interior angles of a polygon = (n2) 180.
7) Each interior angle = \(\frac{(n2)180}{n}\)
8) An isosceles triangle whose perimeter is fixed will have the maximum area when it is equilateral.
9) If medians are equal in a triangle, then their corresponding sides are equal.
10) For any right  angled triangle whose inradius is r and whose circumradius is R , a. Perimeter = (2r+4R) b. Area = ( \(r^2\) + 2Rr)
11) Inradius of a right angled triangle = \(\frac{(a+bc)}{2}\) , where c = Hypotenuse.
12) Longest diagonal of cuboid = \(√(l^2+b^2+h^2 )\)
13) Total surface area of cuboid = 2(lb+bh+lh)
14) Top surface area = lb; Front surface area = lh
15) Number of revolutions = \(\frac{Distance travelled by the wheel}{Circumference of the wheel}\)
16) Area of the field = (Curved surface area of the roller) * Number of revolutions
17) Area of the 4 walls of a room = 2h(l+b)
18) Area of an equilateral triangle = \(\frac{√3}{4 〖Side〗^2}\)
19) Circumradius of a triangle = \(\frac{(a*b*c)}{4*(Area of triangle)}\) , where a, b and c are the sides of the triangle.
