Average Theory

Average is an indicator of a specified characteristics of a group. The average value is arrived at by distributing equally the total available vale of the group among all the members of the group.

Traditionally, average is calculated by dividing the sum of all the numbers by the number of numbers.

$Average = \frac{Sum \, Of\, Numbers}{Number \, of \,Numbers}$

Example, if five articles are bought for Rs.345, Rs.375, Rs.225, Rs.255 and Rs.300. then the average price of the article is

$\frac{345 + 375 + 225 + 255 + 300}{5} = Rs.300$

Point to Remember

1) If the toal value of all items in a group of n items is V, then the average per item of the group = $A = \frac{V}{n}$

2) The average always lies between the greater and the least of the individual value of the items in the group.

3) For a group of items True average = Assumed Average + Average of Individual deviations from the assumed average.

4) For two groups of items of respective average of P1 and P2 and containing Q1 and Q2 items respectively, the weighted average of the two groups = $p =p_{1}q_{1} = \frac{p_{1}q_{1}}{q_{1}+q_{2}}$

Average can also be seen as the central value of all the given values. To facilitate the calculation, the above procedure can be modified and the modified method is called the Method of Deviations.

In this method a number which is between the lowest and the highest values is arbitrarily assumed as the average. Let say we assume 325, the deviation of the individual values are.

(345 - 325), (375 – 325), (225 – 325), (255 – 325) and (300 -325)

Now finding the average of deviations gives us.
Algebraic sum of deviations = 20 + 50 -100 – 70 – 25 = -125

Average of deviations = $\frac{-125}{5} = -25$

So, Average = Assumed Central Value + Average of Deviations = (325) + (-25) = 300

Change in values and in averages:
1) If the value of each element in a group is increased or decreased by the same value (say k), the average of the group will also correspondingly increase or decrease by the same value i.e. k

2) If the value of each element in a group is either multiplied or divided by the same value say m, the average of the group will also correspondingly get multiplied or divided by the same value i.e. m

Combining two or more groups:
When two or more groups are combined to form a single group, then the average of the newly formed group is called the “Weighted Average of the Groups"

For example, if the average weight of a group of 20 people is 50kg and another group of 30 people has an average weight of 60 kg, then the weighted average of the group (i.e. the average weight of the combined group of 50 people) is:

$\frac{Total\, weight\, of\, 50\,people}{50} = 48km/h$

$\frac{(20 \times 50) + (30 \times 60)}{(20 + 30)}$

$=\frac{2800}{50} = 56 Kg$

The same thing can be generalized and algebraically represented as

$p = \frac{(p_{1}q_{1} + p_{2}q_{2})}{(q_{1}+q_{2})}$

Where, p is the weighted average, P1 and P2 are the average of the two groups and Q1, Q2 are the respective number of items in the two group.

Solved Questions of Average asked in previous year exams

Example 1

Average age Dharam, Prem and Praan is 84 years. When Amit joins them the average age of Dharam, Prem, Praan and Amit becomes 80 years. A new person Alok, whose age is 4 years more than Amit replace Dharam and the average age of Prem, Praan, Amit and Alok becomes 78 years. What is the age of Dharam?

Solution: Since the average age of Dharam, Prem and Praan = 84 years, so we can assume that the age of Dharam, Prem and Praan is 84 years.

Now

Decrease 4 years in the age of Dharam, Prem and Praan (total = 4 + 4 + 4 = 12) now the average age of Dharam, Prem and Praan = 80 years, so we can assume that the age of Dharam, Prem and Praan is 80 years.

Now decrease in the age of Dharam, Prem and Praan can be attributed to the increase in the age of Amit. So after getting 12 years in total Amit is at 80 years, The original age of Amit = 80 – 12 = 68 Years.

And the age of Alok is 4 years more than Amit,

So Alok’s age is 68 + 4 = 72 years.

Now the average age of Dharam, Prem, Praan and Amit = 80 years

Dharam, Prem, Praan and Amit = (80 + 80 + 80 + 80) = 320

And the average age of Prem, Praan, Amit and Alok = 78 years

So Prem + Praan + Amit + Alok = (78 + 78 + 78 + 78) = 312

Since the average difference between the age of Dharam and Alok is 2 years)

Difference (Dharam – Alok) = 2 * 4 = 8 years

Since Alok = 78 years, so Dharam = 80 years.

By using Central Value Method of average every question of average can be done easily by mental calculation. Just practice few questions and you will be expert in solving Average questions.

Example 2
The average score of Sachin after 25 innings is 46 runs per innings, if after the 26th innings, his average runs increased by 2 runs, then what is the score in the 26th inning ?

Solution: Run in 26th inning = Total runs after 26th innings – Total runs after 25th innings.

= 26 × 48 – 25 × 46 = 98
Alternatively, this question can be done by the above given central value meaning of average. Since the average increase by 2 runs, we can assume that 2 runs have been added to his score in each of the first 25 innings. Now the total runs added in the 26th inning, which must be equal to 25 × 2 = 50 runs.
Hence, runs scored in the 26th inning = new average + old innings × change in average

= 48 + 25 × 2 = 98.

Some Special cases of Average:

Average involving time, speed and distance.

$Average\,speed = \frac{Total\, Distance}{Total\, Time}$

However, while solving the question involving time, speed and distance, we should assume some distance preferably the LCM of all the given speeds.

Example: Sohan Lal goes to Mumbai from Allahabad at a speed of 40km/h and returns with the speed of 60km/h. what is his average speed during the whole journey?

Solution: Assuming that the total distance between Mumbai and Allahabad is 120Km (LCM of 40 and 60) the total time taken (Mumbai – Allahabad and Allahabad – Mumbai) = 3 + 2 = 5h

So, average speed = $\frac{240}{5}$ = 48km/h

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