Average

An average also known as the arithmetic mean is the sum of the given observations divided by the number of observations. For instance, if we have 3 numbers 10, 20 and 30 then the average will be

Average = (10+20+30)/3 = 20

Formula

Avg = Sum of given observations
Number of observations

Important points

• Average of given observations is always greater than the least number and smaller than the greatest number.
  For example, if the numbers are 10, 20 and 30 then the average is 20 which are greater than 10 and smaller than 30.
• If all the observations are equal then the average is equal.
  For instance, if the numbers are 1, 1 and 1 then average is 1.
• While calculating the average we take into consideration all the observations even if it is zero.
  For example, if the observations are 10, 20 and 0 the average of the 3 numbers is 10.
• If all the observations are increased by X then the average is also increased by X.
• If all the observations are decreased by X then the average is also decreased by X.
• If all the observations are divided by X then the average is also divided by X.
• If all the observations are multiplied by X then the average is also divided by X.
• If average of n1 observations is a1 and average of n2 observations is a2 then average of all the observations is
  Avg = n1a1 + n2a2
  n1+n2
• If average of n1 is a1, n2 is a2, n3 is a3 and so on then average of all the observations is
  Avg = n1a1 + n2a2 + n3a3 + …
  n1 + n2 + n3 …

Average speed

If a car or a train travels D distance in T time then the average speed of the car or the train is S.

Average speed S = Distance D
Time T

If a car travels from point A to point B at a speed of X kph (kilometer per hour) and travels from point B to point A at a speed of Y kph. Then average speed of the car during the whole journey is

Avg speed (S) = 2XY
(X+Y)

Question

If a man drives his Audi from his house to his Office at a speed of 100 kph and returns back from his office back home at a speed of 60kph taking the same route daily then find the average speed.

Answer

Given, X = 100 and Y = 60

Then average speed S = 2XY/(X+Y)

= (2 x 100 x 80)/(100 + 60)

= 16000 / 160

= 100 kph

If a car travels distance D1 at a speed of S1 and travels distance D2 at a speed of S2 then the average speed of the whole journey is calculated in the following manner.

Time taken to cover the first part of the journey T1 = D1/S1

Time taken to cover the second part of the journey T2 = D2/S2

Therefore, average speed S = D1 + D2
T1 + T2

Question

If a car travels 100 km at a speed of 50 kph and the next 80 km at a speed of 40 kph then find the average speed of the whole journey

Answer

Given,

D1 = 100km and S1 = 50kph

D2 = 80 km and S2 = 40kph

Time taken to cover first part of the journey T1 = D1/S1 = 100/50 = 2 hours.

Time taken to cover second part of the journey T2 = D2/S2 = 80/40 = 2 hours.

Therefore, average speed of the car is S = (D1 + D2) / (T1 + T2)

= (100 + 80) / (2 + 2)

= 180 / 4

= 45 kph