Average Concepts and formulae with solved examples
Average
The average is the ratio of the sum of the given observations to the number of observations. It is also known as the arithmetic mean.
Average = Sum of the observations / Number of observations
Properties of average
- The average of a given data lies between the smallest and the greatest observation (it is less than the greatest and greater than the smallest, unless all observations are equal).
- If all the observations of the given data are equal, the average is the same as each observation.
- If 0 is one of the observations of the given data, that 0 is also included while calculating the average.
Average speed
If a person covers a certain distance at a speed of A km/h and covers the same distance again at a speed of B km/h, the average speed of the whole journey is
2AB/(A + B)
If a person covers three equal distances at speeds of A km/h, B km/h and C km/h, the average speed during the whole journey is
3ABC/(AB + BC + CA)
If distance P is covered with speed x, distance Q with speed y and distance R with speed z, the average speed for the whole journey is
(P + Q + R + ...)/(P/x + Q/y + R/z + ...)
If a person covers part P of the total distance with speed x, part Q of the total distance with speed y and part R of the total distance with speed z (P + Q + R = 1), then
Average speed = 1/(P/x + Q/y + R/z)
Points to remember
- Average of the first n natural numbers = (n + 1)/2
- Average of the first n even numbers = n + 1
- Average of the first n odd numbers = n
- Average of consecutive (equally spaced) numbers = (first number + last number)/2
- Average of the odd numbers from 1 to n = (last odd number + 1)/2
- Average of the even numbers from 1 to n = (last even number + 2)/2
- Average of the squares of the first n natural numbers = (n + 1)(2n + 1)/6
- Average of the cubes of the first n natural numbers = n(n + 1)2/4
- Average of n multiples of any number = number × (n + 1)/2
- If the average of n1 observations is a1, the average of n2 observations is a2, and so on, the average of all observations = (n1a1 + n2a2 + ...)/(n1 + n2 + ...)
- If the average of m observations is a and the average of n observations taken out of them is b, the average of the rest of the observations = (ma − nb)/(m − n)
- If the average of n observations is a, but the average becomes b when one observation is removed or added, then
Value of the removed observation = n(a − b) + b
Value of the added observation = n(b − a) + b - If the average of all the components in a group is a, the average of n components is b and the average of the remaining components is c, then the number of remaining components = n(a − b)/(c − a)
Solved examples
Ex.1If the average of m numbers is n2 and that of n numbers is m2, find the average of all (m + n) numbers.
Sum of m numbers = m × n2 = mn2
Sum of n numbers = n × m2 = nm2
Average of (m + n) numbers = (mn2 + nm2)/(m + n)
= mn(m + n)/(m + n) = mn
Ans: mn
Ex.2The average of three numbers is 28. The first number is half of the second number and the third number is twice the second. Find the third number.
Let the first number = x, so the second number = 2x and the third number = 4x.
Total = x + 2x + 4x = 28 × 3, so 7x = 84 and x = 12.
Third number = 4x = 48
Ans: 48
Ex.3Out of three numbers, the first is twice the second and half of the third. If the average of the three numbers is 56, find the difference between the first and the third number.
Let the second number = x, so the first number = 2x and the third number = 4x.
Total = 2x + x + 4x = 56 × 3, so 7x = 168 and x = 24.
Difference between the first and third number = 4x − 2x = 2x = 48
Ans: 48
Ex.4There were 24 students in a class. One of them, aged 18 years, left the class and his place was filled by a new student. If the average age of the class was lowered by 1 month, find the age of the new student.
Method 1
Let the previous average = x and the age of the new student = y.
24x − 18 + y = 24(x − 1/12)
24x − 18 + y = 24x − 2, so y = 16 years.
Method 2
Age of new student = 18 − (24 × 1)/12 = 16 years.
Ans: 16 years
Ex.5The average of six numbers is 3.95. The average of two of them is 3.4 and the average of another two is 3.85. Find the average of the remaining two numbers.
Required average = [(3.95 × 6) − (3.4 × 2 + 3.85 × 2)]/2
= [23.70 − (6.80 + 7.70)]/2 = 9.2/2 = 4.6
Ans: 4.6
Ex.6Out of 9 persons, 8 spent Rs. 30 each on their meal. The 9th person spent Rs. 20 more than the average expenditure of all 9 persons. Find the total money spent by all of them.
Method 1
Let the average of the nine persons = x.
Total = 9x = 8 × 30 + x + 20, so 8x = 260 and x = 32.5.
Total = 9 × 32.5 = Rs. 292.5
Method 2
Since the 9th person spent Rs. 20 more, the share of each person = 30 + 20/8 = 32.5.
Total expenditure = 9 × 32.5 = Rs. 292.5
Ans: Rs. 292.5
Ex.7In a school with 600 students, the average age of boys is 12 years and that of girls is 11 years. If the average age of the school is 11 years 9 months, find the number of girls in the school.
Let the number of boys = x, so the number of girls = 600 − x.
11 years 9 months = 119⁄12 = 11.75 years
12x + 11(600 − x) = 600 × 11.75
x + 6600 = 7050, so x = 450.
Girls = 600 − 450 = 150
Ans: 150 girls
Ex.8What is the average of the squares of the natural numbers from 1 to 41?
Average of squares of the first n natural numbers = (n + 1)(2n + 1)/6
= [(41 + 1)(2 × 41 + 1)]/6 = (42 × 83)/6 = 3486/6 = 581
Ans: 581
Ex.9A train covers 50% of the journey at 30 km/hr, 25% of the journey at 25 km/hr and the remaining at 20 km/hr. Find the average speed of the train during the entire journey.
Average speed = 100/(A/x + B/y + C/z), where A, B, C are the percentages of the distance and x, y, z are the speeds.
A = 50, B = 25, C = 25 and x = 30, y = 25, z = 20
= 100/(50/30 + 25/25 + 25/20) = 100/(47/12) = 1200/47
Ans: 1200/47 km/hr (about 25.53 km/hr)
Ex.10If the mean of a, b, c is M and ab + bc + ca = 0, find the mean of a2, b2, c2.
(a + b + c)/3 = M, so a + b + c = 3M.
(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) = 9M2
Since ab + bc + ca = 0: a2 + b2 + c2 = 9M2
Mean = (a2 + b2 + c2)/3 = 3M2
Ans: 3M2
Ex.11A batsman has a certain average of runs after 12 innings. In his 13th inning he scores 96 runs, thereby increasing his average by 5 runs. What is his average after the 13th inning?
Let the average of the first 12 innings = x.
(12x + 96)/13 = x + 5
12x + 96 = 13x + 65, so x = 31.
New average = 31 + 5 = 36
Ans: 36 runs
Ex.12a, b, c, d, e are five consecutive odd numbers. What is their average?
For consecutive odd numbers, average = first number + (n − 1), where n is the number of terms.
Here n = 5, so average = a + (5 − 1) = a + 4
Check: a, a + 2, a + 4, a + 6, a + 8 have the middle value a + 4.
Ans: a + 4
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