Question

The number of observations in a group is 40 . If the average of first 10 is 4.5 and that of the remaining 30 is 3.5. What is the average of the whole group.

A. \[\dfrac{1}{5}\]
B. \[\dfrac{{15}}{4}\]
C. \[4\]
D. \[8\]

Answer 1

Hint: In this question, we use the average concept. We define the average in this way,if we divide the sum of all observations by total numbers of observations then we get the average of all observations.

Formula used:The formula for finding average for any numbers of observations:
\[\text{Average}=\dfrac{\text{Sum of all observation}}{\text{Total number of observations}}\]
By using this formula,we can find the average of whole group.

Complete step-by-step solution:
In this question given that the total numbers of observations in group is 40 and given average of first 10 observations is 4.5.
So, we can write here average formula for first 10 observations
\[\text{Average(first 10 observations)} = \dfrac{{{X_1} + {X_2} + {X_3} + \ldots + {X_{10}}}}{{10}}\]
And here given in the question average for first 10 observations is 4.5 .Then,

 \[4.5 = \dfrac{{{X_1} + {X_2} + {X_3} + \ldots + {X_{10}}}}{{10}}\]
\[4.5 \times 10 = {X_1} + {X_2} + {X_3} + \ldots + {X_{10}}\]
\[45 = {X_1} + {X_2} + {X_3} + \ldots + {X_{10}}\]
 So,Here we see that sum of first 10 observations is 45.
Now another thing given in the question that the average of remaining next 30 observations is 3.5 .
So, we can write here average formula for next 30 observations
\[\text{Average(next 30 observations)} = \dfrac{{{X_{11}} + {X_{12}} + {X_{13}} + \ldots + {X_{40}}}}{{30}}\]
And here given in the question average for next 30 observations is 3.5 .Then,
\[3.5 = \dfrac{{{X_{11}} + {X_{12}} + {X_{13}} + \ldots + {X_{40}}}}{{30}}\]
\[3.5 \times 30 = {X_{11}} + {X_{12}} + {X_{13}} + \ldots + {X_{40}}\]
 \[105 = {X_{11}} + {X_{12}} + {X_{13}} + \ldots + {X_{40}}\]
So,Here we see that sum of next 30 observations is 105.
Here, \[{X_1},{X_2},{X_3}, \ldots ,{X_n}\] denote the observations from 1 to 40.
Now we can find easily the average of whole group by combine the sum for first 10 observations and next 30 observations.
\[\text{Average for whole group} = \dfrac{\text{Sum of first 10 observations} + \text{Sum of next 30 observations}}{{40}}\]\[\]
\[ = \dfrac{{45 + 105}}{{40}}\]
 \[ = \dfrac{{150}}{{40}}\]
 After converting the lowest form , we get
 \[ = \dfrac{{15}}{4}\]
So,We can say that average of the whole group is \[\dfrac{15}{4}\].
Hence,Option (B) is the correct answer:

Note:In these types of questions, the common mistakes of all students is that maximum students thinks there are given average for first 10 observations is 4.5 and for next 30 observations is 3.5 .so , they think that we combine these average and then divide by 2 and we get average for whole observations \[(\dfrac{\text{Average of first 10 observations} + \text{Average of next 30 observations}}{2})\] but this is wrong way.