Averages

Averages

Always people find it difficult to remember the information given in different groups or in different numbers. In such cases, the approximate number is useful. That is why we often hear remarks such as, the average of Sachin's runs in five innings is 65, in a standard VII class of 65 pupils, the average attendance last week was 60. Let us try to find out the mathematical meaning of the word "Average" first. Have a look at the following example.
All of us know that the speed of a vehicle does not remain constant during a journey. Therefore, the distance traveled is different in each hour. Still taking into consideration the total distance traveled and and total time required for that, the distance traveled by that vehicle per hour is determined for the sake of convenience. In order to find out
the distance traveled in one hour, the total distance is divided by the total period, that is, by the total number of hours. The distance traveled in one hour is the average hourly speed of that vehicle.

Need lil more explanation? Ok, assume that a Car travelled a distance of 62 kms in the first hour, 55 kms in the second hour and 48 kms in the third hour. In this case, the total distance covered = 62 + 55 + 48 = 165 kms

Therefore, distance covered in one hour = 165 / 3 = 55 kms

Have a look at some important points about average.
When several numbers are given to find the average, add them all and divide the addition by the number of numbers i.e. if six different numbers are given, the addition of these six numbers will be, divided by 6 to get the average.
When the average of a group of three numbers is given and the average of other seven numbers is given, first get total of three from the given average. Then, similarly get the total of seven from the given average. Add the total and divide this answer by 10 (the sum of 3 + 7) to get the final average.

1. If the average of 10 numbers and the average of 4 numbers out of 10 are given and you are asked to find out the average of the remaining six, then

• Multiply 10 by the average, then find out the total of 4 by multiplying average of 4 by 4. Then subtract the smaller total from the greater total. Divide the answer by 6 to get the average of six.

• Note : The average number must be always less than the greatest number of the group.
• Average Speed = (Total Distance / Time)
• Note : If a number equal to the average is added to a group, their average remains the same.
• Shortcut : To find the average of numbers who have equal common difference, follow this shortcut method. 

Arrange all the numbers in asscending order. 

1. If there are odd number of numbers, the middle number is the average of all the numbers. 

• Eg. Average of 20, 25, 30, 35 and 40 is 30, which is the middle number of the given five numbers. 

2:     If there are even number of numbers, the average of two middle numbers is the average of all the numbers. 

• Eg. Average of 10, 12, 14, 16, 18 and 20 is average of 14 and 16 which are two middle numbers. Therefore, average of these six numbers 15.

Lets See Some Examples >>

The average income of 3 friends in a group of six is Rs. 6,600 and that of the other two is Rs. 6,500. If the average income of all the six is Rs. 6,400, find the income of the 6th friend.
Sol :
            Average Income of 3 Friends = Rs. 6,600
            So, Total Income of 3 friends = 3 X 6,600 = Rs. 19,800

            Average income of 2 friends = 2 X 6,500
            So, Total income of 2 friends = 2 X 6,500 = 13,000

            Total income of 5 friends = Rs. 19,800 + Rs. 13,000 
                                                                         = Rs. 32,800

            Average income of 6 friends = Rs.6,400
         
            Total income of 6 friends = 6 x 6,400 = Rs. 38,400

       So, income of 6th friend = Total income of 6 friends - Total income of 5 friends
                                              = Rs.38,400 - Rs.32,800 = Rs.5,600

The average of Ramesh's marks in English, Maths and Science is 75 and the average of his marks in English, Maths, Science and Hindi is 70. How many marks did he score in Hindi?
Sol :
            The average marks in English, maths and Science is 75
                  So, Total marks in English, maths and Science are 3 X 75 = 225

            The average marks in English, Maths, Science and Hindi are 70.
                  So, Total marks in English, Maths, Science and Hindi are 4 X 70 = 280
       
              So, Marks in Hindi = 280 - 225 = 55

 
The average age of the husband, his Wife and their Son ten years ago was 25 Years and that of the husband and wife six years ago was 37 years. What is the present age of the son (in years) ?
Sol :
          Average age of the husband, his wife and their son ten years ago was 25 years 

          Total of present ages of husband, wife and their son 
                                                       = (25 x 3) + 10 + 10 + 10 = 105 Years

          Average age of the husband and his wife six years ago was 37 years
          Total of present ages of husband and wife 
                                                                     = (37 X 2) + 6 + 6 = 86

          So, Son's present age = 105 - 86 = 19 Years

The average of A and B is 30 years. If C were to replace A, the average age would be 40 years and if he were to replace B, the average age would be 35 years. What is A's age?
Sol :
           Average age of A and B = 30 Years
           Total age of A and B = 30 x 2 = 60 years ---------- (1)

          Average age of B and C = 40 Years
            Total age of B and C = 40 X 2 = 80 years ---------- (2)

           Average age of A and C = 35 years
              Total age of A and C = 35 X 2 = 70 years ---------- (3)

            After adding (1) and (3) , we get
                       2A + B + C = 130

                  But, we know that (B + C) = 80
                           So, 2A + B + C = 130
                                                    => A = (130 - 80) / 2 = 25

                     So, A's age is 25 years.