**An average or an arithmetic mean of given data is the sum of the given observations divided by number of observations.**

**Important Formulae Related to Average of numbers**

**1. Average of first n natural number=(n+1)/2**

**2. Average of first n even number= (n+1)**

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**3. Average of first n odd number= n**

**4. Average of consecutive number= (Firtst number+Last number)/2**

**5. Average of 1 to n odd numbers= (Last odd number+1)/2**

**6. Average of 1 to n even numbers= (Last even number+2)/2**

**7. Average of squares of first n natural numbers=[(n+1)(2n+1)]/6**

**8. Average of the cubes of first n natural number=[n(n+1)^2]/4**

**9. Average of n multiples of any number=[Number*(n+1)]/2**

**Concept 1**

**If the average of n_1 observations is a_1; the average of n_2 observations is a_2 and so on, then**

**Average of all the observations=(n_1* a_1+n_2 *a_2+……)/(n_1+n_2+……..)**

**Concept 2**

**If the average of m observations is a and the average of n observations taken out of is b, then**

**Average of rest of the observations=(ma-nb)/(m-n)**

**Example 1 : A man bought 20 cows in RS. 200000. If the average cost of 12 cows is Rs. 12500, then what will be the average cot of remaining cows?**

**Here m = 20 , n = 12 , a = 10000 , b = 12500**

**average cost of remaining cows ( 20-8) cows = (20*10000 – 12*12500)/(20-8) =Rs 6250**

**Concept 3**

**If the average of n students in a class is a, where average of passed students is x and average of failed students is y, then**

**Number of students passed=[Total Students (Total average-Average of failed students)]/(Average of passed students-Average of failed students)**

**=[n(a-y)]/(x-y)**

**Example 2: In a class, there are 75 students are their average marks in the annual examination is 35. If the average marks of passed students is 55**

**and average marks of failed students is 30, then find out the number of students who failed.**

**Here , n = 75 , a = 35 , x = 55 , y = 30**

**Number of students who passed = 75(35- 30)/(55- 30) = 15**

**Number of students who failed = 75- 15 = 60**

**Concept 4**

**If the average of total components in a group is a, where average of n components (1st part) is b and average of remaining components (2nd part) is c, then**

**Number of remaining components (2nd part)=**

**[n(a-b)]/(c-a)**

**Example 3 : The average salary of the entire staff in an offfice is Rs. 200 per day. The average salary of officers is Rs. 550 and that of non-officers is Rs. 120.**

**If the number of officers is 16, then find the numbers of non-officers in the office.**

**Here n= 16 , a = 200 , b = 550 , c = 120**

**Number of non – officer = 16(200- 550)/(120- 200) = 70**

**Average Speed**

**Average speed is defined as total distance travelled divided by total time taken.**

**Average speed=Total distance travelled/ Total time taken**

**Case 1**

**If a person covers a certain distance at a speed of A km/h and again covers the same distance at a speed of B km/h, then the average speed during the whole journey**

**will be**

**2AB/A+B**

**Case II**

**If a person covers three equal distances at the speed of A km/h, B km/h and C km/h respectively, then the average speed during the whole**

**Journey will be**

**3ABC/(AB+BC+CA)**

**Case III**

**If distance P is covered with speed x, distance Q is covered with speed y and distance R is covered with speed z, then for the whole journey,**

**Average speed=(P+Q+R+…..)/(P/x+Q/y+R/z+…)**

**Example 4 : A person covers 20 km distance with a speed of 5 km/h, then he covers the next 15 km with a speed of 3 km/h and the last 10 km is covered by him**

**with a speed of 2 km/h. Find out his average speed for the whole journey.**

**Average speed = ( 20 +15 +10)/(20/5+15/3+10/2) = 3(3/14)**

**Case IV**

**If a person covers P part of his total distance with speed of x, Q part of total distance with speed of y and R part of total distance with speed**

**of z,then**

**Average speed=1/(P/x+Q/y+R/z+……)**

*To do the Quizzes based on “Average”, click on the below links –***1. Quiz 1**

**2. Quiz 2**

**3. Quiz 3**

**4. Quiz 4**

**To do the various Mix Quant Quizzes,**

**Click here**

**To do the Quizzes on D.I.,**

**Click here**