Wednesday, June 1, 2011

Solving Problems in Averages without Tears

Average is defined as the Total of data/ Number of Data

Sometimes average of two groups of quantities is given and we are asked to find the combined average. For example The average age of 40 students in section A is 10 years and the average age of students in section B of 30 students is 12 years. Find the average age of the students in both the sections taken together. To solve such problems we go like this: Suppose the assumed average age is 10 years. Total excess over assumed average will be 30 x (12-10) = 60 years for second group.These 60 years will be divided in all 70 (40+30) students. So increase in average will be 60/70= .85 years. So actual average will be 10.85 years ( 10+.85 years).

Or there may be questions where the combined average and average of one group is given and we are asked to find the average of other group. For example if average of 5 quantities is 6. The average of three of them is 4 then what will be average of remaining 2 numbers. Here average of 5 quantities is  and average of 3 quantities is 4. The total deficit in 3 quantities is 3 x (6-4)= +6. Total increase in the remaining 2 numbers is +6/2= 3. Therefore average expenditure of remaining 2 numbers is 6+3=9.

How to solve such problems in averages when some members are added or subtracted and the average increases or decreases. For example, the average weight of 29 students in a class is 48 kg. If the weight of teacher is included, the average weight rises by 500 gms. Find the weight of the teacher. Here total increase in weight after teacher is added is .500 x 30= 15 kg. So teacher's weight must be more than 48 kg by 15 kg. Because only teacher was responsible for increase in weight. So teacher's weight must be 48+15=63 kg.

Or a batsman has certain average runs for 20 innings. In the 21st innings, he made 107 runs, thereby increasing his average by 2. What is his average after 21 innings. In this case, after 21 innings his average increase by 2. Therefore total runs that are more than average will be 21 x 2 = 42. Therefore old average must be 107-42=65 and new average will be more than 2 ie. 65+2=67.

In questions where a group of quantities is replaced by another group of quantities and average we proceed as follows: (sum of) replacing quantities - ( sum of ) replaced quantities = change in average x total number of quantities in the group. For example the average temperature of June, July and August was 31 deg C. The average temperature of July, August and September was 30 deg C. If the temperature of June was 29 deg C, find the temperature of September.Here Temperature of September ( Replacing Qty)- 29 ( Replaced)= -1 x 3 which means Temperature of September= 29-3=26 deg C.

Finally the questions are asked about the mid term quantity. For example average of 11 observations is 50 and average of first 6 is 49 and average of last 6 is 52, what is the 6th quantity. In such questions the decrease in total for 1st 6 observations is -1 x 6= -6. and increase in the total for last 6 observations is +2 x 6= 12 from average therefore total change is 12-6=+6, therefore 6th quantity= 50+6=56.

I hope this theoretical discussion will be able to see you through from the first level of questions in average.

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