Average

Average=(sum / N) where N is total number of objects

Weighted Average: The average between two sets of numbers is closer to the set with more numbers.
If the value of each item is increased by x, then the average of the group will also increase by x.

If the value of each item is decreased by y, then the average of the group of items will also decrease by y.
If the value of each item is multiplied by the same value m, then the average of the group or items will also get multiplied by m.
If the value of each item is multiplied by the same value n, then the average of the group or items will also get divided by n.
If we know only the average of the two groups individually, we cannot find out the average of the combined group of items.

Average of x natural no’s = (x+1)/2

Average of even No’s = (x+1)

Average of odd No’s = x

 Change in the value of a Quantity and its effect on the Average

When one/more than one quantity is not removed but replaced with same no. of quantities of different value,

Change in the no. of quantities and its effect on Average

+ = if quantities are Added, – = if quantities are removed

Time and Work

If a person ‘X’ can complete a piece of work in ‘Y’ days, then

Work Done by X in 1 day = 1/Y

If Utkarsh can complete a piece of work in ‘X’ days and Prashant can complete a piece of work in ‘Y’ days, then

Work done by Utkarsh and Prashant together in 1 day = X+Y/XY

Trick 1: If Adarsh can do a work in D1 days and Bhanu can do the same work in D2 days then Adarsh and Bhanu together can do the same work in (D1 x D2)/ (D1 + D2)

Trick 2: Sometimes the work done is dependent on the number of days worked and efficiency of each worker. Let’s see how the formula changes in this case:

• If person P1 can do the work in D1 days and M2 persons can do the same work in D2 days then we can say M1.D1 = M2.D2
• If the persons work T1 and T2 hours per day respectively then the equation gets modified to M1.D1.T1 = M2.D2.T2
• If the persons has efficiency of E1 and E2 respectively then: M1.D1.T1.E1 = M2.D2.T2.E2

Pipes & Cisterns

Trick: If a pipe A can fill a tank in ‘x’ hours and Pipe B can fill a tank in ‘y’ hours, then

Time Taken by both the pipes to fill the tank together = xy/(x+y)

Time, Speed and Distance

Trick 1: Speed = Distance/ Time and the unit of Speed is Km/hr or m/s

Trick 2: If an object covers a certain distance with a speed of ‘p’ Km/hr and again covers the same distance with a speed of ‘q’ Km/hr, then

Average Speed for the Entire Journey = 2 pq/(p+q)

Boats and Streams

Trick: If the speed of stream is ‘p’ Km/hr and the speed of boat in still water is ‘q’ Km/hr, then

Speed of Boat Downstream = (p + q) Km/hr

Speed of Boat Upstream = (q – p) Km/hr

Mixtures and Alligations

Trick 1: When two different commodities are mixed such that one is cheaper than the other, then:

Here Mean Price is CP of mixture per unit quantity.

Trick 2: When x amount of commodity 1 is replaced with commodity 2 of y quantity n times, then:

Quadratic Equations

ax2 + bx + c = 0.

py2 + qy + c = 0

Trick 1:

Sum of roots = -b/a

Product of roots = c/a

Trick 2: A quadratic equation ax2 + bx + c = 0 will have reciprocal roots when a = c

Approximation

Trick 1: Use the BODMAS rule to solve the simplification questions. This is the basis of simplification questions.

Trick 2: Round of decimal numbers to the nearest whole numbers to get the approximate value of the equation.

Ratio Proportion

Ratio is a fraction of two values. It can be represented in any of the following ways:

=> x/y,   x : y , x÷y

• In ratio of the form x : y,  x is called as the antecedent/first term and y is the consequent/second term.
• Generally, ratio is a handy way to compare two terms.
• For example : 4 / π > 1 , it is clear that 4 > π
• One thing that has to be remembered while comparing two numbers in a ratio is that they should be represented in same units. For example, if x is in meters and y in litres they cannot be compared by using ratio as they are expressed in different units – meters vs litres.

A proportion is the equality of two ratios/ fractions.

• If x : y = a : b, it can be written as  x : y :: a : b and it is said that x, y, a, b are in proportion.
• Here x and b are called extremes, while y and a are called mean terms.
• Product of means=Product of extremes
  Thus,
  x :y :: a : b  => (y∗a)=(x∗b)
• If x : y=a : b

b is called the fourth proportional to x, y and a.
a is called the third proportional to x and y.

• Sub-duplicate: Sub-duplicate ratio of (a:b) is (a^1/2: b^1/2)
• Duplicate ratio of (a:b) is (a^2:b^2)
• Triplicate Ratio:Triplicate ratio of (a:b) is (a^3:b^3)
• Sub-triplicate Ratio: Sub-triplicate ratio of (a:b) is (a^1/3:b^1/3)
• If a/b=c/d then, a+b/a−b=c+d/c−d This is known as Componendo and Dividendo.
• We say that x is directly proportional to y, if x=ky for some constant k and we write, x∝y
• We say that x is inversely proportional to y, if xy=k for some constant k and we write, x∝1y

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