Monday, November 2, 2015

Quantitative Aptitude: Average


Welcome Friends, Average is a very simple topic, and weights 1 to 3 marks in almost every
examinations.
Let's start with a definition, Average is simply the central value or mean value of given set of numbers. An example will make the point much clear.

Suppose a kid in primary school got his result, he got 80 in math, 68 in English and 92 in Social Science. Can you tell me average marks of all the subjects?
It's Simple, Take a total of all the marks and then divide the sum with a total number of subjects.
Formula = Total  of all the Values / Number of values      OR   Average= Sum of Elements/No of Elements
               =80+68+92/3 = 240/3 =80.
You might have noticed an important property of average that the average would always be greater than the lowest value and smaller than the highest value among all numbers in a list.
One more type of simple example asked often in exams is like, Average marks obtained by 5 students is 90 then total marks of all students is……. we can simply calculate by multiplying the average by a number of students to find the answer. (450)
We have already understood the basics of Averages. Let's move deeper!

Combined Average to two Groups

Suppose in a class of 30, the average weight of 10 girls is 12kg, and an average weight of remaining 20 boys is 14kg. Then to find the average weight of the whole class, we need a sum of the weight of all the students. Multiply average weight with the number of students in both the group and add them which is (12*10)+(14*20) = 400kg. So the average weight of a class is 400/30 = 13.33 Table below will help in understanding the concept in a better way.

Group 1 (Girls)
Group 2(Boys)
New combined group (1+2)(Whole Class)
Average
A1 (12kg)
A2 (14kg)
S1+S2/M1+M2
Number of members
M1 (10)
M2 (20)
M1+M2 (30)
Sum(Total)
S1=A1*M1 (120kg)
S2=A2*M2 (280kg)

All members get the same amount of gain/loss

Suppose in your family currently there are four members and the average age of family is 35 year. Now after five years means in 2020 what will be average family age? Here in the next five years each family member age will increase by 5 years, so average will also get an increase of 5 years, and the new average will be 40 years.
Remember always, if all the numbers in a list gain or lose by a particular amount, then the average also increases or decreases by the same amount. You do not need to perform any further calculations for such questions.
Similarly, if all of the numbers in a list are multiplied by a certain amount, then the average also gets multiplied by the same amount. These concepts would help you solve questions on averages much faster.

Average Speed

My husband’s office is 30km from my house, now suppose in the morning he goes at a speed of 60 kph and in the evening (due to the temptation of getting home early) he drives faster at a speed of 90 kph then what will be the average speed of his total journey? The simple answer that come in mind is 75 (60+90/2) but it's not right. Let's see how. The average speed of total journey should be calculated with the formula of speed.
Speed = Distance Covered / Time taken
So to calculate average speed we already have Total distance = 30 + 30 = 60km. Now the time taken in the morning = 30km/60kmph = 1/2 hour. while time in evening is =30km/90kmph = 1/3 hour. So total time is 1/2+1/3=5/6 hours (50 minutes).
Now Avg Speed = 60km/(5/6)kmph = 60*6/5 = 72kmph.
Don’t worry how don’t always have to take this long way, here is a shortcut formula.
Average Speed while covering same distance with different speeds = 2uv / (u+v), where u and v are speed.

But make sure you apply this formula only to problems that have the same distance in both journeys, otherwise use the earlier mentioned procedure. Apply the above formula to above example and check whether you get the same result.
You have come to the last part of the Averages course!
Here are some important results to remember. Note them down or if you use Google chrome then press Ctrl+P and save this page as pdf.

1. If a person Joins/Leaves  a group and

         New Average Increases then

           Age of person Joined/Left = Old Average + (Updated Group Size * Increase In Avg Age)

         New Average decreases then

           Age of person Joined/Left = Old Average - (Updated Group Size * decrease In Avg Age)

    2. Average of n consecutive natural numbers is ( n + 1 )/2.

    3. Average of squares of n consecutive natural numbers is [( n + 1 ) ( 2n + 1 )] /6

    4. Average of cubes of first n consecutive natural numbers is [ n ( n + 1 )2  ]/4

    5. Average of first n consecutive even numbers is ( n+1 )

    6. Average of first n consecutive odd numbers is ( n )


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